NCERT Physics Question Paper (Class - 12)
:: Chapter 1 - Electric Charges And Fields ::
List of Physics Formula
EXERCISE
Question 1: What is the force between two small charged
spheres having charges of 2 × 10−7 C and 3 × 10−7 C placed 30 cm apart in air?
Question 2: The electrostatic force on a small sphere of
charge 0.4 µC due to another small sphere of charge − 0.8 µC in air is 0.2 N.
(a) What is the distance between the two spheres? (b) What is the force on the
second sphere due to the first?
Question 3: Check that the ratio ke2/G memp is dimensionless.
Look up a Table of physical constants and determine the value of this ratio.
What does the ratio signify?
Question 4: (a) Explain the meaning of the statement
‘electric charge of a body is quantised’.
(b) Why can one ignore quantisation of electric charge when dealing with
macroscopic i.e., large scale charges?
Question 5: When a glass rod is rubbed with a silk cloth,
charges appear on both. A similar phenomenon is observed with many other pairs
of bodies. Explain how this observation is consistent with the law of
conservation of charge.
Question 6: Four point charges qA = 2 µC, qB = −5 µC, qC = 2
µC, and qD = −5 µC are located at the corners of a square ABCD of side 10 cm.
What is the force on a charge of 1 µC placed at the centre of the square
Question 7: (a) An electrostatic field line is a continuous
curve. That is, a field line cannot have sudden breaks. Why not? (b) Explain why
two field lines never cross each other at any point?
Question 8: Two point charges qA = 3 µC and qB = −3 µC are
located 20 cm apart in vacuum. (i) What is the electric field at the midpoint O
of the line AB joining the two charges? (ii) If a negative test charge of
magnitude 1.5 × 10−9 C is placed at this point, what is the force experienced by
the test charge?
Question 9: A system has two charges qA = 2.5 × 10−7 C and qB
= −2.5 × 10−7 C located at points A: (0, 0, − 15 cm) and B: (0, 0, + 15 cm),
respectively. What are the total charge and electric dipole moment of the
system?
:: Chapter 2 - Electrostatic Potential and Capacitance ::
EXERCISE
Question 2.1 Two charges 5 × 10–8 C and –3 × 10–8 C are
located 16 cm apart. At what point (s) on the line joining the two charges is
the electric potential zero? Take the potential at infinity to be zero.
Question 2.2 A regular hexagon of side 10 cm has a charge 5
μC at each of its vertices. Calculate the potential at the centre of the
hexagon.
Question 2.3 Two charges 2 μC and –2 μC are placed at
points A and B 6 cm apart.
(a) Identify an equipotential surface of the system.
(b) What is the direction of the electric field at every point on this surface?
Question 2.4 A spherical conductor of radius 12 cm has a
charge of 1.6 × 10–7C distributed uniformly on its surface. What is the electric
field
(a) inside the sphere
(b) just outside the sphere
(c) at a point 18 cm from the centre of the sphere?
Question 2.5 A parallel plate capacitor with air between the
plates has a capacitance of 8 pF (1pF = 10–12 F). What will be the capacitance
if the distance between the plates is reduced by half, and the space between
them is filled with a substance of dielectric constant 6?
Question 2.6 Three capacitors each of capacitance 9 pF are
connected in series.
(a) What is the total capacitance of the combination?
(b) What is the potential difference across each capacitor if the combination is
connected to a 120 V supply?
Question 2.7 Three capacitors of capacitances 2 pF, 3 pF
and 4 pF are connected in parallel.
(a) What is the total capacitance of the combination? (b) Determine the
charge on each capacitor if the combination is connected to a 100 V supply.
Question 2.8 In a parallel plate capacitor with air between
the plates, each plate has an area of 6 × 10–3 m2 and the distance between the
plates is 3 mm. Calculate the capacitance of the capacitor. If this capacitor is
connected to a 100 V supply, what is the charge on each plate of the capacitor?
Question 2.9 Explain what would happen if in the capacitor
given in Exercise2.8, a 3 mm thick mica sheet (of dielectric constant = 6) were
inserted between the plates,
(a) while the voltage supply remained connected.
(b) after the supply was disconnected.
Question 2.10 A 12pF capacitor is connected to a 50V battery.
How much electrostatic energy is stored in the capacitor?
Question 2.11 A 600pF capacitor is charged by a 200V supply.
It is then disconnected from the supply and is connected to another uncharged
600 pF capacitor. How much electrostatic energy is lost in the process?
ADDITIONAL EXERCISES QUESTIONS
Question 2.12 A charge of 8 mC is located at the origin.
Calculate the work done in taking a small charge of –2 × 10–9 C from a point P
(0, 0, 3 cm) to a point Q (0, 4 cm, 0), via a point R (0, 6 cm, 9 cm).
Question 2.13 A cube of side b has a charge q at each of its
vertices. Determine the potential and electric field due to this charge array at
the centre of the cube.
Question 2.14 Two tiny spheres carrying charges 1.5 μC and
2.5 μC are located 30 cm apart. Find the potential and electric field: (a) at
the mid-point of the line joining the two charges, and (b) at a point 10 cm from
this midpoint in a plane normal to the line and passing through the mid-point.
Question 2.15 A spherical conducting shell of inner radius
r1 and outer radius r2 has a charge Q.
(a) A charge q is placed at the centre of the shell. What is the surface
charge density on the inner and outer surfaces of the shell?
(b) Is the electric field inside a cavity (with no charge) zero, even if the
shell is not spherical, but has any irregular shape? Explain.
Question 2.16 (a) Show that the normal component of
electrostatic field has a discontinuity from one side of a charged surface to
another given by 2 1 0 ( ) ˆ σ ε E − E n = where ˆn is a unit vector normal to
the surface at a point and σ is the surface charge density at that point. (The
direction of ˆn is from side 1 to side 2.) Hence show that just outside a
conductor, the electric field is σ ˆn /ε0.
(b) Show that the tangential component of electrostatic field is continuous from
one side of a charged surface to another. [Hint: For (a), use Gauss’s law. For,
(b) use the fact that work done by electrostatic field on a closed loop is
zero.]
Question 2.17 A long charged cylinder of linear charged
density λ is surrounded by a hollow co-axial conducting cylinder. What is the
electric field in the space between the two cylinders?
Question 2.18 In a hydrogen atom, the electron and proton
are bound at a distance of about 0.53 Å:
(a) Estimate the potential energy of the system in eV, taking the zero of
the potential energy at infinite separation of the electron from proton.
(b) What is the minimum work required to free the electron, given that its
kinetic energy in the orbit is half the magnitude of potential energy obtained
in (a)?
(c) What are the answers to (a) and (b) above if the zero of potential energy is
taken at 1.06 Å separation?
Question 2.19 If one of the two electrons of a H2 molecule is
removed, we get a hydrogen molecular ion H+ 2.In the ground state of an H+ 2,
the two protons are separated by roughly 1.5 Å, and the electron is roughly 1 Å
from each proton. Determine the potential energy of the system. Specify your
choice of the zero of potential energy.
Question 2.20 Two charged conducting spheres of radii a and b
are connected to each other by a wire. What is the ratio of electric fields at
the surfaces of the two spheres? Use the result obtained to explain why charge
density on the sharp and pointed ends of a conductor is higher than on its
flatter portions.
Question 2.21 Two charges –q and +q are located at points
(0, 0, –a) and (0, 0, a), respectively.
(a) What is the electrostatic potential at the points (0, 0, z) and (x, y,
0) ?
