Kamal A.A. 1000 Solved Problems in Classical Physics: An Exercise Book
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Problem Index 791
11.50 Application of Gauss’ law to calculate E inside and outside a charged
sphere.
11.51 In prob. (11.50) to find E on the surface of charged sphere with cavity.
11.52 In prob. (11.50) E ∝ r for r < R and that V (0) =
3
2
V (R).
11.53 E in three regions of a non-conducting charged sphere.
11.54 E for two regions of a long charged cylinder.
11.55 Net charge within the sphere’s surface given E. σ from E on a football field.
Total electric flux.
11.56 Derivation of Coulomb’s formula from Gauss’ law. Electric flux through
spherical surface concentric with a charged sphere.
11.57 Application of Gauss’ law to find E in the three regions of two charged con-
centric spherical shells.
11.58 Two insulated spheres positively charged at large distance are brought into
contact and separated by the same distance as before. To compare force of
repulsion before and after contact.
11.59 Maximum charge a sphere can withstand given the breakdown voltage.
11.60 (a) Capacitance of a conducting sphere and (b) U when two charged
spheres are connected by a wire and wire is removed.
11.61 When two spherical charged conductors are brought in contact and separated
σ ∝ 1/r.
11.62 To find V if E on balloon is given. Pressure in balloon which would produce
the same effect. Total electrostatic energy of the balloon.
11.63 A soap bubble of radius R
1
when charged expands to radius R
2
. To derive an
expression for the charge.
11.64 E in the three regions when an insulating shell is charged to specified charged
density.
11.65 (a) Electrostatic field is conservative and (b) U when a charged soap bubble
collapses to a smaller size.
11.66 Form of E generated by a long charged cylinder. Speed of electron circling
around the axis of the cylinder.
11.67 To find σ for non-conducting and conducting infinite sheets.
11.68 Capacitance of a parallel plate capacitor. Modification of C when a thin metal
is introduced.
11.69 Values of C
1
and C
2
, given their combined values in series and parallel
arrangement.
11.70 Energy in two capacitors: (a) singly; (b) in series; and (c) in parallel, given
P.D.
11.71 U when a charged air capacitor is submerged in oil of given ε
r
.
11.72 Value of K when A, d and C are given.
11.73 Resulting voltage when a charged capacitor is connected to an uncharged
one.
11.74 To find the equivalent capacitance of capacitors in the given arrangement.
11.75 To calculate q, V and U for three capacitors connected in series to a battery
of 260 V.
792 Problem Index
11.76 Two capacitors are charged to a battery and connected in parallel. Find P.D
of the combination if (a) positive ends are connected and (b) positive end is
connected to negative terminal of the other.
11.77 Two capacitors are charged to P.D V
1
and V
2
and connected in parallel. U
when (a) positive ends are joined and (b) positive end of one is joined to
negative end of the other.
11.78 Effect of dielectric on V , E, q, C and U, when the battery (a) remains con-
nected and (b) is disconnected.
11.79 In prob. (11.78) dependence of the given quantities on the distance of sepa-
ration of plates.
11.80 Force of attraction between the plates of a parallel plate capacitor.
11.81 n identical droplets each of radius r and charge q coalesce to form a large
drop. To find relations of radius, C, V , σ and U for the large drop and
droplet.
11.82 Half of the stored U
E
of a cylindrical capacitor of radii a and b lies within a
radius
√
ab of the cylinder.
11.83 Capacitor of capacitance C
1
withstands maximum voltage V
1
and C
2
with-
stands maximum voltage V
2
. Maximum voltage that the system of C
1
and C
2
can withstand when connected in series.
11.84 Application of Gauss’ law to calculate t he capacitance of Geiger–Muller
counter.
11.85 Capacitance of a capacitor formed by two spherical metallic shells.
11.86 For two concentric shells the capacitance reduces to that of a parallel plate
capacitor in the limit of large radii.
11.87 In the R – C circuit shown to find (i) time for charge to reach 90% of its final
value; (ii) U stored in the capacitor at t = τ ; and (iii) Joule heating in R at
t = τ .
11.88 In prob. (11.87) number of time constants after which energy in capacitor
will reach half of equilibrium value.
