Video Transcript
In this video, we’re talking about
inductance. This term refers to the way that
current-carrying conductors resist change. In particular, they resist a change
in their own current. We’ll see how this all works in a
bit. But to get started, let’s consider
that inductance basically involves the interaction of two physical properties. It involves magnetic flux,
symbolized Φ sub 𝑚, and it involves current, symbolized capital 𝐼. Now, we know that current is charge
that flows through a closed circuit. And we can remind ourselves of what
magnetic flux means.
If we have some area, and we can
call that area capital 𝐴, and there’s a magnetic field 𝐵 passing uniformly through
that area, then we would say the magnetic flux through this loop of area 𝐴 is equal
to 𝐵 times 𝐴. This shows us that magnetic flux is
a magnetic field spread over some area. Now, it’s worth saying a quick word
about the units of magnetic flux because this will come up in example exercises. Considering how magnetic flux is
defined, we know that the SI base units of magnetic field are teslas, capital 𝑇,
and that the base unit of area is meter squared. So magnetic flux is measured in
teslas meter squared. And the name for that simplified
unit is the weber.
So magnetic flux is measured in
webers, abbreviated this way. And each weber is equal to one
tesla times a meter squared. Knowing this about magnetic flux,
let’s get back to this idea that inductance is basically about these two physical
quantities, magnetic flux and current. Say that we have a loop of wire,
like this, that’s carrying a current 𝐼 clockwise as we see it. We can remember that a
current-carrying wire produces a magnetic field around itself. So in the case of our wire, this
would produce a magnetic field that points into the screen inside this loop of
wire. And if we combine the effects of
all of these magnetic field lines, what we have is some total magnetic field, we
could call it 𝐵, within the area of the cross section of this wire. In other words, we have a magnetic
flux through this loop.
Whatever this value is, if we take
the ratio of that magnetic flux to the current 𝐼 that moves through the loop, then
this is equal to what’s called the loop’s inductance, symbolized capital 𝐿. We mentioned that inductance is a
measure of how much a current-carrying wire resists a change in its current. So the higher 𝐿 — the inductance
of this circuit — is, the harder it is to change the current 𝐼 running through
it. Now, it may seem obvious, but when
we talk about the inductance of this current-carrying wire, we’re talking about how
the current in this wire responds to the magnetic flux created by that current.
All that to say, when we talk about
inductance symbolized this way, we’re really talking about the self-inductance of a
wire, how its current responds to the flux that it creates. We bring all this up because
there’s another kind of inductance, called mutual inductance. This has to do with more than one
current-carrying wire, and we’ll talk about it soon. So far, though, we just need to
know that the inductance of a current-carrying wire describes quantitatively how
much it resists a change in current and that it’s equal to the ratio of the magnetic
flux through a current-carrying loop divided by that current.
Now, talking about magnetic flux
may remind us of the law known as Faraday’s law. This law tells us that if we have a
conducting loop, even if that loop has no current in it, so long as the magnetic
flux Φ sub 𝑚 through that loop is changing in time, then, believe it or not, an
electromotive force, an emf, will be induced in the loop. Now, as far as the rest of this
equation goes, this capital 𝑁 refers to the number of turns in the loop or loops in
a coil. And then this minus sign here just
refers to the direction of induced current movement. So Faraday’s law involves a
magnetic flux, really a change in magnetic flux, and then an induced emf, which
creates a current in a conducting loop.
So we can see some elements in
Faraday’s law that also appear in this equation for inductance. And in fact, we can write an
expression for the emf induced in a conducting loop in terms of the inductance of
that loop. It looks like this. Given a conducting loop with
inductance capital 𝐿, if we change the current in that loop over some amount of
time, then that time rate of change times the loop’s inductance is equal to the emf
induced in it.
Knowing this, we can actually
combine this relationship with Faraday’s law. And as we do, let’s say that we’re
talking about the magnitude of the emf induced. So that way, we can just neglect
this minus sign here. Okay, so if emf is equal to 𝐿
times Δ𝐼 over Δ𝑡 and its magnitude is also equal to the number of turns in a coil
multiplied by the change in magnetic flux through one of those loops divided by the
change in time, then we can equate the right sides of those two equations. And notice that on both sides we
have Δ𝑡 in the denominator, which means that if we multiply both sides by Δ𝑡, then
that time interval cancels out.