(b) Obtain the dependence of potential on the distance r of a point from the
origin when r/a >> 1.
(c) How much work is done in moving a small test charge from the point (5,0,0)
to (–7,0,0) along the x-axis? Does the answer change if the path of the test
charge between the same points is not along the x-axis?
Question 2.22 Figure2.34 shows a charge array known as an
electric quadrupole. For a point on the axis of the quadrupole, obtain the
dependence of potential on r for r/a >> 1, and contrast your results with that
due to an electric dipole, and an electric monopole (i.e., a single charge).
Question 2.23 An electrical technician requires a capacitance
of 2 μF in a circuit across a potential difference of 1 kV. A large number of 1
μF capacitors are available to him each of which can withstand a potential
difference of not more than 400 V. Suggest a possible arrangement that requires
the minimum number of capacitors.
Question 2.24 What is the area of the plates of a 2 F
parallel plate capacitor, given that the separation between the plates is 0.5
cm? [You will realise from your answer why ordinary capacitors are in the range
of μF or less. However, electrolytic capacitors do have a much larger
capacitance (0.1 F) because of very minute separation between the conductors.]
Question 2.25 Obtain the equivalent capacitance of the
network in Fig. 2.35. For a 300 V supply, determine the charge and voltage
across each capacitor.
Question 2.26 The plates of a parallel plate capacitor have
an area of 90 cm2 each and are separated by 2.5 mm. The capacitor is charged by
connecting it to a 400 V supply.
(a) How much electrostatic energy is stored by the capacitor?
(b) View this energy as stored in the electrostatic field between the plates,
and obtain the energy per unit volume u. Hence arrive at a relation between u
and the magnitude of electric field E between the plates.
Question 2.27 A 4 μF capacitor is charged by a 200 V supply.
It is then disconnected from the supply, and is connected to another uncharged 2
μF capacitor. How much electrostatic energy of the first capacitor is lost in
the form of heat and electromagnetic radiation?
Question 2.28 Show that the force on each plate of a parallel
plate capacitor has a magnitude equal to (½) QE, where Q is the charge on the
capacitor, and E is the magnitude of electric field between the plates. Explain
the origin of the factor ½.
Question 2.29 A spherical capacitor consists of two
concentric spherical conductors, held in position by suitable insulating
supports (Fig2.36). Show that the capacitance of a spherical capacitor is given
by 0 1 2 1 2 4 – r r C r r πε = where r1 and r2 are the radii of outer and inner
spheres, respectively.
Question 2.30 A spherical capacitor has an inner sphere of
radius 12 cm and an outer sphere of radius 13 cm. The outer sphere is earthed
and the inner sphere is given a charge of 2.5 μC. The space between the
concentric spheres is filled with a liquid of dielectric constant 32.
(a) Determine the capacitance of the capacitor.
(b) What is the potential of the inner sphere?
(c) Compare the capacitance of this capacitor with that of an isolated sphere of
radius 12 cm. Explain why the latter is much smaller.s the magnitude of
electrostatic force between them exactly given by Q1 Q2/4πε0r 2, where r is the
distance between their centres?
(b) If Coulomb’s law involved 1/r3 dependence (instead of 1/r2), would Gauss’s
law be still true ?
(c) A small test charge is released at rest at a point in an electrostatic field
configuration. Will it travel along the field line passing through that point?
(d) What is the work done by the field of a nucleus in a complete circular orbit
of the electron? What if the orbit is elliptical?
(e) We know that electric field is discontinuous across the surface of a charged
conductor. Is electric potential also discontinuous there?
(f ) What meaning would you give to the capacitance of a single conductor?
(g) Guess a possible reason why water has a much greater dielectric constant (=
80) than say, mica (= 6).
Question 2.32 A cylindrical capacitor has two co-axial
cylinders of length 15 cm and radii 1.5 cm and 1.4 cm. The outer cylinder is
earthed and the inner cylinder is given a charge of 3.5 μC. Determine the
capacitance of the system and the potential of the inner cylinder. Neglect end
effects (i.e., bending of field lines at the ends).
Question 2.33 A parallel plate capacitor is to be designed
with a voltage rating 1 kV, using a material of dielectric constant 3 and
dielectric strength about 107 Vm–1. (Dielectric strength is the maximum electric
field a material can tolerate without breakdown, i.e., without starting to
conduct electricity through partial ionisation.) For safety, we should like the
field never to exceed, say 10% of the dielectric strength. What minimum area of
the plates is required to have a capacitance of 50 pF?
Question 2.34 Describe schematically the equipotential
surfaces corresponding to
(a) a constant electric field in the z-direction,
(b) a field that uniformly increases in magnitude but remains in a constant
(say, z) direction,
(c) a single positive charge at the origin, and
(d) a uniform grid consisting of long equally spaced parallel charged wires in a
plane .
Question 2.35 In a Van de Graaff type generator a spherical
metal shell is to be a 15 × 106 V electrode. The dielectric strength of the gas
surrounding the electrode is 5 × 107 Vm–1. What is the minimum radius of the
spherical shell required? (You will learn from this exercise why one cannot
build an electrostatic generator using a very small shell which requires a small
charge to acquire a high potential.)
Question 2.36 A small sphere of radius r1 and charge q1 is
enclosed by a spherical shell of radius r2 and charge q2. Show that if q1 is
positive, charge will necessarily flow from the sphere to the shell (when the
two are connected by a wire) no matter what the charge q2 on the shell is.
Question 2.37 Answer the following:
(a) The top of the atmosphere is at about 400 kV with respect to the surface
of the earth, corresponding to an electric field that decreases with altitude.
Near the surface of the earth, the field is about 100 Vm–1. Why then do we not
get an electric shock as we step out of our house into the open? (Assume the
house to be a steel cage so there is no field inside!)
(b) A man fixes outside his house one evening a two metre high insulating slab
carrying on its top a large aluminium sheet of area 1m 2.Will he get an electric
shock if he touches the metal sheet next morning?
(c) The discharging current in the atmosphere due to the small conductivity of
air is known to be 1800 A on an average over the globe. Why then does the
atmosphere not discharge itself completely in due course and become electrically
neutral? In other words, what keeps the atmosphere charged?
(d) What are the forms of energy into which the electrical energy of the
atmosphere is dissipated during a lightning? (Hint: The earth has an electric
field of about 100 Vm–1 at its surface in the downward direction, corresponding
to a surface charge density = –10–9 C m– 2. Due to the slight conductivity of
the atmosphere up to about 50 km (beyond which it is good conductor), about +
1800 C is pumped every second into the earth as a whole. The earth, however,
does not get discharged since thunderstorms and lightning occurring continually
all over the globe pump an equal amount of negative charge on the earth.)
:: Chapter 3 - Current Electricity ::
Question 3.1 The storage battery of a car has an emf of 12 V.
If the internal resistance of the battery is 0.4 Ω, what is the maximum current
that can be drawn from the battery?
Question 3.2 A battery of emf 10 V and internal resistance 3 Ω is connected to a
resistor. If the current in the circuit is 0.5 A, what is the resistance of the
resistor? What is the terminal voltage of the battery when the circuit is
closed?
Question 3.3 (a) Three resistors 1 Ω, 2 Ω, and 3 Ω are combined in series. What
is the total resistance of the combination? (b) If the combination is connected
to a battery of emf 12 V and negligible internal resistance, obtain the
potential drop across each resistor.