11.89 Capacitance of a parallel plate capacitor whose plates are slightly inclined.
11.90 and 11.91 In the given arrangements of capacitor to find P.D, q and U in the
capacitors.
11.92 To find the effective capacitance between two points in the given arrangement
of capacitors.
11.93 To obtain an expression for q(t) for an R − C circuit.
11.94 For the given R −C circuit, to find battery current at t = 0 and t =∞when
switch is closed. To find the current through R when switch is open after a
long time.
11.95 A charged capacitor is discharged through a resistance. To find U, i, V
c
at
given time, τ and equation for t when q drops to half of its value.
11.96 Charge q is uniformly distributed in a sphere of radius R. (i) To find div E
inside the sphere; (ii) electric force on a proton at r < R; and (iii) work done
on proton to move it from infinity to a point at r < R.
11.97 The electric displacement D is uniform in a parallel plate capacitor. To obtain
E(x) for a non-uniform relative permittivity, using Gauss’ law.
11.98 To obtain the differential Gauss’ law for gravitation.
Problem Index 793
Chapter 12
Electric Circuits
12.1 Effective resistance between two points A and B in the given arrangement.
12.2 Change in resistance when a wire is stretched.
12.3 Equivalent resistance is p for two resistors in series and q for parallel, mini-
mum value of n where p = nq .
12.4 Equivalent resistance for five resistors in the given arrangement.
12.5 Effective resistance of five resistors in the given arrangement.
12.6 Resistance between two terminals in the given network.
12.7 Equivalent resistance between two terminals in the given network.
12.8 Wheatstone bridge.
12.9 Five resistors are connected in the form of a square and a diagonal. To find
R
eq
across a side.
12.10 Effective resistance for a network of infinite number of resistors.
12.11 What equal length of an iron wire and a constantan wire of equal diameter
must be joined in parallel to give equivalent resistance of 2 ?
12.12 Temperature coefficient of resistance.
12.13 A square ABCD is formed by bending a wire. B and D are joined by a similar
wire and a battery of negligible internal resistance is included between A and
C. To find R
eq
and power dissipated.
12.14 Temperature dependence of resistance.
12.15 Equivalent resistance of a network in the form of a skeleton cube across the
body diagonal.
12.16 Brightness of two bulbs in series.
12.17 Brightness of two bulbs in parallel.
12.18 Three resistors in parallel are connected with two in series. If a PD of
120 V is applied across the ends of circuit, to find PD drop across parallel
arrangement.
12.19 Maximum power delivered to the external resistor.
12.20 Boiling of water by two heater coils when they are connected (a) in series
and (b) in parallel.
12.21 Calculation of power expended in two resistors connected in series and in
parallel.
12.22 Rate of energy loss in power transmission.
12.23 Power dissipated in three resistors in (a) series and (b) parallel.
12.24 The value of resistance in parallel with a heater so that the 1000 W heater
operates at 62.5 W.
12.25 Number of cells wrongly connected in a battery.
12.26 Condition for maximum current in m external resistances connected to m
rows of cells with each row containing n cells in series, each cell with internal
resistance r.
12.27 Internal resistances of two cells in series and the external resistance.
12.28 Rate of heat production in resistors in series and in parallel.
12.29 Charging of a battery.
794 Problem Index
12.30 Comparison of power dissipated in resistance with the value for power sup-
plied by battery.
12.31 Internal resistance of a cell by potentiometer.
12.32 Emf of a cell by potentiometer method.
12.33 Condition for null deflection and i
G
/i in galvanometer of Wheatstone bridge.
12.34 To find current in an ammeter in a network.
12.35 Internal resistance of a battery.
12.36 Internal resistance of a cell by potentiometer method.
12.37 External resistance by potentiometer method.
12.38 Reading in the voltmeter.
12.39 Metre bridge.
12.40 A moving coil meter reading up to 1 mA to be converted into (a) 100 mA full
scale and (b) 80 V full scale.
12.41 Pocket voltmeter.
12.42 Resistance of a galvanometer.
12.43 A battery of ξ
1
and r
1
with a second battery of ξ
2
and r
2
in parallel is joined
to an external resistance R. To calculate i
1
, i
2
, P
1
, P
2
and P.
12.44 Emf of batteries by applying the loop theorem.
12.45 Application of Kirchhoff’s rules to determine currents in branches of a
circuit.
