Video Transcript
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.