Question 3.4 (a) Three resistors 2 Ω, 4 Ω and 5 Ω are combined in parallel. What
is the total resistance of the combination?
(b) If the combination is connected to a battery of emf 20 V and negligible
internal resistance, determine the current through each resistor, and the total
current drawn from the battery.
Question 3.5 At room temperature (27.0 °C) the resistance of a heating element
is 100 Ω. What is the temperature of the element if the resistance is found to
be 117 Ω, given that the temperature coefficient of the material of the resistor
is 1.70 × 10–4 °C–1.
Question 3.6 A negligibly small current is passed through a wire of length 15 m
and uniform cross-section 6.0 × 10–7 m2, and its resistance is measured to be
5.0 Ω. What is the resistivity of the material at the temperature of the
experiment?
Question 3.7 A silver wire has a resistance of 2.1 Ω at 27.5 °C, and a
resistance of 2.7 Ω at 100 °C. Determine the temperature coefficient of
resistivity of silver.
Question 3.8 A heating element using nichrome connected to a 230 V supply draws
an initial current of 3.2 A which settles after a few seconds to a steady value
of 2.8 A. What is the steady temperature of the heating element if the room
temperature is 27.0 °C? Temperature coefficient of resistance of nichrome
averaged over the temperature range involved is 1.70 × 10–4 °C–1.
Question 3.9 Determine the current in each branch of the network shown in Fig.
3.30:
Question 3.10 (a) In a metre bridge [Fig. 3.7], the balance point is found to
be at 39.5 cm from the end A, when the resistor Y is of 12.5 Ω. Determine the
resistance of X. Why are the connections between resistors in a Wheatstone or
meter bridge made of thick copper strips?
(b) Determine the balance point of the bridge above if X and Y are
interchanged.
(c) What happens if the galvanometer and cell are interchanged at the balance
point of the bridge? Would the galvanometer show any current?
Question 3.11 A storage battery of emf 8.0 V and internal resistance 0.5 Ω is
being charged by a 120 V dc supply using a series resistor of 15.5 Ω. What is
the terminal voltage of the battery during charging? What is the purpose of
having a series resistor in the charging circuit?
Question 3.12 In a potentiometer arrangement, a cell of emf 1.25 V gives a
balance point at 35.0 cm length of the wire. If the cell is replaced by another
cell and the balance point shifts to 6 3.0 cm, what is the emf of the second
cell?
Question 3. 13 The number density of free electrons in a copper conductor
estimated in Example 3.1 is 8.5 × 1028 m–3. How long does an electron take to
drift from one end of a wire 3.0 m long to its other end? The area of
cross-section of the wire is 2.0 × 10–6 m2 and it is carrying a current of 3.0
A.
ADDITIONAL EXERCISES QUESTIONS
Question 3. 14 The earth’s surface has a negative surface
charge density of 10–9 C m–2. The potential difference of 400 kV between the top
of the atmosphere and the surface results (due to the low conductivity of the
lower atmosphere) in a current of only 1800 A over the entire globe. If there
were no mechanism of sustaining atmospheric electric field, how much time
(roughly) would be required to neutralise the earth’s surface? (This never
happens in practice because there is a mechanism to replenish electric charges,
namely the continual thunderstorms and lightning in different parts of the
globe). (Radius of earth = 6.37 × 106 m.)
Question 3.15 (a) Six lead-acid type of secondary cells each of emf 2.0 V and
internal resistance 0.015 Ω are joined in series to provide a supply to a
resistance of 8.5 Ω. What are the current drawn from the supply and its terminal
voltage?
(b) A secondary cell after long use has an emf of 1.9 V and a large internal
resistance of 380 Ω. What maximum current can be drawn from the cell? Could the
cell drive the starting motor of a car?
Question 3.16 Two wires of equal length, one of aluminium and the other of
copper have the same resistance. Which of the two wires is lighter? Hence
explain why aluminium wires are preferred for overhead power cables. (ρAl = 2.63
× 10–8 Ω m, ρCu = 1.72 × 10–8 Ω m, Relative density of Al = 2.7, of Cu = 8.9.)
Question 3.17 What conclusion can you draw from the following observations on a
resistor made of alloy manganin?
Question 3.18 Answer the following questions:
(a) A steady current flows in a metallic conductor of non-uniform
cross-section. Which of these quantities is constant along the conductor:
current, current density, electric field, drift speed?
(b) Is Ohm’s law universally applicable for all conducting elements? If not,
give examples of elements which do not obey Ohm’s law.
(c) A low voltage supply from which one needs high currents must have very low
internal resistance. Why?
(d) A high tension (HT) supply of, say, 6 kV must have a very large internal
resistance. Why?
Question 3.19 Choose the correct alternative:
(a) Alloys of metals usually have (greater/less) resistivity than that of
their constituent metals.
(b) Alloys usually have much (lower/higher) temperature coefficients of
resistance than pure metals.
(c) The resistivity of the alloy manganin is nearly independent of/ increases
rapidly with increase of temperature.
(d) The resistivity of a typical insulator (e.g., amber) is greater than that of
a metal by a factor of the order of (1022/103).
Question 3.20 (a) Given n resistors each of resistance R, how will you
combine them to get the
(i) maximum
(ii) minimum effective resistance? What is the ratio of the maximum to minimum
resistance? (b) Given the resistances of 1 Ω, 2 Ω, 3 Ω, how will be combine them
to get an equivalent resistance of (i) (11/3) Ω (ii) (11/5) Ω,
(iii) 6 Ω, (iv) (6/11) Ω? (c) Determine the equivalent resistance of networks
shown in Fig. 3.31.
Question 3.22 Figure 3.33 shows a potentiometer with a cell of 2.0 V and
internal resistance 0.40 Ω maintaining a potential drop across the resistor wire
AB. A standard cell which maintains a constant emf of 1.02 V (for very moderate
currents upto a few mA) gives a balance point at 67.3 cm length of the wire. To
ensure very low currents drawn from the standard cell, a very high resistance of
600 kΩ is put in series with it, which is shorted close to the balance point.
The standard cell is then replaced by a cell of unknown emf ε and the balance
point found similarly, turns out to be at 82.3 cm length of the wire.
(c) Is the balance point affected by this high resistance?
(d) Is the balance point affected by the internal resistance of the driver cell?
(e) Would the method work in the above situation if the driver cell of the
potentiometer had an emf of 1.0V instead of 2.0V? (f ) Would the circuit work
well for determining an extremely small emf, say of the order of a few mV (such
as the typical emf of a thermo-couple)? If not, how will you modify the circuit?
Question 3.23 Figure 3.34 shows a potentiometer circuit for comparison of two
resistances. The balance point with a standard resistor R = 10.0 Ω is found to
be 58.3 cm, while that with the unknown resistance X is 68.5 cm. Determine the
value of X. What might you do if you failed to find a balance point with the
given cell of emf ε ?
Question 3.24 Figure 3.35 shows a 2.0 V potentiometer used for the determination
of internal resistance of a 1.5 V cell. The balance point of the cell in open
circuit is 76.3 cm. When a resistor of 9.5 Ω is used in the external circuit of
the cell, the balance point shifts to 64.8 cm length of the potentiometer wire.
Determine the internal resistance of the cell.
:: Chapter 4 - Moving Charges and Magnetism ::
Question 4.1 A circular coil of wire consisting of 100 turns,
each of radius 8.0 cm carries a current of 0.40 A. What is the magnitude of the
magnetic field B at the centre of the coil?
Question 4.2 A long straight wire carries a current of 35 A.