12.46 Currents in various resistors and PD across the cells in a network.
12.47 Equivalent resistance of the circuit and power dissipated.
12.48 Given the internal resistance of one cell to calculate the internal resistance of
the other cell.
12.49 Equivalent resistance, voltages, currents and power dissipated in a series par-
allel resistive circuit.
12.50 Current in the circuit and terminal voltage of battery under load conditions
and power dissipated in R and r.
12.51 Equivalent resistance, currents, voltages and power dissipated in a series-
parallel circuit.
12.52, 12.53, 12.54 Application of Kirchhoff’s rules to the circuit to produce three
equations with three unknown branch currents.
12.55 Application of Kirchhoff’s rules to the given circuit.
12.56 Application of Kirchhoff’s rules to the circuit to calculate currents in various
branches.
Chapter 13
Electromagnetism I
13.1 Cyclotron frequency for alpha particles.
13.2 Energy and oscillator frequency for p and α
13.3 Mass spectrometer, velocity filter.
13.4 Identification of pion.
13.5 A particle of mass m and charge q travelling along x-axis is acted by E along
y-axis. The trajectory.
Problem Index 795
13.6 The radius of a circular orbit of an electron of K = 5keVin B = 0.01 T.
13.7 Crossed E and B fields.
13.8 v and T for electron moving with R = 1.9min B = 3 × 10
−5
wb/m
2
.
13.9 B and K for D in a cyclotron with V = 5 × 10
4
V, f = 5MC/r and d =
1.524 m.
13.10 f = 11.5MC/s, R = 30
for D. K
max
and f for p.
13.11 Given B = 15,000 G and R = 50 cm, f and K for D.
13.12 Current from charge and time.
13.13 Condition for no deflection in E and B fields.
13.14 Radius of curvature of a particle of mass m and charge q which enters a
magnetic field.
13.15 Separation of isotopes of uranium.
13.16 (a) Radius of curvature in a magnetic field and (b) emf produced in a time-
varying B.
13.17 (a) No. of electrons in charge q and (b) E
min
to prevent a droplet from falling.
13.18 Time period and pitch in a magnetic field.
13.19 A change −q released from the plate of a capacitor in which E and B fields
are set up.
13.20 In prob. (13.19) condition that the electrons are able t o reach the positive
plate
13.21 Current in a long wire deduced from observed B.
13.22 B for a wire of finite length. Limiting case for infinite wire.
13.23 B at the centre of a conducting circular wire.
13.24 B at the centre of square conducting loop of side a with current i.
13.25 Two semicircular wires of resistance R and 4R are joined. To find B at the
centre of the circle.
13.26 B at the centre of three-fourths of a conducting circular wire.
13.27 (a) B from a hair pin conducting wire and (b) B at the centre of a semicircular
wire.
13.28 B at the centre of a conducting wire in the form of a polygon.
13.29 Ratio of B at the centres of a conducting wire in the form of a circle and
square of the same length.
13.30 (a) B at the common centre of circular arcs of a circuit and (b) B at the
common centre of semicircular arcs of radii R
1
and R
2
.
13.31 B at the centre of a circular conducting wire plus straight portion as in the
diagram.
13.32 B on the axis of a circular ring carrying current.
13.33 B inside a 1 m long tube wound by 500 turns of wire carrying 5 A.
13.34 B between parallel current-carrying wires at distance x from one of the wires.
13.35 B at p in a hollow copper cylinder with radii a and b (a < R < b) carrying
current I.
13.36 Magnetic field midway between Helmholtz coils.
13.37 B at the centre of a charged rotating disc
13.38 In prob. (13.36) the magnetic field is fairly uniform.
13.39 B due to cylinder carrying current of 100 A at R = 1.0 m and R = 6 mm.
796 Problem Index
13.40 Application of ampere’s l aw to find B due to a current-carrying cylinder for
r < R and r > R.
13.41 To find B at the centre of two concentric arcs of radii r and 2r as in the
diagram.
13.42 (a) To find B distance x from the midpoint of wire of length L. (b) To com-
pare B at the centre of a loop when it is bent into (i) a square and (ii) an
equilateral triangle.
13.43 To calculate B midway between Helmholtz coils, given the values of N , R
and I.