So this is another way of
expressing inductance based on these two relationships here. Now, earlier, we said that
self-inductance isn’t the only kind of inductance there is. Mutual inductance, and we saw an
example of this on our opening screen, is another way for induction to happen. It works like this. Say that we have a current in a
conducting loop such that it creates a magnetic field through that loop that looks
this way. Now, say we put a second conducting
loop in a position so that this magnetic field went through it. In that case, there will be a
magnetic field moving through the cross-sectional area of this loop. And that would mean a magnetic flux
in the loop.
Faraday’s law says that a change in
magnetic flux happening over some amount of time induces an emf. And that means that charge would
start to flow in this loop. Now, the change in magnetic flux
we’re talking about could be the initial movement of this conducting loop into the
magnetic field. But of course, once the loop is
there, assuming there’s no change in the magnetic field through it, there’d also be
no change in magnetic flux.
But what if we did this? What if we gradually increased the
magnitude of the current in this loop over here? That would create an increasing
magnetic field, which would lead to an increase in magnetic flux through this
smaller loop. So long as current is changing
then, in this primary loop we could call it, then it will drive the creation of
current in this secondary loop. That induced current direction will
be such that it creates a magnetic field which opposes the change in magnetic flux
experienced by this current loop.
This overall process is referred to
as mutual induction. And rather than symbolizing this
kind of induction with a capital 𝐿, it’s typically a capital 𝑀 that is used. So, for example, in this equation
over here, instead of an 𝐿, we might see an 𝑀. But it would still mean essentially
the same thing, that some inductance, whether self-inductance or mutual, multiplied
by a change in current over a change in time is equal to the induced emf in a
current-carrying loop. The same thing is true for this
equation here, or sometimes instead of an 𝐿, we would see the 𝑀 of mutual
inductance.
It’s worth pointing out that rather
than being a rare occurrence, mutual inductance is the operating principle for
electrical transformers. Say we had a transformer here
consisting of a core and a primary and secondary circuit. If the current in the primary
circuit changes in time, then that will create a change in magnetic field
transmitted by the core through the coils of the secondary circuit. Through mutual induction between
these circuits, emf and then current will be induced in the secondary circuit. Now, whether we’re talking about
mutual or self-inductance, the unit for inductance is the same. It’s the henry, named after Joseph
Henry, and it’s abbreviated capital H. To understand this topic better,
let’s now try out an example exercise.
A transformer with an iron core has
a primary coil with 25 turns and a secondary coil that also has 25 turns. The current in the primary coil
increases the magnetic flux through the core by 0.15 webers per second. The current in the secondary coil
increases at 0.075 amperes per second. What is the mutual inductance of
the coils?
To get started here, let’s draw a
sketch of this transformer with the primary and secondary coils. Okay, so here’s our transformer
core. And this is the primary coil; we’ll
call this the secondary. And even though we can see that
neither of these has 25 turns to it, like our problem statement tells us, we can
pretend they do. The idea with the transformer is
that as current passes through the primary coil and specifically as it changes and
passes through this coil, then this creates a change in magnetic field that the core
directs through the loops of the secondary coil. When the secondary coil experiences
this change in magnetic flux, current is induced in it.
In our scenario, the current in the
primary coil is increasing. We don’t know the rate it’s
increasing by, but we do know that it’s affecting the magnetic flux in the core and
that that flux is increasing at a steady rate, given as 0.15 webers per second,
where a weber is the unit of magnetic flux. Now, just as a reminder, if we have
some area 𝐴 and there’s a magnetic field 𝐵 passing through that area, then we can
say there’s a magnetic flux Φ sub 𝑚 through that loop. And the units of magnetic flux, as
we’ve seen, are webers.
So going back to our transformer,
in the core, this iron material that connects our primary and secondary coils, the
magnetic flux is increasing at this given rate. And that, our problems statement
tells us, drives an increasing current in the secondary coil. We have then a change in current in
one coil, creating a change in magnetic flux, which induces a change in current in
another coil. This means we have mutual
inductance going on. And we want to calculate the value
of that inductance.