What is the magnitude of the field B at a point 20 cm from the wire?
Question 4.3 A long straight wire in the horizontal plane
carries a current of 50 A in north to south direction. Give the magnitude and
direction of B at a point 2.5 m east of the wire.
Question 4.4 A horizontal overhead power line carries a
current of 90 A in east to west direction. What is the magnitude and direction
of the magnetic field due to the current 1.5 m below the line?
Question 4.5 What is the magnitude of magnetic force per unit
length ?
Question 4.6 A 3.0 cm wire carrying a current of 10 A is
placed inside a solenoid perpendicular to its axis. The magnetic field inside
the solenoid is given to be 0.27 T. What is the magnetic force on the wire?
Question 4.7 Two long and parallel straight wires A and B
carrying currents of 8.0 A and 5.0 A in the same direction are separated by a
distance of 4.0 cm. Estimate the force on a 10 cm section of wire A.
Question 4.8 A closely wound solenoid 80 cm long has 5 layers
of windings of 400 turns each. The diameter of the solenoid is 1.8 cm. If the
current carried is 8.0 A, estimate the magnitude of B inside the solenoid near
its centre.
Question 4.9 A square coil of side 10 cm consists of 20 turns
and carries a current of 12 A. The coil is suspended vertically and the normal
to the plane of the coil makes an angle of 30º with the direction of a uniform
horizontal magnetic field of magnitude 0.80 T. What is the magnitude of torque
experienced by the coil?
Question 4.10 Two moving coil meters, M1 and M2 have the
following particulars: R1 = 10 Ω, N1 = 30, A1 = 3.6 × 10–3 m2, B1 = 0.25 T R2 =
14 Ω, N2 = 42, A2 = 1.8 × 10–3 m2, B2 = 0.50 T (The spring constants are
identical for the two meters). Determine the ratio of (a) current sensitivity
and (b) voltage sensitivity of M2 and M1.
Question 4.11 In a chamber, a uniform magnetic field of 6.5 G
(1 G = 10–4 T) is maintained. An electron is shot into the field
with a speed of 4.8 × 106 m s–1 normal to the field. Explain why the path of the
electron is a circle. Determine the radius of the circular orbit. (e = 1.6 ×
10–19 C, me = 9.1×10–31 kg)
Question 4.12 In Exercise 4.11 obtain the frequency of
revolution of the electron in its circular orbit. Does the answer depend on the
speed of the electron? Explain.
Question 4.13 (a) A circular coil of 30 turns and radius
8.0 cm carrying a current of 6.0 A is suspended vertically in a uniform
horizontal magnetic field of magnitude 1.0 T. The field lines make an angle of
60º with the normal of the coil. Calculate the magnitude of the counter torque
that must be applied to prevent the coil from turning.
(b) Would your answer change, if the circular coil in
(a) were replaced by a planar coil of some irregular shape that encloses the
same area? (All other particulars are also unaltered.)
ADDITIONAL EXERCISES QUESTION
Question 4.14 Two concentric circular coils X and Y of radii
16 cm and 10 cm, respectively, lie in the same vertical plane containing the
north to south direction. Coil X has 20 turns and carries a current of 16 A;
coil Y has 25 turns and carries a current of 18 A. The sense of the current in X
is anticlockwise, and clockwise in Y, for an observer looking at the coils
facing west. Give the magnitude and direction of the net magnetic field due to
the coils at their centre .
Question 4.15 A magnetic field of 100 G (1 G = 10–4 T) is
required which is uniform in a region of linear dimension about 10 cm and area
of cross-section about 10–3 m2. The maximum current-carrying capacity of a given
coil of wire is 15 A and the number of turns per unit length that can be wound
round a core is at most 1000 turns m–1. Suggest some appropriate design
particulars of a solenoid for the required purpose. Assume the core is not
ferromagnetic.
Question 4.16 For a circular coil of radius R and N turns
carrying current I, the magnitude of the magnetic field at a point on its axis
at a distance x from its centre is given by, ( ) 2 0 2 2 3/2 2 IR N B x R μ = +
(a) Show that this reduces to the familiar result for field at the centre of
the coil.
(b) Consider two parallel co-axial circular coils of equal radius R, and number
of turns N, carrying equal currents in the same direction, and separated by a
distance R. Show that the field on the axis around the mid-point between the
coils is uniform over a distance that is small as compared to R, and is given
by, 0.72 0 NI B R μ = , approximately. [Such an arrangement to produce a nearly
uniform magnetic field over a small region is known as Helmholtz coils.]
Question 4.17 A toroid has a core (non-ferromagnetic) of
inner radius 25 cm and outer radius 26 cm, around which 3500 turns of a wire are
wound. If the current in the wire is 11 A, what is the magnetic field (a)
outside the toroid, (b) inside the core of the toroid, and (c) in the empty
space surrounded by the toroid.
Question 4.18 Answer the following questions:
(a) A magnetic field that varies in magnitude from point to point but has a
constant direction (east to west) is set up in a chamber. A charged particle
enters the chamber and travels undeflectedalong a straight path with constant
speed. What can you say about the initial velocity of the particle?
(b) A charged particle enters an environment of a strong and non-uniform
magnetic field varying from point to point both in magnitude and direction, and
comes out of it following a complicated trajectory. Would its final speed equal
the initial speed if it suffered no collisions with the environment?
(c) An electron travelling west to east enters a chamber having a uniform
electrostatic field in north to south direction. Specify the direction in which
a uniform magnetic field should be set up to prevent the electron from
deflecting from its straight line path.
Question 4.19 An electron emitted by a heated cathode and accelerated through
a potential difference of 2.0 kV, enters a region with uniform magnetic field of
0.15 T. Determine the trajectory of the electron if the field
(a) is transverse to its initial velocity,
(b) makes an angle of 30º with the initial velocity.
Question 4.20 A magnetic field set up using Helmholtz coils
(described in Exercise 4.16) is uniform in a small region and has a magnitude of
0.75 T. In the same region, a uniform electrostatic field is maintained in a
direction normal to the common axis of the coils. A narrow beam of (single
species) charged particles all accelerated through 15 kV enters this region in a
direction perpendicular to both the axis of the coils and the electrostatic
field. If the beam remains undeflected when the electrostatic field is 9.0 ×
10–5 V m–1, make a simple guess as to what the beam contains. Why is the answer
not unique?
Question 4.21 A straight horizontal conducting rod of length 0.45 m and mass
60 g is suspended by two vertical wires at its ends. A current of 5.0 A is set
up in the rod through the wires.
(a) What magnetic field should be set up normal to the conductor in order
that the tension in the wires is zero?
(b) What will be the total tension in the wires if the direction of current is
reversed keeping the magnetic field same as before? (Ignore the mass of the
wires.) g = 9.8 m s–2.
Question 4.22 The wires which connect the battery of an
automobile to its starting motor carry a current of 300 A (for a short time).
What is the force per unit length between the wires if they are 70 cm long and
1.5 cm apart? Is the force attractive or repulsive?
Question 4.23 A uniform magnetic field of 1.5 T exists in
a cylindrical region of radius10.0 cm, its direction parallel to the axis along
east to west. A wire carrying current of 7.0 A in the north to south direction
passes through this region. What is the magnitude and direction of the force on
the wire if,
(a) the wire intersects the axis,
(b) the wire is turned from N-S to northeast-northwest direction,
(c) the wire in the N-S direction is lowered from the axis by a distance of 6.0
cm?