13.44 B(r) for the current-carrying long cylinder.
13.45 (a) L of a solenoid; (b) magnetic energy; and (c) application of Faraday’s
law.
13.46 (a) B at a given distance from a long straight wire carrying current and (b)
dB at the given values of x, y and z from Idl at the origin.
13.47 B and H for a torus with and without magnetic material.
13.48 Current required to circulate the earth’s core to produce the known dipolar
magnetic field.
13.49 Emf generated by a revolving disc midway between Helmholtz coils.
13.50 Variation of voltage developed across a conductor moving with velocity v in
a known field B.
13.51 A proton travelling with velocity v = (
ˆ
i +3
ˆ
j)10
4
m/s is located at x = 2m
and y = 3 m at some instant t. To find B.
13.52 To find F/l between two long straight wires separated by distance d carrying
i
1
and i
2
in opposite directions.
13.53 Two parallel wires of distance d apart attract each other with a force F/l.If
i
1
is given, to find i
2
and its direction.
13.54 Equilibrium between two current-carrying parallel wires separated by d
vertically.
13.55 Three long parallel wires carrying 20 A are placed in the same plane with
equal spacing of 10 cm. To find F/l for an outer wire and the central wire.
13.56 To calculate the force acting on a bent wire in a uniform magnetic field as in
the diagram.
13.57 A rectangular coil of given dimensions is placed parallel to a long wire car-
rying current. To find force on each segment of rectangular coil and net force
on it.
13.58 In a set-up similar to prob. (13.57) to derive an expression for the resultant
force on the coil and find its value from the given data.
13.59 A loop is formed by two parallel rails, a resistor and a rod across the rails in
a magnetic field. A force F drags the r od at velocity v. To find the current,
total power and F.
13.60 In prob. (13.37) to calculate the magnetic moment of the disc.
13.61 To show that the magnetic dipole moment of the earth can be produced by
wire carrying a current of 5 × 10
7
A around the magnetic equator.
13.62 Energy density at the centre of a current-carrying loop.
13.63 Magnetic energy density at the centre of H atom due to circulating electron.
13.64 Maximum torque of a circular coil in a magnetic field.
Problem Index 797
13.65 The ratio μ/L for a charged sphere rotating with constant angular velocity.
13.66 Potential energy of an electric dipole is an electric field.
13.67 Emf induced in an expanding flexible circular wire placed in a magnetic field.
13.68 Emf developed between the wing tips of an aeroplane flying over earth’s
magnetic field.
13.69 Potential difference between the centre and the outer edge of a spinning disc
in the horizontal component of earth’s magnetic field.
13.70 A coil in the given magnetic field is suddenly withdrawn from the field and
a galvanometer in series with the coil records the charge passed around the
circuit. To find the resistance of the coil and the galvanometer.
13.71 The induced emf and current in a wire loop when the magnetic field is
reduced to zero in the given time.
13.72 Amplitude of the induced current when a square loop of wire rotates in a
magnetic field.
13.73 A bar slides on parallel rods in a magnetic field when a current flows through
the resistor. To find the speed of the bar.
13.74 A uniform magnetic field of induction fills a cylindrical volume. To calculate
the emf produced at the end of a rod placed in it when B changes
13.75 A square wire of length l,massm and resistance R slides on frictionless
inclined rails. Magnetic field exists within the frame. To show that the wire
frame acquires steady velocity.
13.76 A copper disc spins in a magnetic field and induced emf is recorded. To
find B.
13.77 To verify that Faraday’s law is dimensionally correct.
13.78 Given the equation of B waves, to find emf in a coil.
13.79 Ratio of ξ
max
(television)/ξ
max
(radio) in a loop antenna.
13.80 Two rails are connected by a wire and are connected to a wire and a slider to
form a loop. To find F on the slider to move it with velocity v.
13.81 Application of Faraday’s law.
13.82 A wire carrying current is placed across a rectangular coil. To obtain the
magnetic flux and emf induced.
13.83 To obtain magnetic flux in a betatron.
13.84 The magnetic and direction of the Hall field and concentration of free
electrons.
