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Wize University Physics Textbook (Master) > Magnetic Induction

Inductance

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Mutual Inductance


In this section, we describe how systems can induce changes in each other by electromagnetic induction.
  • Consider two sets of coils that are beside each other (we'll call them set 1 and set 2).
  • If the current in set 1 of coils changes, this will induce a magnetic field from set 1.
  • This new magnetic field is detected in set 2.
  • An induced current appears in set 2, reacting to the change in magnetic flux.
  • Because the two sets of coils affect each other, we say that the system has a mutual inductance (M):
ε1=−MΔI2Δt         ε2=−MΔI1Δt\boxed{\varepsilon_1=-M\frac{\Delta I_2}{\Delta t}~~~~~~~~~\varepsilon_2=-M\frac{\Delta I_1}{\Delta t}}
  • Mutual inductance is measured in SI units of Henry (H).
Wize Tip
The negative sign in these formulas is a statement of Lenz's Law.

Wize Concept
A high mutual inductance means the two sets of coils react strongly in response to changes in the other set, or that the system is strongly coupled.

Likewise, a low mutual inductance means the sets do not react as strongly, or the system is weakly coupled.

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Example: Mutual Inductance


Consider two sets of coils that are near each other. In one set of coils, a change in current from zero to 6.0 A in two seconds results in a detected voltage of 0.5 V in the other set of coils.

a) What is the mutual inductance of this system?
b) Qualitatively, what do you think would happen to the mutual inductance if the two systems were brought much closer together?

Part a)

We simply plug the values into the mutual equation formula:
ε1=−MΔI2ΔtM=∣ε1ΔtΔI2∣M=∣(0.5V)(2.0s)(6.0A)∣M=0.167H\begin{aligned} \varepsilon_1&=-M\frac{\Delta I_2}{\Delta t} \\ M&=\bigg |\varepsilon_1\frac{\Delta t}{\Delta I_2} \bigg | \\ M&=\bigg |(0.5V)\frac{(2.0s)}{(6.0A)} \bigg | \\ M&= 0.167 H\\ \end{aligned}
Part b)

The magnetic field that reaches the second set of coils from the first set of coils would realistically be stronger now that the systems are much closer together. As a result, we might also expect a stronger reaction than 0.5 V from the second set of coils as a result of this current change.

In other words, the mutual inductance of the system has increased.
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Self-Inductance and Inductors


It turns out that having materials that can resist changes to magnetic flux have widespread applications in science and engineering.
  • Consider one circular set of coils.
  • If the current through the coils changes, an induced magnetic field will appear.
  • By Lenz's Law, the coils themselves will produce a new magnetic field in the opposite direction.
  • This results in resisting the change in current, and producing a back emf.
  • The relationship between the change in current and emf produced depends on the self-inductance (L) of the system:
ε=−LΔIΔt\boxed{\varepsilon=-L\frac{\Delta I}{\Delta t}}
  • Self-inductance (L) is measured in SI units of Henry (H).
  • Devices that are designed to resist changes in circuits are called inductors.
Wize Tip
In most cases, self-inductance is simply called inductance.

Wize Concept
Systems with high self-inductance produce very strong back emfs and have a stronger resistance to changes in current.

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Inductance of a device
  • The inductance of a device depends only on geometric and material properties (e.g. shape, size, composition).
  • Let's apply Faraday's Law to the inductor equation:
−NΔΦmΔt=−LΔIΔtNΔΦm=LΔIL=NΔΦmΔI\begin{aligned} -N\frac{\Delta \Phi_m}{\Delta t}&=-L\frac{\Delta I}{\Delta t}\\ N\Delta \Phi_m&=L\Delta I \end{aligned}\\ \boxed{L=N\frac{\Delta \Phi_m}{\Delta I}}
  • This can also be written as LΔI=NΔΦL \Delta I = N \Delta \Phi.
Wize Tip
This result tells us that a high inductance (L) means that even for a small change in current, there is a large magnetic flux produced to counter the change - exactly as we would expect.

Practice: Self-Inductance


Consider a system that is safely designed to carry high currents. It carries 300 A of current and has an inductance of 0.30 H.

a) If we were to abruptly shut off this current (over a time of one millisecond), what would be the induced emf through the system?
b) What is the fastest we could shut off the current to prevent the induced emf from exceeding 100 V?

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Example: Inductance of a Solenoid


Recall that the magnetic field through a long solenoid is given by B=μoNIl=μonIB=\frac{\mu_oNI}{l}=\mu_o nI.

What is the inductance of a circular solenoid of radius rr and NN turns of wire?

It turns out that solenoids are one of very few circuit arrangements where we can actually find an equation for inductance. This is because solenoids have a constant and uniform magnetic field that is easy to work with.

We start with the relationship between magnetic flux, current, and inductance.
L=NΔΦΔIL=\frac{N\Delta \Phi}{\Delta I}
Consider a change in magnetic flux through the solenoid. The flux for a solenoid is given as follows:
Φ=BAΦ=μoNIlπr2\begin{aligned} \Phi&=BA\\ \Phi&=\frac{\mu_o N I}{l}\pi r^2 \end{aligned}
The change in this flux would be caused by a change in the current in the solenoid (all of the other variables are geometric only):
ΔΦ=μoNΔIlπr2\Delta \Phi=\frac{\mu_o N \Delta I}{l}\pi r^2
Plugging this back into the relationship above, we see that the change in current cancels out, and we find an expression for the inductance of a solenoid which depends only on geometric variables (as it should!)
L=NΔΦΔIL=N[μoNΔIlπr2]ΔIL=πμoN2r2l\begin{aligned} L&=\frac{N\Delta \Phi}{\Delta I}\\ L&=\frac{N[\frac{\mu_o N \Delta I}{l}\pi r^2]}{\Delta I} \end{aligned}\\ \boxed{L=\frac{\pi\mu_o N^2 r^2 }{l}}