Eddy Currents and Inductance

Eddy Currents

When a bulk piece of conducting material (like a metal plate or core) is subjected to a changing magnetic flux, induced currents are produced within the body of the conductor itself. These currents flow in closed loops within the conductor, resembling swirling eddies in water, hence they are called eddy currents.

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Mechanism of Eddy Currents

The phenomenon of eddy currents is a direct consequence of Faraday's Law of electromagnetic induction. When the magnetic flux linked with a large piece of conductor changes, an EMF is induced in various parts of the conductor. Since the conductor is a continuous conducting medium, these induced EMFs cause circulating currents to flow within the material itself.

The direction of these eddy currents is given by Lenz's Law: they flow in such a direction as to oppose the change in magnetic flux that produces them.

Consider a metallic plate moved into or out of a magnetic field, or placed in a changing magnetic field (e.g., the field of an electromagnet with varying current). As the magnetic flux through different regions of the plate changes, EMFs are induced, driving circulating currents within the plate.

Eddy currents induced in a metal plate moving into a magnetic field.

In the diagram, as the metal plate moves to the right into the magnetic field (directed inwards), the inward magnetic flux through the region entering the field increases. Eddy currents are induced in an anti-clockwise direction (by Lenz's Law) to create an outward magnetic field opposing this increase. Similarly, in the region leaving the field, inward flux decreases, inducing clockwise currents to create an inward field.

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Effects of Eddy Currents

Eddy currents have both undesirable and desirable effects:

Undesirable Effects:

1. Energy Loss (Joule Heating): Eddy currents flow through the resistance of the bulk conductor. This results in the dissipation of electrical energy as heat ($H = I^2Rt$). This heating is undesirable in many applications, such as transformer cores or motor armatures operating in changing magnetic fields, as it leads to energy loss and reduces efficiency.
2. Damping of Motion: According to Lenz's Law, eddy currents oppose the cause that produces them. If the cause is the motion of a conductor in a magnetic field, the induced eddy currents will interact with the magnetic field to produce a force that opposes the motion. This effect is called magnetic damping.

Reducing Eddy Currents: To minimise undesirable eddy currents and the associated power loss and damping, the bulk conducting material is often laminated. This involves slicing the conductor into thin sheets or laminations, insulated from each other by a thin layer of varnish or oxide. The plane of the laminations is kept parallel to the magnetic field lines. This significantly increases the resistance offered to the flow of circulating eddy currents, thereby reducing their magnitude and the consequent heating. Transformer cores and armatures of DC motors are typically laminated.

Laminated core used to reduce eddy currents. Insulating layers between thin sheets break the circulating paths.

Desirable Applications:

1. Magnetic Damping: This effect is utilised in some instruments, like moving coil galvanometers, to bring the coil to rest quickly without oscillations. The coil is wound on a metallic frame, and as it oscillates in the magnetic field, eddy currents are induced in the frame, which oppose the motion and provide damping.
2. Induction Furnaces: High-frequency eddy currents are deliberately induced in metallic objects in an induction furnace to produce very high temperatures for melting metals.
3. Electromagnetic Brakes: Eddy currents can be used to provide braking force in some trains (e.g., Maglev trains) or roller coasters. A conducting plate moves through a magnetic field, inducing eddy currents which oppose the motion, providing a braking effect without mechanical friction.
4. Speedometers: In some older vehicle speedometers, a rotating magnet induces eddy currents in a nearby metal disc, causing it to deflect by an amount proportional to the speed.
5. Induction Cooktops: An alternating magnetic field induces large eddy currents in the metallic base of a cooking pan, generating heat directly within the pan.

Eddy currents, although sometimes a source of energy loss, are a fascinating demonstration of electromagnetic induction and find useful applications when controlled and harnessed appropriately.

Inductance

We have seen that a changing magnetic flux through a coil induces an EMF in the coil (Faraday's Law). The change in magnetic flux can be caused by an external source (like a nearby magnet or another coil with varying current) or by the current flowing through the coil itself changing with time. The property of a coil or circuit that opposes changes in the current flowing through it is called inductance. Inductance is a measure of the coil's ability to store energy in a magnetic field and resist changes in current.

