280 Week 8: Faraday’s Law and Induction
Nothing to it! Now suppose thatI(t) =I 0 sin(ωt) (a reasonable assumption for harmonic alter-
native voltages such as those we will shortly study). We can easily find:
∆VL=−LdI
dt=I 0 (ωL) cos(ωt) (609)where the field of the induced voltageopposesthe increasing current during that part of its harmonic
oscillation andreinforcesthe decreasing current during that part of its oscillation. As we indicate
on the figure, ifI, directed into the page at the top of the coils and out at the bottom, is increasing,
then the inducedE-field points out of the page at the top and in at the bottom and the induced
potential decreases right-to-left, opposing the increasing left-to-right current.
This may be tricky for you to see! The direction of the potential difference ultimately depends
onwhich way the coil was wound– if the helix spirals from left to right (in at the top) as drawn then
the net current transport is left to right and the induced voltage from an increasing current decreases
from right to left. If it is wound right to left (in at the top) so that the net current transport is
right to left as well, then the induced voltage for an increasing current will be left to right. It all
makes perfect sense in terms of Lenz’s Law either way – the voltagedecreases in the direction that
opposes the flow of the increasing current either way, and reverses tosupportit if and when the
current decreases instead.
Before we move on, it is indeed worth pointing out thatωLin the expression for ∆VLabove has
units ofresistance(sinceI 0 ωLhas units of volts). Next week we will nameωLinductive reactance
as it will be a very important quantity in AC circuits.
Example 8.7.2: Toroidal Solenoid
N turns
a
b
r
C
h
z
I
dr
dA for flux
Figure 103: A tightly-wrapped toroidal solenoid withNturns produces a magnetic field inside that
varies withr, but is approximately constant everywhere in a narrow strip of heighthand widthdr.
The field is, of course, in the direction determined by the right hand rule, meaning that it pointsin
to the page through the shaded strip we need to use to find the flux.
In figure 103 we see the same toroidal solenoid that we saw in week 7,where we evaluated
the magnetic field inside using Ampere’s Law. We will follow exactly the same rubric as before,
except that this time I won’t actually do the steps for you; they arepart of this week’s homework.
Remember:
a) Evaluate the field (magnitude)B(r) using Ampere’s Law. Only refer back to week 7 if you
must, as by now youshouldbe able to do this on your own without looking!b) Evaluate the flux through a single turn of the toroidal solenoid. This will involve setting up an
integral that is almostexactlythe same as the integral in the example of finding the mutual