Faraday's Law, Inductance, and RL Circuits

Faraday’s Law, Inductance, and RL Circuits
NAME:
NAME:
NAME:
RECITATION SECTION:
INSTRUCTOR:
DATE:
____________________________________
____________________________________
____________________________________
__________________________
__________________________
__________________________
This activity is based on the following concepts:
Faraday's Law of Induction: If the magnetic flux Φ B through a closed loop C changes
with time, an induced electric field E is created. The rate of change of the magnetic flux
and the electric field are related by:
d
B
E ds
dt
C
Self-induction: If a current i is a coil changes with time, an emf is induced in the coil.
o This self-induced emf is
di
L
L ,
dt
where L is the inductance of the coil (inductor).
o Inductance is measured in units of henries (H)
Rules of thumb for RL circuits:
o When you make a change in an RL circuit, initially, an inductor acts to oppose the
change in current. A long time later, it acts like an ordinary piece of wire.
o Circuits with inductors resist any changes in current; so, if you throw a switch, the
current through an inductor cannot change instantaneously (unless, of course, you
break the circuit…)
The current in a series RL circuit is
t
L
Rise of Current i ( 1 e )
R
Decay of Current t
L
i i0
e
o In questions Q5 – Q7, the circuit is not a series RL circuit

Exercise 1:
The figure below shows the cross-sectional view of an ideal solenoid with n turns per unit length
and a radius R 1 . At some instant in time, the current through the coil is increasing at a rate di/dt
and the instantaneous value of the current is i going clockwise as viewed in this picture. A point
charge +Q is located as shown at a distance R 2 from the axis of the solenoid.
i
+Q
R 1
R 2
Your aim is to use Faraday’s Law to determine the instantaneous force experienced by the
charge Q. The following steps are meant to help you work through the necessary reasoning.
Q1. First, determine the instantaneous magnetic field B(r) created by the solenoid. (Note: you
may use Ampere’s Law to derive this, or refer to the result derived in your textbook or lecture
notes). Be sure to write explicitly: what is the magnitude and direction of B(r) for r < R 1 ? what
about r > R 1 ?
Q2. Next, imagine a circular path of radius R 2 concentric with the solenoid and determine the
instantaneous magnetic flux through the area enclosed by this circle. From this, then determine
the instantaneous rate of change of magnetic flux through this area with time.

As was described in the introduction, Faraday’s Law connects the rate of change of the magnetic
flux through a closed loop (which you just found) to the integral of the electric field around that
loop. If you look at the figure again, you’ll notice that (aside from the charge +Q outside the
solenoid), the entire situation looks exactly the same if you rotate the figure about the center of
the solenoid. We say, then, that the problem is rotationally symmetric. We can use this symmetry
to argue that (i) the electric field induced by the changing current in the solenoid must be
oriented tangent to any circular path concentric with the solenoid and (ii) the component of the
electric field along our imaginary circular path must be constant all along that circle.
Q3. So, now that we know that the induced electric field must be tangent to our imaginary circle
of radius R 2 , in which sense is the electric field oriented: clockwise or counter-clockwise? One
way to answer this is to consider what would happen if there were an actual conducting wire
placed along our imaginary loop. Use Lenz’ law to figure out the direction of the current that
would be induced if you did this, and from this, determine the direction of the electric field.
Q4. Given that the magnitude of the electric field must be the same at every point along this
imaginary loop, use the results you have obtained so far to finally determine the instantaneous
force on the point charge Q.

Exercise 2:
In the circuit shown adjacent, V = 60.0 V, R 1 = 10.0 Ω, R 2 = 20.0 Ω, and L = 3 H.
Q5. Immediately after the switch is closed, what
is the value of the current through R 2 ? Make
sure you justify your answer.
R 1
R 2 L
V
+
−
Q6. Immediately after the switch is closed, what is the rate of change of current through L? Also,
specify the direction (up/down) that the current flows through L. Justify your answer.
Q7. A long time after the switch is closed, what is the value of the current through R 2 ? Justify
your answer.
Q8. Once the circuit has reached a steady state (a long time after the switch was initially closed),
the switch is reopened. Immediately after the switch is reopened, what is the value of the current
through R 2 ? After the switch is reopened, how much time does it take for the current in R 2 to
reach approximately 37% of its initial value? Justify your answers.