ÿŸ3æ(February 24, 2005)
(1): A certain toroidal solenoid has a rectangular cross section, as show in
Fig-1. It has N uniformly spaced turns, with air inside. (a) Use Ampere’s
oN I
. (b)
law show that the magnetic field inside the solenoid is B = µ2πr
Show that the magnetic flux through the cross section of the toroid is
b
µo NIh
ln( )
2π
a
(c)Show that the self inductance of the toroidal solenoid is
Φ=
L=
b
µo N 2 h
ln( )
2π
a
(d) The fraction b/a may be written as ab = a+b−a
= 1 + b−a
. The power
a
a
series expansion for ln(1 + x) is ln(1 + x) = x + x2 /2 + · · ·. Hence show
that when b−a is much less than a, the self inductance i s approximately
equal to
µo N 2 h(b − a)
L=
2πa
(2): A coaxial cable consists of a small solid conductor of radius ra supported
by insulating disks on the axis of a thin walled tube of inner radius rb .
Show that the self inductance of a length of the cable is
rb
µo
L = ln( )
2π
ra
Assume the inner and outer conductors carry equal currents in opposite
directions.
(3): An inductor with an inductance of 15H and resistance of 30Ω is connected across a 100-V battery. (a)What is the initial rate of increase of
current in the circuit? (b)At what rate is current change at t = 1.5s?
(c)How long will it take the current to reach 50% of its final value?
(d)How long will it take the current to reach 90% of its final value?
(Ans:(a)6.67 A/s (b)0.33 A/s (c)0.347 s (d)1.151 s)
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(4): In Fig-2 let V = 200 V, R = 500 Ω, and L = 0.1 H. With switch S2
open, switch S1 is closed and left until a constant current is established.
The S2 is closed and S1 opened, so the battery is take out of the circuit.
(a)What is the initial current in the resistor?(0.4A)
(b)What is the current in the resistor at t = 0.2 × 10−3 s? (0.147A)
(c)What is the potential difference between points b and c at t = 0.2 ×
10−3 s? Which point is at higher potential? (Vcb = −73.6V , b is at
higher potential)
(d)How long does it takes the current to decrease to half its initial value?
(Ans:1.39 × 10−4 s)
(e)What is the total energy initially stored in the inductor? (0.008J)
(f)At t = 0.2 × 10−3 s, what rate is the energy stored in the inductor
decrease? (Ans:-10.83J/s)
(g)At t = 0.2 × 10−3 s, what rate is the electrical energy being convert
into heat in the resistor?(Ans:+10.83J/s)
(h)Obtain an expression for the rate at which electrical energy is being
covert in the resistor, as a function of time. Integrate this expression
from t = 0 to t = ∞ to obtain the total electrical energy dissipated in
the resistor. Compare your result to that of part (e). (Ans:0.008 J)
(5): An inductor of resistance R and self inductance L is connected series with
a resistor of resistance Ro to a constant potential difference V (see Fig-3)
(a)Find the expression for the potential difference vcb across the inductor
o
RV
Ro V − R+R
at any time t after switch S1 is closed. (Ans: R+R
+ R+R
e L t)
o
o
(b)Let V = 20 V, Ro = 50 Ω, R = 150 Ω, L = 5 H. Compute vac
and vcb for t = 0, 0.5τ, τ, 1.5τ and 2.5τ , where τ is the time constant of the circuit. Also calculate vac and vcb for t → ∞. (vcb (0) =
20V, vcb(0.5τ ) = 18.03V, vcb (τ ) = 16.84V, vcb(1.5τ ) = 16.12V, vcb(2.5τ ) =
15.41V, vcb(∞) = 15V )
After the current in the Fig-3 has reach its final steady value, the switch
S2 is closed, thus short-circuiting the inductor. (Switch S1 remain closed.)
(c)Derive an expression for the currents through R, Ro , and S2 as funcR
R
V
V
tions of time. Ans:(V /Ro , R+R
e− L t , RVo − R+R
e− L t )
o
o
(d)What will be the magnitude and the direction of the current in S2 ,
0.01 s after S2 is closed? (Ans:0.326A)
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(6): At t = 0, the switch in Fig-4 is closed. By using Kirchhoff’s law, show
that the current through the inductor is given by
I(t) =
ε
[1 − e−(R /L)t ]
R1
where R = R1 R2 /(R1 + R2 ).
(7): If the switch S shown in the Fig-5 is close at t = 0, Find (a) the initial(t =
0+ ) rate of change of the current in the inductor. (ε/3L) (b) the steadystate (t −→ ∞) current in the inductor. (ε/2R) (c) find the current in
−2R
ε
)(1 − e 3L t)
the emf after switch had closed for a time t. Ans:( 2R
(8): The battery in the circuit shown in the Fig-6 has an emf ε = 24 V .
(a)What current is being delivered by the battery 1 ms after the switch
is closed? (b) determine the potential difference across the 5 − Ω resistor
3 ms after the switch is closed. (Neglect the mutual inductance of the
coils.) (Ans:(a)2.36 A,(b)7.77 V )
(9): An L − C series circuit contains an 82 mH inductor and a 17 µF capacitor, which initially carries a charge 180 µC. The switch i s closed at
t = 0. (a) Find the frequency of resulting oscillations. At t = 1.0 ms,
find (b) the charge on the capacitor and (c) the current in the circuit.
(Ans:(a)134.8Hz(b)−1.04 × 10−4C (c)0.116A)
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