13.85 Mobility of the electron given Hall coefficient and electrical conductivity.
Chapter 14
Electromagnetism II
14.1 Current, P.D and energy in an RC circuit.
14.2 P.D and phase difference in an LR circuit.
14.3 Given current and energy for an inductor, to find its value.
14.4 To find frequency and wavelength for an LC circuit.
14.5 Impedance and power dissipated in an RLC circuit.
798 Problem Index
14.6 L and C have equal X,at f = 600 Hz, to find X
C
/X
L
at 60 Hz.
14.7 A capacitance has X
C
= 4 at 250 Hz. To find C and X
C
at 100 Hz and at
220 V, 50 Hz line.
14.8 In an LR series circuit across a 12 V–50 Hz supply, i = 0.05 A flows with
φ = 60
◦
with V . To find R and L, and to find C when connected in series
to produce to φ = 0.
14.9 i
rms
and i
m
when 0.6 H inductor is connected to 220 V–50 Hz AC line.
14.10 i
rms
, Joule heat, V
rms
in RL circuit for each component when f = 50 Hz
and i
rms
are available.
14.11 An ac applied to R = 100 given V = 0.5 V
m
at t = 1/300 s. To find f .
14.12 To write an equation for the given LRC parallel circuit similar to that for
standard equation.
14.13 To verify the formula for velocity of light.
14.14 Quantities RC, L/R and
√
LC have units of time.
14.15 Fractional decrement of the resonance frequency in an RLC circuit.
14.16 Current in a damped LC circuit for low damping.
14.17 RLC circuit in series.
14.18 RLC circuit in parallel.
14.19 Differential equation for the charge of the given RLC circuit.
14.20 Solution of differential equation in prob. (14.19)
14.21 X
L
, X
C
, Z, I
T
, φ, C
R
, V
C
, V
L
and f
0
for the given RLC circuit.
14.22 A40 resistor and 50 μF capacitor in s eries and an AC of 5 V–300 Hz
current in the circuit.
14.23 Z for the two networks involving L, R, C.
14.24 Definition of electric current, current density and quantization of charge,
drift speed.
14.25 f
0
for two LC circuits in series is identical with individual circuits if
L
1
C
1
= L
2
C
2
.
14.26 Values of L and C if X
C
at f
1
and X
L
at f
2
are given.
14.27 A condenser of 0.01 μF is charged to 100 V. To find i
m
when condenser is
connected to an inductor of L = 10 mH.
14.28 An inductance of 1 mH has resistance of 5 . Value of R and C to yield
f
0
= 500 kHz in series and Q = 150.
14.29 f
0
and Q of a parallel RLC circuit with L = 1 mH, R = 10 and C =
0.005 μf.
14.30 In the discharge of a capacitor in an RC circuit to find error on V (t),given
error on R and C.
14.31 (a) Time for voltage on a condenser to fall to 1/e of initial value through a
resister and (b) percentage error introduced if thermal effects are ignored.
14.32 Fractional half-width of resonance curve of an RLC circuit.
14.33 Dielectric constant from a plane em wave equation.
14.34 Speed of a Lorentz frame in which a pure magnetic field is observed.
14.35 Wave equation for B wave.
14.36 To show that ∇
j +
1
ε
0
∂ E
∂ t
= 0.
Problem Index 799
14.37 Prediction that em waves propagate with velocity c.
14.38 Depth to which an em wave penetrates in aluminium.
14.39 To show that E
0
= cB
0
14.40 em wave equation for a conduction medium.
14.41 Capacitance and inductance per unit length for a coaxial cylinder.
14.42 E, B and S for a coaxial cable in the region between the central wire and
tube.
14.43 To obtain the equation u
B
= B
2
/2μ
0
14.44 In em field u
B
= u
E
.
14.45 To obtain an expression for R if u
B
= u
E
in a coaxial cable.
14.46 Radiation energy loss of a low energy proton in a cyclotron.
14.47 Amplitude of an E wave at a distance from a point source.
14.48 Intensity, E
0
and B
0
of a laser beam.
14.49 Calculation of irradiance of a laser beam.
14.50 Time-averaged power per unit area carried by a plane em wave.
14.51 Irradiance of a plane em wave.
14.52 E × H is in the propagation direction.
14.53 Given the E field equation to find f , λ, v, E
0
and polarization.