Question 4.24 A uniform magnetic field of 3000 G is
established along the positive z-direction. A rectangular loop of sides 10 cm
and 5 cm carries a current of 12 A. What is the torque on the loop in the
different cases shown in Fig. 4.28? What is the force on each case? Which case
corresponds to stable equilibrium?
Question 4.25 A circular coil of 20 turns and radius 10 cm
is placed in a uniform magnetic field of 0.10 T normal to the plane of the coil.
If the current in the coil is 5.0 A, what is the
(a) total torque on the coil,
(b) total force on the coil,
(c) average force on each electron in the coil due to the magnetic field? (The
coil is made of copper wire of cross-sectional area 10–5 m2, and the free
electron density in copper is given to be about 1029 m–3.)
Question 4.26 A solenoid 60 cm long and of radius 4.0 cm has
3 layers of windings of 300 turns each. A 2.0 cm long wire of mass 2.5 g lies
inside the solenoid (near its centre) normal to its axis; both the wire and the
axis of the solenoid are in the horizontal plane. The wire is connected through
two leads parallel to the axis of the solenoid to an external battery which
supplies a current of 6.0 A in the wire. What value of current (with appropriate
sense of circulation) in the windings of the solenoid can support the weight of
the wire? g = 9.8 m s–2.
Question 4.27 A galvanometer coil has a resistance of 12 Ω
and the metre shows full scale deflection for a current of 3 mA. How will you
convert the metre into a voltmeter of range 0 to 18 V?
Question 4.28 A galvanometer coil has a resistance of 15 Ω
and the metre shows full scale deflection for a current of 4 mA. How will you
convert the metre into an ammeter of range 0 to 6 A?
:: Chapter 5 - Magnetism and Matter ::
Question 5.1 Answer the following questions regarding
earth’s magnetism:
(a) A vector needs three quantities for its specification. Name the three
independent quantities conventionally used to specify the earth’s magnetic
field.
(b) The angle of dip at a location in southern India is about 18º. Would you
expect a greater or smaller dip angle in Britain?
(c) If you made a map of magnetic field lines at Melbourne in Australia, would
the lines seem to go into the ground or come out of the ground?
(d) In which direction would a compass free to move in the vertical plane point
to, if located right on the geomagnetic north or south pole?
(e) The earth’s field, it is claimed, roughly approximates the field due to a
dipole of magnetic moment 8 × 1022 J T–1 located at its centre. Check the order
of magnitude of this number in some way.
(f ) Geologists claim that besides the main magnetic N-S poles, there are
several local poles on the earth’s surface oriented in different directions. How
is such a thing possible at all?
Question 5.2 Answer the following questions:
(a) The earth’s magnetic field varies from point to point in space. Does it
also change with time? If so, on what time scale does it change appreciably?
(b) The earth’s core is known to contain iron. Yet geologists do not regard this
as a source of the earth’s magnetism. Why?
(c) The charged currents in the outer conducting regions of the earth’s core are
thought to be responsible for earth’s magnetism. What might be the ‘battery’
(i.e., the source of energy) to sustain these currents?
(d) The earth may have even reversed the direction of its field several times
during its history of 4 to 5 billion years. How can geologists know about the
earth’s field in such distant past?
(e) The earth’s field departs from its dipole shape substantially at large
distances (greater than about 30,000 km). What agencies may be responsible for
this distortion?
(f ) Interstellar space has an extremely weak magnetic field of the order of
10–12 T. Can such a weak field be of any significant consequence?
Question 5.2 is meant mainly to arouse your curiosity.
Answers to some questions above are tentative or unknown. Brief answers wherever
possible are given at the end. For details, you should consult a good text on
geomagnetism.]
Question 5.3 A short bar magnet placed with its axis at 30º
with a uniform external magnetic field of 0.25 T experiences a torque of
magnitude equal to 4.5 × 10–2 J. What is the magnitude of magnetic moment of the
magnet?
Question 5.4 A short bar magnet of magnetic moment m = 0.32
JT–1 is placed in a uniform magnetic field of 0.15 T. If the bar is free to
rotate in the plane of the field, which orientation wou
Question 5.5 A closely wound solenoid of 800 turns and area
of cross section 2.5 × 10–4 m2 carries a current of 3.0 A. Explain the sense in
which the solenoid acts like a bar magnet. What is its associated magnetic
moment?
Question 5.6 If the solenoid in xercise 5.5 is free to turn
about the vertical direction and a uniform horizontal magnetic field of 0.215 T
is applied, what is the magnitude of torque on the solenoid when its axis makes
an angle of 30° with the direction of applied field?
Question 5.7 A bar magnet of magnetic moment 1.5 J T–1
lies aligned with the direction of a uniform magnetic field of 0.22 T. (a) What
is the amount of work required by an external torque to turn the magnet so as to
align its magnetic moment:
(i) normal to the field direction , (ii) opposite to the field direction?
(b) What is the torque on the magnet in cases (i) and (ii)?
Question 5.8 A closely wound solenoid of 2000 turns and
area of cross-section 1.6 × 10–4 m2, carrying a current of 4.0 A, is suspended
through its centre allowing it to turn in a horizontal plane.
(a) What is the magnetic moment associated with the solenoid?
(b) What is the force and torque on the solenoid if a uniform horizontal
magnetic field of 7.5 × 10–2 T is set up at an angle of 30º with the axis of the
solenoid?
Question 5.9 A circular coil of 16 turns and radius 10 cm
carrying a current of 0.75 A rests with its plane normal to an external field of
magnitude 5.0 × 10–2 T. The coil is free to turn about an axis in its plane
perpendicular to the field direction. When the coil is turned slightly and
released, it oscillates about its stable equilibrium with a frequency of 2.0
s–1. What is the moment of inertia of the coil about its axis of rotation?
Question 5.10 A magnetic needle free to rotate in a vertical
plane parallel to the magnetic meridian has its north tip pointing down at 22º
with the horizontal. The horizontal component of the earth’s magnetic field at
the place is known to be 0.35 G. Determine the magnitude of the earth’s magnetic
field at the place.
Question 5.11 At a certain location in Africa, a compass
points 12º west of the geographic north. The north tip of the magnetic needle of
a dip circle placed in the plane of magnetic meridian points 60º above the
horizontal. The horizontal component of the earth’s field is measured to be 0.16
G. Specify the direction and magnitude of the earth’s field at the location.
Question 5.12 A short bar magnet has a magnetic moment of
0.48 J T–1. Give the direction and magnitude of the magnetic field produced by
the magnet at a distance of 10 cm from the centre of the magnet on (a) the
axis,(b) the equatorial lines (normal bisector) of the magnet.
Question 5.13 A short bar magnet placed in a horizontal plane
has its axis aligned along the magnetic north-south direction. Null points are
found on the axis of the magnet at 14 cm from the centre of the magnet. The
earth’s magnetic field at the place is 0.36 G and the angle of dip is zero. What
is the total magnetic field on the normal bisector of the magnet at the same
distance as the null–point (i.e., 14 cm) from the centre of the magnet? (At null
points, field due to a magnet is equal and opposite to the horizontal component
of earth’s magnetic field.)
Question 5.14 If the bar magnet in exercise 5.13 is turned
around by 180º, where will the new null points be located?
Question 5.15 A short bar magnet of magnetic moment 5.25 ×
10–2 J T–1 is placed with its axis perpendicular to the earth’s field direction.
