30.7. Identify: 
and 
Set Up: 
Execute: (a) 
(b) The average flux through each turn is 
Evaluate: The self-induced emf depends on the rate of change of flux and
therefore on the rate of change of the current, not on the value of the
current.
30.9. Identify and Set Up: Apply 
Apply Lenz’s law to
determine the direction of the induced emf in the coil.
Execute: (a) 
(b) Terminal 
is at a higher potential since the coil pushes current
through from 
to and if replaced by a battery it would have the 
terminal at 
Evaluate: The induced emf is directed so as to oppose the decrease in the
current.
30.11. Identify and Set Up: Use Eq.(30.6) to relate L to the flux through each turn of the
solenoid. Use Eq.(28.23) for the magnetic field through the solenoid.
Execute: 
If the magnetic field is
uniform inside the solenoid 
From
Eq.(28.23), 
Then 
Evaluate: Our result is the same as L for a torodial solenoid calculated in
Example 30.3, except that the average circumference 
of the toroid is
replaced by the length l of the straight solenoid.
30.13. Identify and Set Up: Use
Eq.(30.9) to relate the energy stored to the inductance. Example 30.3 gives the
inductance of a toroidal solenoid to be 
so once we know L we
can solve for N.
Execute: 
Evaluate: L and hence U increase according to the
30.15. Identify: A current-carrying inductor has a magnetic field inside of itself and hence stores magnetic energy.
(a) Set Up: The magnetic field inside a solenoid is 
Execute: 
(b) Set Up: The energy density in a magnetic field is 
Execute: 
(c) Set Up: The total stored energy is U = uV.
Execute: 
(d) Set Up: The energy stored in an inductor is 
Execute: Solving for L and putting in the numbers gives
Evaluate: An inductor stores its energy in the magnetic field inside of it.
30.19. Identify: Apply Kirchhoff’s loop rule to the circuit. i(t) is given by Eq.(30.14).
Set Up: The circuit is sketched in Figure 30.19.
|
|
increases from its initial value of zero. |
|
Figure 30.19 |
|
Execute: 
(a) Initially (t
= 0), i = 0 so 
(b) 
(Use this equation rather than Eq.(30.15)
since i rather than t is given.)
Thus 
(c) 
(d) Final steady state means 
Evaluate: Our results agree
with Fig.30.12 in the textbook. The current is initially zero and increases to
its final value of 
The slope of the
current in the figure, which is di/dt, decreases with t.
30.21. Identify: 
with 
The energy stored in
the inductor is 
Set Up: The maximum current occurs after a long time and is equal to
Execute: (a) 
so 
when 
and 
(b) 
so 
Evaluate: 
The time in part (a) is 
and the time in part (b) is 