14.54 Equation for the B field associated with the E wave of prob. (14.53).
14.55 Speed of an approaching car by radar.
14.56 Beat frequency registered by a radar from a receding car.
14.57 Application of ampere’s law of a coaxial cylinder to calculate magnetic field
in four regions.
14.58 Energy stored in the magnetic field in the large Hadron collider’s magnet.
14.59 Electric and magnetic field amplitude at the surface of the sun.
14.60 Amplitude of the electric field at the orbit of the earth.
14.61 Amplitude of B field at the surface of Mars and flux of radiation.
14.62 To show that |E|/|H|=377 .
14.63 Skin depth in copper.
14.64 Penetration of microwave in a copper screen.
14.65 Derivation of formula for skin depth.
14.66 Energy transported per square centimetre area for a light wave having E
m
=
10
−3
V/m.
14.67 Proof and interpretation of Poynting’s theorem.
14.68 Dispersion relation from Maxwell’s equations in dielectric.
14.69 To show that the identity ∇×(∇×E) =−∇
2
E +∇(∇·E) is true for the
vector field F = x
2
z
3
ˆ
i.
14.70 Use of Poynting vector to determine power flow in a coaxial cable.
14.71 Application of Poynting’s theorem to show that power dissipated in a con-
ducting wire is given by i
2
R.
14.72 Using Maxwell’s equations to show that the equation for a super-conductor
leads to the stated equation for B.
14.73 Ratio of high-frequency resistance to direct resistance.
14.74 Derivation of the expression curl E =−∂ B/∂t using Stokes’ theorem.
14.75 Derivation of the expression B = μ
0
(H + M).
800 Problem Index
14.76 Gauss and Ampere’s laws in free space subject to the Lorentz condition.
14.77 To show E
y
= Z
0
H
x
for propagation of em wave where Z
0
=
√
μ
0
/ε
0
14.78 Given E wave’s direction of propagation and polarization of the wave.
Boundary condition for media of different magnetic properties.
14.79 Use of boundary conditions to calculate the reflectance and transmittance
of em waves at the dielectric discontinuity.
14.80 To obtain an expression for r eflectance for an em wave in terms of angle of
incidence and refraction and the refraction indices of two dielectrics.
14.81 Using the result in prob. (14.80), to show that reflectance is zero when tan
θ = n
2
/n
1
.
14.82 To sketch R and T against n
1
/n
2
for normal incidence.
14.83 (a) B is perpendicular to E; (b) B is in phase with E; and (c) B = E/c.
14.84 Charge density induced on the surface of uncharged dielectric cube contain-
ing electric field.
14.85 Fractional difference between phase and group velocity at λ = 5000 Å.
14.86 Given the dispersion relation ω = ak
2
, to calculate v
ph
and v
g
.
14.87 To show that v
p
v
g
= c
2
.
14.88 v
g
= v
ph
+ kdv
p
/dk.
14.89 v
g
= c/n +
λc/n
2
dn/dλ.
14.90 If v
ph
∝ 1/λ then v
g
= 2v
ph
.
14.91 v
g
= c/
n + ω(∂n/∂ω)
.
14.92 Value of v
p
and free space wavelength for radiation to traverse a length of a
rectangular waveguide in given time.
14.93 1/v
g
= 1/v
ph
+ (ω/c) dn/dω.
14.94 v
g
= cdv/d(nv).
14.95 v
g
= v for a non-relativistic classical particle.
14.96 Given a relation between refractive index and λ, to calculate v
g
.
14.97 v
ph
, v
g
and λ for rectangular wave guide of given dimensions.
14.98 In a rectangular guide of width a = 3cm,valueofλ if λ
g
= 3λ.
14.99 To calculate λ
g
and λ
c
given a and λ
14.100 Number of states of em radiation between 5000 and 6000 Å in a cube of
side 0.5 m.
14.101 Possibility of AM radio waves propagating in a tunnel of given dimensions.
14.102 Calculation of least cut-off frequency for TE
mn
wave for guide of given
dimension
14.103 Variation of v
ph
and v
g
of TE
01
wave in a wave guide of given dimensions.
14.104 Given the wave equation for E
z
, to find E
z
and f
c
.
14.105 Given the wave equation for H
z
, to find H
z
and f
c
.
Chapter 15
Optics
15.1 Fraction of light from a point source in a medium escaping across a flat sur-
face.