At what distance from the centre of the magnet, the resultant field is inclined
at 45º with earth’s field on (a) its normal bisector and (b) its axis. Magnitude
of the earth’s field at the place is given to be 0.42 G. Ignore the length of
the magnet in comparison to the distances involved.
ADDITIONAL EXERCISES QUESTIONS
Question 5.16 Answer the following questions:
(a) Why does a paramagnetic sample display greater magnetisation (for the
(c) If a toroid uses bismuth for its core, will the field in the core be
(slightly) greater or (slightly) less than when the core is empty?
(d) Is the permeability of a ferromagnetic material independent of the magnetic
field? If not, is it more for lower or higher fields?
(e) Magnetic field lines are always nearly normal to the surface of a
ferromagnet at every point. (This fact is analogous to the static electric field
lines being normal to the surface of a conductor at every point.) Why?
(f ) Would the maximum possible magnetisation of a paramagnetic sample be of the
same order of magnitude as the magnetisation of a ferromagnet?
Question 5.17 Answer the following questions:
(a) Explain qualitatively on the basis of domain picture the irreversibility
in the magnetisation curve of a ferromagnet.
(b) The hysteresis loop of a soft iron piece has a much smaller area than that
of a carbon steel piece. If the material is to go through repeated cycles of
magnetisation, which piece will dissipate greater heat energy?
(c) ‘A system displaying a hysteresis loop such as a ferromagnet, is a device
for storing memory?’ Explain the meaning of this statement.
(d) What kind of ferromagnetic material is used for coating magnetic tapes in a
cassette player, or for building ‘memory stores’ in a modern computer?
(e) A certain region of space is to be shielded from magnetic fields. Suggest a
method .
Question 5.18 A long straight horizontal cable carries a
current of 2.5 A in the direction 10º south of west to 10º north of east. The
magnetic meridian of the place happens to be 10º west of the geographic
meridian. The earth’s magnetic field at the location is 0.33 G, and the angle of
dip is zero. Locate the line of neutral points (ignore the thickness of the
cable). (At neutral points, magnetic field due to a current-carrying cable is
equal and opposite to the horizontal component of earth’s magnetic field.)
Question 5.19 A telephone cable at a place has four long
straight horizontal wires carrying a current of 1.0 A in the same direction east
to west. The earth’s magnetic field at the place is 0.39 G, and the angle of dip
is 35º. The magnetic declination is nearly zero. What are the resultant magnetic
fields at points 4.0 cm below the cable?
Question 5.20 A compass needle free to turn in a
horizontal plane is placed at the centre of circular coil of 30 turns and radius
12 cm. The coil is in a vertical plane making an angle of 45º with the magnetic
meridian. When the current in the coil is 0.35 A, the needle points west to
east.
(a) Determine the horizontal component of the earth’s magnetic field at the
location.
(b) The current in the coil is reversed, and the coil is rotated about its
vertical axis by an angle of 90º in the anticlockwise sense looking from above.
Predict the direction of the needle. Take the magnetic declination at the places
to be zero.
Question 5.21 A magnetic dipole is under the influence of two
magnetic fields. The angle between the field directions is 60º, and one of the
fields has a magnitude of 1.2 × 10–2 T. If the dipole comes to stable
equilibrium at an angle of 15º with this field, what is the magnitude of the
other field?
Question 5.22 A monoenergetic (18 keV) electron beam
initially in the horizontal direction is subjected to a horizontal magnetic
field of 0.04 G normal to the initial direction. Estimate the up or down
deflection of the beam over a distance of 30 cm (me = 9.11 × 10–19 C). [Note:
Data in this exercise are so chosen that the answer will give you an idea of the
effect of earth’s magnetic field on the motion of the electron beam from the
electron gun to the screen in a TV set.]
Question 5.23 A sample of paramagnetic salt contains 2.0 ×
1024 atomic dipoles each of dipole moment 1.5 × 10–23 J T–1. The sample is
placed under a homogeneous magnetic field of 0.64 T, and cooled to a temperature
of 4.2 K. The degree of magnetic saturation achieved is equal to 15%. What is
the total dipole moment of the sample for a magnetic field of 0.98 T and a
temperature of 2.8 K? (Assume Curie’s law
Question 5.24 A Rowland ring of mean radius 15 cm has 3500
turns of wire wound on a ferromagnetic core of relative permeability 800. What
is the magnetic field B in the core for a magnetising current of 1.2 A?
Question 5.25 The magnetic moment vectors μs and μl
associated with the intrinsic spin angular momentum S and orbital angular
momentum l, respectively, of an electron are predicted by quantum theory (and
verified experimentally to a high accuracy) to be given by: μs = –(e/m) S, μl =
–(e/2m)l Which of these relations is in accordance with the result expected
classically? Outline the derivation of the classical result
:: Chapter 6 - Electromagnetic Induction ::
Question 6.1 Predict the direction of induced current in
the situations described by the following Figs. 6.18(a) to (f ).
Question 6.2 Use Lenz’s law to determine the direction of
induced current in the situations described by Fig. 6.19:
(a) A wire of irregular shape turning into a circular shape;
(b) A circular loop being deformed into a narrow straight wire.
Question 6.3 A long solenoid with 15 turns per cm has a small
loop of area 2.0 cm2 placed inside the solenoid normal to its axis. If the
current carried by the solenoid changes steadily from 2.0 A to 4.0 A in 0.1 s,
what is the induced emf in the loop while the current is changing?
Question 6.4 A rectangular wire loop of sides 8 cm and 2 cm
with a small cut is moving out of a region of uniform magnetic field of
magnitude 0.3 T directed normal to the loop. What is the emf developed across
the cut if the velocity of the loop is 1 cm s–1 in a direction normal to the (a)
longer side, (b) shorter side of the loop? For how long does the induced voltage
last in each case?
Question 6.5 A 1.0 m long metallic rod is rotated with an
angular frequency of 400 rad s–1 about an axis normal to the rod passing through
its one end. The other end of the rod is in contact with a circular metallic
ring. A constant and uniform magnetic field of 0.5 T parallel to the axis exists
everywhere. Calculate the emf developed between the centre and the ring.
Question 6.6 A circular coil of radius 8.0 cm and 20 turns is
rotated about its vertical diameter with an angular speed of 50 rad s–1 in a
uniform horizontal magnetic field of magnitude 3.0 × 10–2 T. Obtain the maximum
and average emf induced in the coil. If the coil forms a closed loop of
resistance 10 Ω, calculate the maximum value of current in the coil. Calculate
the average power loss due to Joule heating. Where does this power come from?
Question 6.7 A horizontal straight wire 10 m long
extending from east to west is falling with a speed of 5.0 m s–1, at right
angles to the horizontal component of the earth’s magnetic field, 0.30 × 10–4 Wb
m–2.
(a) What is the instantaneous value of the emf induced in the wire?
(b) What is the direction of the emf?
(c) Which end of the wire is at the higher electrical potential?
Question 6.8 Current in a circuit falls from 5.0 A to 0.0 A
in 0.1 s. If an average emf of 200 V induced, give an estimate of the
self-inductance of the circuit.
Question 6.9 A pair of adjacent coils has a mutual inductance
of 1.5 H. If the current in one coil changes from 0 to 20 A in 0.5 s, what is
the change of flux linkage with the other coil?
