# HW#13 Inductor and RLC circuits Inductor and RLC circuits

```1. The current in a coil changes from Ii to If in the same direction in time t. If the
average emf induced in the coil is , what is the inductance of the coil?
  L
| I  Ii |
dI
 |  | L f
t
dt
|  | t
L
| If  Ii |
2. A technician wraps wire around a tube of length  having a diameter of D. When the
windings are evenly spread over the full length of the tube, the result is a solenoid
containing N turns of wire.
(a) Find the inductance of this solenoid.
(b) If the current in this solenoid increases at the rate of r, find the self-induced emf in the
solenoid?
(a)
For a solenoid, B   0 nI
  B  NBA   0 nNIA
d
d
d
 B  -  0 nNIA  - 0 nNA I
dt
dt
dt
d
By definition,   -L I
dt
 L   0 nNA
 -
D
1
N
But A   ( ) 2  D 2 and n 

2
4

N
1
 L   0 ( )N( D 2 )  0 N 2 D 2

4
4
(b)  -L
d
d
I  | |L| I|
dt
dt
 |  |  Lr
 | |
 0
4
N2D2r
3. A N turn solenoid has a radius of r and an overall length of  .
(a) What is its inductance? (b) If the solenoid is connected in series with a resistor R and
a battery, what is the time constant of the circuit?
(a)
For a solenoid, B   0 nI
  B  NBA   0 nNIA
d
d
d
 B  -  0 nNIA  - 0 nNA I
dt
dt
dt
d
By definition,   -L I
dt
 L   0 nNA
 -
But A  r 2 and n 
N


N
 L   0 ( )N(r 2 )  0 N 2 r 2


 0
L
(b)   
R

N2r 2
R

 0
R
N2r 2
4. A battery of emf V is connected into a series circuit containing a resistor R and an
inductor L.
(a) In what time interval will the current reach 50.0% of its final value?
(b) In what time interval will the current reach 90.0% of its final value?
L
Solution
V
I
V
(1 - e - t/ )
R

R
L
R
V
when t  
R
1V
(a) We want to find t when I 
2R
1V V
1

 (1 - e - t/ )   1 - e - t/
2R R
2
1
 e - t/ 
2
t
1
   n

2
t
  n2
Final value of I 

 t  n2
L
 t  n2
R
(b) We want to find t when I 

9 V
10 R
9 V V
9
 (1 - e - t/ ) 
 1 - e - t/
10
10 R R
1
 e - t/ 
10
t
1
   n

10
t
  n10

 t  n10
t
L
n10
R
5. Consider the circuit shown in the figure.
V
(a) When the switch is in position a, for what value of R1 will the circuit have a time
constant of ?
(b) What is the current in the inductor at the instant the switch is thrown to position b?
(a)  
L
L
L


R 1R 2
R R 1 // R 2
R1  R 2

L(R 1  R 2 )
R 1R 2
 R 1 R 2  L(R 1  R 2 )
 R 1 (R 2  L)  LR 2
 R1 
LR 2
R 2  L
(b) Current in the inductor at the instant the switch is thrown to position b
 Current in the inductor before the switch is thrown to position b
V

R 1 // R 2


L
R 1R 2
R1  R 2
V(R 1  R 2 )
R 1R 2
6. Consider the circuit in the figure below.
(a) What is the inductive time constant of the circuit?
Ans. =L/R
(b) Calculate the current in the circuit t after the switch is closed.
I

R
-
t
(1  e ) 


R
(1  e
-
Rt
L
)
(c) What is the value of the final steady-state current?
Ans. Final steady-state current = /R
(d) After what time interval does the current reach a factor f of its maximum value?
I

-
t
-
t
(1  e  )  I max (1  e  ) where I max 
R
When I  fI max
-
t
 fI max  I max (1  e )  f  1  e

e
-
t

t

 1 f
t
 n(1  f)

1
)

1- f
1
 t   n(
)
1- f
L
1
t 
n(
)
R
1- f

t
-
-
 n(

R
7. A inductor L and a resistor R are connected with a switch to a battery  as shown in
the figure below.
(a) After the switch is first thrown to a (connecting the battery), what time interval
elapses before the current reaches I?
I
 e
-
Rt
L

R
-
(1  e ) 
 1

t

R
(1  e
-
Rt
L
)
RI

  RI

  RI
Rt
 n(
)

L

Rt

 n(
)
  RI
L
L

 t  n(
)
  RI
R
 e
Rt
L

(b) What is the current in the inductor t after the switch is closed?
I

-

t
-
Rt
L
(1  e )  (1  e )
R
R
(c) Now the switch is quickly thrown from a to b. What time interval elapses before the
current in the inductor falls to I?
