Week 8: Faraday’s Law and Induction 283
forI(t) that is at least approximately correct, we would then guess:
I(t) =V 0
R
(
1 −e−tτ)
=V 0
R
(
1 −e−RLt)
(611)before starting the problem!
Although perhaps it will be a bit anticlimactic, let’s solve it the more difficult but formally
correct way. We start, as usual, with Kirchoff’s Loop Rule, some arbitrary time after the switch is
closed:
V 0 −IR−LdI
dt= 0 (612)
We rearrange this to put it in the standard form of a first order, linear, inhomogeneous ordinary
differential equation:
dI
dt+
R
LI=
V 0
L (613)
At this point I shouldn’t have to help you. We’ve now solved this equation several times over two
semesters^75 – it is directly integrable after some rearrangement and is clearly an important equation
to be able to effortlessly solve if you want to understand Nature, not only in the context of physics
but in biology and chemistry and medicine as well. If you remember how,stop reading here, get
out a piece of paper, and do so, verifying that you get the solution Ialready deduced above without
using algebra or calculus. Work neatly, as this is a straight up homework problem so your efforts
won’t be wasted.
But what the heck, you’re learning, you’ve forgotten, so I’ll solve ithere again. Butpay attention
this time – reallylearn to recognize this kind of equation and solve it when yousee it! Practice it
a bit, thenwait a dayand try working through this section again, this time solving the FOLIODE
above without looking.
So here we go:
dI
dt+R
L
I = V^0
L
dI
dt=
V 0
L
−
R
L
I
dI
dt= −
R
L
(
I−
V 0
R
)
dI
(
I−VR^0) = −
R
L
dt
∫
dI
(
I−VR^0) =
∫(
−
R
L
)
dtln(
I−
V 0
R
)
= −
(
R
L
)
t+Cexp{
ln((
I−
V 0
R
)}
= exp{
−
(
R
L
)
t+C}
I−
V 0
R = e−(RL)teCI =
V 0
R
+Ae−(RL)tI(t) = V^0
R(
1 −e−(RL)t)
(614)where we’ve used the fact that the natural log and exponential are inverse functions of one another
and where we set the (exponential of) the constant of integration from the indefinite integralsAto
−V 0 /Rin order thatI(0) = 0 (the initial condition, recall).
(^75) Approach to terminal velocity with a linear drag force, approach to a terminal velocity for a rod on rails with a
battery or gravity, charging a capacitor in a DCRCcircuit, for example.