Question 6.10 A jet plane is travelling towards west at a
speed of 1800 km/h. What is the voltage difference developed between the ends of
the wing
ADDITIONAL EXERCISES QUESTIONS
Question 6.11 Suppose the loop in Exercise Question 6.4 is
stationary but the current feeding the electromagnet that produces the magnetic
field is gradually reduced so that the field decreases from its initial value of
0.3 T at the rate of 0.02 T s–1. If the cut is joined and the loop has a
resistance of 1.6 Ω, how much power is dissipated by the loop as heat? What is
the source of this power?
Question 6.12 A square loop of side 12 cm with its sides
parallel to X and Y axes is moved with a velocity of 8 cm s–1 in the positive
x-direction in an environment containing a magnetic field in the positive
z-direction. The field is neither uniform in space nor constant in time. It has
a gradient of 10 –3 T cm–1 along the negative x-direction (that is it increases
by 10 – 3 T cm–1 as one moves in the negative x-direction), and it is decreasing
in time at the rate of 10 –3 T s–1. Determine the direction and magnitude of the
induced current in the loop if its resistance is 4.50 mΩ.
Question 6.13 It is desired to measure the magnitude of field
between the poles of a powerful loud speaker magnet. A small flat search coil of
area 2 cm2 with 25 closely wound turns, is positioned normal to the field
direction, and then quickly snatched out of the field region. Equivalently, one
can give it a quick 90° turn to bring its plane parallel to the field
direction). The total charge flown in the coil (measured by a ballistic
galvanometer connected to coil) is 7.5 mC. The combined resistance of the coil
and the galvanometer is 0.50 Ω. Estimate the field strength of magnet.
Question 6.14 Figure 6.20 shows a metal rod PQ resting on
the smooth rails AB and positioned between the poles of a permanent magnet. The
rails, the rod, and the magnetic field are in three mutual perpendicular
directions. A galvanometer G connects the rails through a switch K. Length of
the rod = 15 cm, B = 0.50 T, resistance of the closed loop containing the rod =
9.0 mΩ. Assume the field to be uniform.
(a) Suppose K is open and the rod is moved with a speed of 12 cm s–1 in the
direction shown. Give the polarity and magnitude of the induced emf. experience
magnetic force due to the motion of the rod. Explain.
(d) What is the retarding force on the rod when K is closed?
(e) How much power is required (by an external agent) to keep the rod moving at
the same speed (=12 cm s–1) when K is closed? How much power is required when K
is open?
(f ) How much power is dissipated as heat in the closed circuit? What is the
source of this power?
(g) What is the induced emf in the moving rod if the magnetic field is parallel
to the rails instead of being perpendicular?
Question 6.15 An air-cored solenoid with length 30 cm, area
of cross-section 25 cm2 and number of turns 500, carries a current of 2.5 A. The
current is suddenly switched off in a brief time of 10–3 s. How much is the
average back emf induced across the ends of the open switch in the circuit?
Ignore the variation in magnetic field near the ends of the solenoid.
Question 6.16 (a) Obtain an expression for the mutual
inductance between a long straight wire and a square loop of side a as shown in
Fig. 6.21.
(b) Now assume that the straight wire carries a current of 50 A and the loop is
moved to the right with a constant velocity, v = 10 m/s. Calculate the induced
emf in the loop at the instant when x = 0.2 m. Take a = 0.1 m and assume that
the loop has a large resistance.
Question 6.17 A line charge λ per unit length is lodged
uniformly onto the rim of a wheel of mass M and radius R. The wheel has light
non-conducting spokes and is free to rotate without friction about its axis
(Fig. 6.22). A uniform magnetic field extends over a circular region within the
rim. It is given by, B = – B0 k (r ≤ a; a < R) = 0 (otherwise) What is the
angular velocity of the wheel after the field is suddenly switched off?
:: Chapter 7 - Alternating Current ::
Question 7.1 A 100 Ω resistor is connected to a 220 V, 50
Hz ac supply.
(a) What is the rms value of current in the circuit?
(b) What is the net power consumed over a full cycle?
Question 7.2(a) The peak voltage of an ac supply is 300 V.
What is the rms voltage?
(b) The rms value of current in an ac circuit is 10 A. What is the peak current?
Question 7.3 A 44 mH inductor is connected to 220 V, 50 Hz ac
supply. Determine the rms value of the current in the circuit.
Question 7.4 A 60 μF capacitor is connected to a 110 V, 60 Hz
ac supply. Determine the rms value of the current in the circuit.
Question 7.5 In Exercises 7.3 and 7.4, what is the net power
absorbed by each circuit over a complete cycle. Explain your answer.
Question 7.6 Obtain the resonant frequency ωr of a series LCR
circuit with L = 2.0H, C = 32 μF and R = 10 Ω. What is the Q-value of this
circuit?
Question 7.7 A charged 30 μF capacitor is connected to a 27
mH inductor. What is the angular frequency of free oscillations of the circuit?
Question 7.8 Suppose the initial charge on the capacitor in
Exercise
Question 7.7 is 6 mC. What is the total energy stored in the
circuit initially? What is the total energy at later time?
Question 7.9 A series LCR circuit with R = 20 Ω, L = 1.5 H
and C = 35 μF is connected to a variable-frequency 200 V ac supply. When the
frequency of the supply equals the natural frequency of the circuit, what is the
average power transferred to the circuit in one complete cycle?
Question 7.10 A radio can tune over the frequency range of a
portion of MW broadcast band: (800 kHz to 1200 kHz). If its LC circuit has an
effective inductance of 200 μH, what must be the range of its variable
capacitor? [Hint: For tuning, the natural frequency i.e., the frequency of free
oscillations of the LC circuit should be equal to the frequency of the radiowave.]
Question 7.11 Figure 7.21 shows a series LCR circuit
connected to a variable frequency 230 V source. L = 5.0 H, C = 80μF, R = 40 Ω.
(a) Determine the source frequency which drives the circuit in resonance.
(b) Obtain the impedance of the circuit and the amplitude of current at the
resonating frequency.
(c) Determine the rms potential drops across the three elements of the circuit.
Show that the potential drop across the LC combination is zero at the resonating
frequency.
ADDITIONAL EXERCISES QUESTIONS
Question 7.12 An LC circuit contains a 20 mH inductor and
a 50 μF capacitor with an initial charge of 10 mC. The resistance of the circuit
is negligible. Let the instant the circuit is closed be t = 0 .
(a) What is the total energy stored initially? Is it conserved during LC
oscillations?
(b) What is the natural frequency of the circuit?
(c) At what time is the energy stored
(i) completely electrical (i.e., stored in the capacitor)?
(ii) completely magnetic (i.e., stored in the inductor)?
(d) At what times is the total energy shared equally between the inductor and
the capacitor?
(e) If a resistor is inserted in the circuit, how much energy is eventually
dissipated as heat?
Question 7.13 A coil of inductance 0.50 H and resistance
100 Ω is connected to a 240 V, 50 Hz ac supply.
(a) What is the maximum current in the coil?
(b) What is the time lag between the voltage maximum and the current maximum?
Question 7.14 Obtain the answers (a) to (b) in Exercise 7.13
if the circuit is connected to a high frequency supply (240 V, 10 kHz). Hence,
explain the statement that at very high frequency, an inductor in a circuit
nearly amounts to an open circuit. How does an inductor behave in a dc circuit
after the steady state?
Question 7.15 A 100 μF capacitor in series with a 40 Ω
resistance is connected to a 110 V, 60 Hz supply.
(a) What is the maximum current in the circuit?
(b) What is the time lag between the current maximum and the voltage maximum?
