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PHYS 196 Home Work 13
1. How many turns of wire would be required to make a 130-mH inductance out of a 30.0-cm long airfilled coil with a diameter of 4.2cm, assuming the length can be considered very large in comparison
with the diameter?
2. You are given a length  of wire that has radius a and are told to wind it into an inductor in the shape of
a helix that has a circular cross-section of radius r . The windings are to be as close to each other as
possible without overlapping. Show that the self-inductance of this coil is L = µ 0 r (4a ).
3. A toroid of mean radius 25cm and circular cross-section of radius 2cm is wound with a superconducting
wire of length 1000m that carries a current of 400A.
(a) What is the number of turns in the coil?
(b) What is the magnetic field at the mean radius?
(c) Assuming B is constant over the area of the coil, calculate the magnetic energy density and the
total energy stored in the toroid
(d) Calculate the self inductance of the toroid
4. Show that the inductance of a toroid of rectangular cross-section as shown is given by
µ N 2 H ln(b a )
L= 0
2π
5. A coaxial cable consists of two thin-walled conducting cylinders of radii a and b . The currents on the
inner and outer cylinders are equal in magnitude but opposite in directions. Calculate the flux through a
rectangular area of length  and b − a between the conductors as shown and hence find the self
inductance per unit length.
2b
2a

6. When the current in an 8.00-H coil is equal to 3.00A and is increasing at 200 A/s, find (a) the magnetic
flux through the coil, and (b) the induced emf in the coil.
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7. The current through a coil with inductance 8.0mH drops from 4.0A to zero in 5.0µs. What is the
magnitude and direction (relative to the current) of the average induced emf?
8. A current of 4.0A runs through a coil with resistance 20mΩ and inductance 8.0mH. The current is
decaying at a rate of 12A/s. Find the potential difference ΔV = Vb − Va between the inlet a and the outlet
b of the current.
9. A coil that has self-inductance of 2.00-H and a resistance of 12.0-Ω is connected to an ideal 24.0-V
battery. (a) What is the steady current? (b) How much energy is stored in the inductor when the steadystate current is established?
10. A 200-turn solenoid has a cross-sectional area equal to 4.0cm2 and a length equal to 30 cm. The solenoid
carries a current of 4.0A. (a) Calculate the magnetic energy stored in the solenoid using
U = LI 2 2 where L = µ 0 n 2 A . (b) Divide your answer in (a) by the volume of the region inside the
solenoid to find the magnetic energy per unit volume in the solenoid. (c) Check your result in (b) by
computing the magnetic field energy density using u m = B 2 (2µ 0 ) using B = µ 0 nI
11. Typical large values for electric and magnetic fields attained in laboratories are 1.0 × 10 4 V / m and 2.0T .
(a) Determine the energy density for each field and compare. (b) What magnitude of electric field would
be needed to produce the same energy density as the 2.0T magnetic field?
12. At t=0, an emf of 500 V is applied to a coil that has an inductance of 0.800 H and a resistance of 30.0Ω.
(a) Find the energy stored in the magnetic field when the current reaches half its maximum value
(b) After the emf is connected, how long does it take the current to reach this value?
13. In the circuit shown, the throw of the make-before-break switch has been at contact a for a long time
and the current in the 1.00mH coil is equal to 2.00A. At t = 0 the throw is quickly moved to contact b .
The total resistance R + r of the coil and resistor is 10.0Ω. Find the current when t = 0.200ms
14. A circuit consists of a 4.00-mH inductor, a 150-Ω resistor, a 12.0-V ideal battery, and an open switch,
all connected in series. After the switch is closed: (a) What is the initial rate of increase of the current?
(b) What is the rate of increase of the current when the current is equal to half of its steady state value?
(c) How long does it take for the current to reach 99% of its steady-state value?
15. (a) What is the period of oscillations of an LC circuit consisting of a 2.0mH coil and a 20-nF capacitor?
(b) A circuit that oscillates consists solely of an 80-µF capacitor and a variable ideal inductor. What
inductance is needed in order to tune this circuit to oscillate at 60 Hz?
16. The variable capacitor in the tuner of an AM radio has a capacitance of 1350 pF when the radio is tuned
to a station at 550 kHz. (a) What must be the capacitance for a station at 1600 kHz? (b) What is the
inductance (assumed constant)?
17. An LC circuit consists of a 3.30-H inductor and an 840-nF capacitor, initially carrying a charge of 105µC.What is the frequency of the oscillation?
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At t=0 the switch is closed. Compute the following quantities at t=2.00ms.
(a) the energy stored in the capacitor
(b) the energy stored in the inductor
(c) the total energy
Calculate the initial stored energy to verify that energy is conserved.
18. A 5.0-µF capacitor is charged to 30V and is then connected across an ideal 10-mH inductor. (a) How
much energy is stored in this system? (b) What is the frequency of oscillation of the circuit? (c) What is
the peak current in the circuit?
19. A 2.44-m long coil containing 225 loops is wound on an iron core (average µ = 1850µ0).along with a
second coil of 115 loops. The loops of each coil have a radius of 2.00cm. If the current in the first coil
drops uniformly from 12.0A to zero in 98.0ms, determine (a) the mutual inductance M (b) the emf
induced in the second coil.
20. Determine the mutual inductance per unit length between two long solenoids, one insude the other,
whose radii are r1 and r2 (r1 < r2 )and whose turns per unit length are n1 and n2 .
Answers:
1. 4733
2.
3. 7937, 2.54T , 5.06kJ , 0.063H
4.
µ
b
5. 0 ln
2π a
6. 24Wb, 1600V
7. 6.4kV same direction as current
8. 16mV
9. 2 A, 4 J
10. 0.536mJ , 4.47 J / m 3 , 4.47 J / m 3
11. 0.44mJ / m 3, 1.59MJ / m 3, 6 !108 V / m
12. 27.9 J , 0.019s
13. 0.27A
14. 3000 A / s, 1500 A / s, 0.123ms
15. 40 µs, 88mH
16. 160 pF, 62µ H
17. 0.86 mJ , 5.74mJ , 6.6mJ
18. 2.25mJ , 712Hz, 0.67 A
19. 0.030H , 3.7V
20. µ 0πn1 n2 r22
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