To begin to do this, we can recall
Faraday’s law. The reason we’re thinking of this
law is because we have a magnetic flux that changes in time. Faraday’s law tells us that this
change in flux, ΔΦ sub 𝑚, over some change in time, Δ𝑡, multiplied by the number
of turns in whatever coil we’re considering is equal to the magnitude of the emf
induced in that coil. Now, in terms of mutual inductance,
induced emf is equal to something else as well. We can recall that induced emf is
equal to the mutual inductance between two conducting loops multiplied by the time
rate of change of current and what we can call the secondary loop or, in the example
of our transformer, the secondary coil.
So it’s 𝑀, the mutual inductance,
that we want to solve for. And we see that that value
multiplied by Δ𝐼 divided by Δ𝑡 is equal to the emf induced in a secondary coil,
which by Faraday’s law is also equal to negative the number of turns or loops in the
coil multiplied by the change in magnetic flux divided by the change in time. Since we’re given this time rate of
change of magnetic flux, let’s combine these two equations for induced emf by
equating the right sides with one another.
When we do this though, we’ll leave
off this minus sign here because in our case we’re only concerned with the magnitude
of the emf induced. So then we get this. 𝑀, the mutual inductance of the
coils, times Δ𝐼 divided by Δ𝑡, the time rate of change of current in the secondary
coil, is equal to the number of loops in that coil multiplied by the time rate of
change of magnetic flux that each loop in the coil experiences.
What we want to do is isolate 𝑀
since we’re solving for the mutual inductance. And to do that, we’ll divide both
sides of the equation by Δ𝐼 divided by Δ𝑡. When we do, on the right-hand side
of the resulting equation, we have 𝑁 — the number of turns in the secondary coil,
that’s 25 — multiplied by the time rate of change of magnetic flux in the core,
that’s given as 0.15 webers per second, divided by the time rate of change of
current in the secondary coil. This is the induced current, and
its rate of change is 0.075 amperes per second. When we calculate this fraction and
keep two significant figures, we find a result of 50 henrys. That’s the mutual inductance of the
two coils.
Let’s look now at a second example
exercise.
A loop of wire carries a current of
180 milliamperes. The magnetic flux produced by the
current is 0.77 webers. What is the self-inductance of the
loop?
So then, in this example, we have a
loop of wire. Let’s say this is it. And it’s carrying a current of 180
milliamperes. Because of this, a magnetic field
is produced that passes through the loop. Let’s just say the magnetic field
points like this. And because we have this field
passing through the interior area of the loop, that means we have a magnetic flux
through the loop. And that’s the value we have given
to us here in our problem statement, 0.77 webers. We can refer to that symbolically
as Φ sub 𝑚, magnetic flux.
Knowing all this, we want to solve
for the self-inductance of the loop. And we can recall that the
self-inductance of a loop, capital 𝐿, is equal to the ratio of the magnetic flux Φ
sub 𝑚 produced by a current 𝐼 in the loop to that current 𝐼. For our particular loop then, we
have exactly that information about it. Our current 𝐼 is a current in the
loop. And the magnetic flux is produced
thanks to a magnetic field generated by that current. So to solve for the self-inductance
of our loop, we only need to divide Φ sub 𝑚 by 𝐼.
With these values plugged in, the
one change we’ll want to make before dividing is to convert the units of our current
from milliamperes to amperes. Now, since 1000 milliamperes is
equal to one amp, we can say that 180 milliamperes is equal to 0.180 amps. And now we’re ready to divide. And when we do, to two significant
figures, we find a result of 4.3 henrys. That’s the self-inductance of this
loop.
Let’s summarize now what we’ve
learned in this lesson on inductance. Earlier on, we saw that inductance
is the tendency of a current-carrying conductor to resist a change in current. The inductance of a circuit,
sometimes called its self-inductance, is equal to the ratio of the magnetic flux
through the current loop divided by that current. We also learned that Faraday’s law
led us to another relationship describing inductance, that the inductance in a
circuit multiplied by the time rate of change of current in that circuit is equal to
the number of loops or turns in the circuit times the time rate of change of
magnetic flux.
We learned further about mutual
inductance, symbolized using a capital 𝑀, which occurs when a change in current, in
what we could call a primary coil, creates a change in magnetic field, which is then
experienced by a secondary coil where that change in flux through the secondary coil
induces a current. This, we saw, is the basic
principle behind electrical transformers and can be described by saying that the
mutual inductance between two coils or two loops of wire multiplied by the time rate
of change of current in the secondary wire is equal to the emf induced there. This is a summary of
inductance.