Question 7.16 Obtain the answers to (a) and (b) in Exercise
7.15 if the circuit is connected to a 110 V, 12 kHz supply? Hence, explain the
statement that a capacitor is a conductor at very high frequencies. Compare this
behaviour with that of a capacitor in a dc circuit after the steady state.
Question 7.17 Keeping the source frequency equal to the
resonating frequency of the series LCR circuit, if the three elements, L, C and
R are arranged in parallel, show that the total current in the parallel LCR
circuit is minimum at this frequency. Obtain the current rms value in each
branch of the circuit for the elements and source specified in Exercise 7.11 for
this frequency.
Question 7.18 A circuit containing a 80 mH inductor and a
60 μF capacitor in series is connected to a 230 V, 50 Hz supply. The resistance
of the circuit is negligible.
(a) Obtain the current amplitude and rms values
(b) Obtain the rms values of potential drops across each element.
(c) What is the average power transferred to the inductor?
(d) What is the average power transferred to the capacitor?
(e) What is the total average power absorbed by the circuit? [‘Average’ implies
‘averaged over one cycle’.]
Question 7.19 Suppose the circuit in Exercise 7.18 has a
resistance of 15 Ω. Obtain the average power transferred to each element of the
circuit, and the total power absorbed. Physics 268
Question 7.20 A series LCR circuit with L = 0.12 H, C =
480 nF, R = 23 Ω is connected to a 230 V variable frequency supply.
(a) What is the source frequency for which current amplitude is maximum.
Obtain this maximum value.
(b) What is the source frequency for which average power absorbed by the circuit
is maximum. Obtain the value of this maximum power.
(c) For which frequencies of the source is the power transferred to the circuit
half the power at resonant frequency? What is the current amplitude at these
frequencies?
(d) What is the Q-factor of the given circuit?
Question 7.21 Obtain the resonant frequency and Q-factor of a
series LCR circuit with L = 3.0 H, C = 27 μF, and R = 7.4 Ω. It is desired to
improve the sharpness of the resonance of the circuit by reducing its ‘full
width at half maximum’ by a factor of 2. Suggest a suitable way.
Question 7.22 Answer the following questions:
(a) In any ac circuit, is the applied instantaneous voltage equal to the
algebraic sum of the instantaneous voltages across the series elements of the
circuit? Is the same true for rms voltage?
(b) A capacitor is used in the primary circuit of an induction coil.
(c) An applied voltage signal consists of a superposition of a dc voltage and an
ac voltage of high frequency. The circuit consists of an inductor and a
capacitor in series. Show that the dc signal will appear across C and the
ac signal across L.
(d) A choke coil in series with a lamp is connected to a dc line. The lamp is
seen to shine brightly. Insertion of an iron core in the choke causes no change
in the lamp’s brightness. Predict the corresponding observations if the
connection is to an ac line.
(e) Why is choke coil needed in the use of fluorescent tubes with ac mains? Why
can we not use an ordinary resistor instead of the choke coil?
Question 7.23 A power transmission line feeds input power at
2300 V to a stepdown transformer with its primary windings having 4000 turns.
What should be the number of turns in the secondary in order to get output power
at 230 V?
Question 7.24 At a hydroelectric power plant, the water
pressure head is at a height of 300 m and the water flow available is 100 m3s–1.
If the turbine generator efficiency is 60%, estimate the electric power
available from the plant (g = 9.8 ms–2 ).
Question 7.25 A small town with a demand of 800 kW of
electric power at 220 V is situated 15 km away from an electric plant generating
power at 440 V. The resistance of the two wire line carrying power is 0.5 Ω per
km. The town gets power from the line through a 4000-220 V step-down transformer
at a sub-station in the town.
(a) Estimate the line power loss in the form of heat.
(b) How much power must the plant supply, assuming there is negligible power
loss due to leakage?
(c) Characterise the step up transformer at the plant.
Question 7.26 Do the same exercise as above with the
replacement of the earlier transformer by a 40,000-220 V step-down transformer
(Neglect, as before, leakage losses though this may not be a good assumption any
longer because of the very high voltage transmission involved). Hence, explain
why high voltage transmission is preferred?
:: Chapter 8 - Electromagnetic Waves ::
Question 8.1 Figure 8.6 shows a capacitor made of
two circular plates each of radius 12 cm, and separated by 5.0 cm. The capacitor
is being charged by an external source (not shown in the figure). The charging
current is constant and equal to 0.15A.
(a) Calculate the capacitance and the rate of charge of
potential difference between the plates.
(b) Obtain the displacement current across the plates.
(c) Is Kirchhoff’s first rule (junction rule) valid at each plate of the
capacitor? Explain
Question 8.2 A parallel plate capacitor (Fig. 8.7) made of circular plates
each of radius R = 6.0 cm has a capacitance C = 100 pF. The capacitor is
connected to a 230 V ac supply with a (angular) frequency of 300 rad s–1.
(a) What is the rms value of the conduction current?
(b) Is the conduction current equal to the displacement current?
(c) Determine the amplitude of B at a point 3.0 cm from the axis between the
plates.
Question 8.3 What physical quantity is the same for X-rays of
wavelength 10–10 m, red light of wavelength 6800 Å and radiowaves of wavelength
500m?
Question 8.4 A plane electromagnetic wave travels in vacuum
along z-direction. What can you say about the directions of its electric and
magnetic field vectors? If the frequency of the wave is 30 MHz, what is its
wavelength?
Question 8.5 A radio can tune in to any station in the 7.5
MHz to 12 MHz band. What is the corresponding wavelength band?
Question 8.6 A charged particle oscillates about its mean
equilibrium position with a frequency of 109 Hz. What is the frequency of the
electromagnetic waves produced by the oscillator?
Question 8.7 The amplitude of the magnetic field part of a
harmonic electromagnetic wave in vacuum is B0 = 510 nT. What is the amplitude of
the electric field part of the wave?
Question 8.8 Suppose that the electric field amplitude of
an electromagnetic wave is E0 = 120 N/C and that its frequency is ν = 50.0 MHz.
(a) Determine, B0,ω, k, and λ
(b) Find expressions for E and B.
Question 8.9 The terminology of different parts of the
electromagnetic spectrum is given in the text. Use the formula E = hν (for
energy of a quantum of radiation: photon) and obtain the photon energy in units
of eV for different parts of the electromagnetic spectrum. In what way are the
different scales of photon energies that you obtain related to the sources of
electromagnetic radiation?
Question 8.10 In a plane electromagnetic wave, the
electric field oscillates sinusoidally at a frequency of 2.0 × 1010 Hz and
amplitude 48 V m–1.
(a) What is the wavelength of the wave?
(b) What is the amplitude of the oscillating magnetic field? (c) Show that the
average energy density of the E field equals the average energy density of the B
field. [c = 3 × 108 m s–1.]
ADDITIONAL EXERCISES
Question 8.11 Suppose that the electric field part of an
electromagnetic wave in vacuum is E = {(3.1 N/C) cos [(1.8 rad/m) y + (5.4 × 106
rad/s)t]}ˆi .
(a) What is the direction of propagation?
(b) What is the wavelength λ ?
(c) What is the frequency ν ?
(d) What is the amplitude of the magnetic field part of the wave?
(e) Write an expression for the magnetic field part of the wave.
Question 8.12 About 5% of the power of a 100 W light bulb
is converted to visible radiation. What is the average intensity of visible
radiation
(a) at a distance of 1m from the bulb?
(b) at a distance of 10 m? Assume that the radiation is emitted isotropically
and neglect reflection.