I
 e
Rt
L

R
-
t
e 



R
RI

Rt
RI
 n( )
L

Rt


 n( )
L
RI
L

 t  n( )
R
RI
-
e
-
Rt
L
8. An air-core solenoid with N turns length  and has a diameter of D. When the solenoid
carries a current of I, how much energy is stored in its magnetic field?
For a solenoid, B   0 nI
  B  NBA   0 nNIA
d
d
d
 B  -  0 nNIA  - 0 nNA I
dt
dt
dt
d
By definition,   -L I
dt
 L   0 nNA
 -
D
1
But A   ( ) 2  D 2
2
4

N
1
 L   0 ( )N( D 2 )  0 N 2 D 2

4
4
1
Energy stored  LI 2
2
1 
 ( 0 N 2 D 2 )I 2
2 4

 0
8
N2D2I2
9. A battery V, a resistor R, and a inductor L are connected in series. After the current in
the circuit has reached its maximum value, calculate the following.
(a) the power being supplied by the battery (b) the power being delivered to the resistor
(c) the power being delivered to the inductor (d) the energy stored in the magnetic field of
the inductor
(a) Power supplied by the battery = power dissipated by R = V2/R
(b) Power dissipated by R = V2/R
(c) At equilibrium, potential difference across the inductor = 0
 Power delivered to the inductor = 0
(d) Energy stored in the magnetic field =
1 2 1 V 2 LV 2
LI  L( ) 
2
2 R
2R
10. The magnetic field inside a superconducting solenoid is B. The solenoid has an inner
diameter of D and a length of  .
(a) Determine the magnetic energy density in the field.
(b) Determine the energy stored in the magnetic field within the solenoid.
(a) Energy density 
1
B2
2 0
(b) Energy stored  Energy density  Volume
1
D

B2   ( ) 2 
2 0
2

 2 2
B D 
8 0
11. In the circuit of Figure P32.48, the battery emf is , the resistance is R, and the
capacitance is C. The switch S is closed for a long time, and no voltage is measured
across the capacitor. After the switch is opened, the potential difference across the
capacitor reaches a maximum value of V. What is the value of the inductance L?
Figure P32.48
1
Just before the switch is opened, energy stored in inductor  LI 2
2
1 
 L( ) 2
2 R
After switched is closed :
1  2
L( )  Energy stored in inductor  energy stored in capacitor
2 R
When the potential difference across the capacitor is maximum, energy stored in capacitor is maximum
and energy stored in the inductor is 0.

1  2
1

L( )  0  CV 2  L( ) 2  CV 2
2 R
2
R
R
 L  CV 2 ( ) 2

L  (
VR

)2 C
12. Calculate the inductance of an LC circuit that oscillates at frequency f when the
capacitance is C.
0 
1
LC
 0  2 f  2 f 
1
LC
1
 4 2 f 2 
LC
1
 L
2 2
4 f C
13. LC circuit consists of an inductor L and a capacitor C. If the maximum instantaneous
current is I, what is the greatest potential difference across the capacitor?
1
Energy stored in the circuit  LI 2
2
When the potential difference across the capacitor is maximum, energy stored in capacitor is maximum
and energy stored in the inductor is 0.
1
1
 LI 2  0  CV 2  LI 2  CV 2
2
2
1
 L  CV 2  2
I
V
 L  ( )2 C
I
14. The switch in the figure below is connected to position a for a long time interval. At t
= 0, the switch is thrown to position b. After this time, what are the following?
R
L

(a) the frequency of oscillation of the LC circuit. (b) the maximum charge that appears
on the capacitor. (c) the maximum current in the inductor. (d) the total energy the circuit
possesses at t .
(a)

1
LC
  2f  f 

1
 f
2
2 LC
(b) Maximum charge = charge in C just before the switch is thrown to position b
= C
(c) Consivation of energy :
1 Q 2max 1 2
Q 2max
2
 LImax  I max 
2 C
2
LC
Q
C
C
 I max  max or
 
L
LC
LC
(d)
Consivation of energy  Total energy is constant, independent of time.
1 Q 2max 1 (C ) 2
1
U 

 C 2
2 C
2 C
2
15. An LC circuit like the one in the figure below contains an inductor L and a capacitor
C that initially carries a charge Qmax. The switch is open for t &lt; 0 and is then thrown
closed at t = 0.
(a) Find the frequency (in hertz) of the resulting oscillations. (b) Find the charge on the
capacitor at t. (c) Find the current in the circuit at t.
(a)

1
LC
  2f  f 

1
 f
2
2 LC
(b)
Since Q  Q max at t  0 and it is going to decease in the near future t.
 Q  Q max cos  t with  
1
LC
t
 Q max cos
LC
(c)
Q  Q max cos
t 
t
LC
d
d
Q  [Q max cos
dt
dt
t
LC
]  Q max
 -
Q max
LC
d
cos
dt
sin
t
LC
t
LC
```