There are two types of inductance: Mutual Inductance and Self-Inductance.

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Mutual Inductance ($ M = \frac{\Phi_2}{I_1} $)

Mutual inductance is the property of a pair of coils or circuits due to which a change in current in one coil (called the primary coil) induces an EMF in the other coil (called the secondary coil).

Mutual inductance between two nearby coils. Change in current in P induces EMF in S.

Consider two coils, a primary coil P and a secondary coil S, placed near each other. Let $I_1$ be the current flowing in the primary coil. This current produces a magnetic field, and some of the magnetic flux lines from coil P pass through coil S. Let $\Phi_{21}$ be the magnetic flux linked with the secondary coil due to the current $I_1$ in the primary coil.

It is found experimentally that the magnetic flux $\Phi_{21}$ linked with the secondary coil is directly proportional to the current $I_1$ in the primary coil:

$ \Phi_{21} \propto I_1 $

Introducing a constant of proportionality, we write:

$ \Phi_{21} = M_{21} I_1 $

Where $M_{21}$ is a constant called the mutual inductance of coil S with respect to coil P. Its value depends on the geometry of the two coils (size, shape, number of turns) and their relative position and orientation, as well as the properties of the medium between them.

If the current in the primary coil changes with time, say at the rate $dI_1/dt$, the magnetic flux linked with the secondary coil also changes. According to Faraday's Law, an EMF ($\mathcal{E}_2$) is induced in the secondary coil:

$ \mathcal{E}_2 = -N_2 \frac{d\Phi_{B2}}{dt} $ (where $\Phi_{B2}$ is flux per turn in S)

If $\Phi_{21}$ is the total flux linked with $N_2$ turns of coil S due to $I_1$ in P, then $\Phi_{21} = N_2 \Phi_{B2}$. So $\mathcal{E}_2 = -\frac{d\Phi_{21}}{dt}$.

Substituting $\Phi_{21} = M_{21} I_1$:

$ \mathcal{E}_2 = -\frac{d}{dt}(M_{21} I_1) $

If $M_{21}$ is constant (which is true for fixed geometry and medium):

$ \mathcal{E}_2 = -M_{21} \frac{dI_1}{dt} $

Similarly, if a current $I_2$ flows in the secondary coil, it produces a magnetic flux $\Phi_{12}$ linked with the primary coil. We can define the mutual inductance of coil P with respect to coil S ($M_{12}$) by $\Phi_{12} = M_{12} I_2$. A change in $I_2$ induces an EMF $\mathcal{E}_1 = -M_{12} \frac{dI_2}{dt}$ in coil P.

A remarkable result is that the mutual inductance is reciprocal: $M_{12} = M_{21}$. We simply denote it by $M$.

Thus, the mutual inductance $M$ between two coils is given by:

$ M = \frac{\Phi_2}{I_1} $ (where $\Phi_2$ is the total flux linked with secondary coil due to $I_1$)

And the induced EMF in the secondary coil due to a changing current in the primary is:

$ \mathcal{E}_2 = -M \frac{dI_1}{dt} $

The SI unit of mutual inductance is the henry (H), named after Joseph Henry. One henry is the mutual inductance if a change of 1 ampere per second in the primary coil induces an EMF of 1 volt in the secondary coil.

$ 1 \text{ Henry} = \frac{1 \text{ Volt}}{1 \text{ Ampere/second}} = 1 \text{ V} \cdot s/A $

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Self-Inductance ($ L = \frac{\Phi}{I} $)

Self-inductance is the property of a single coil or circuit by which it opposes any change in the current flowing through it. This opposition arises because a changing current in the coil itself produces a changing magnetic flux through the same coil, which induces an EMF in the coil. This induced EMF is called the back EMF because, according to Lenz's Law, it always opposes the change in current (the cause of the change in flux).

Consider a coil carrying a current $I$. This current produces a magnetic field, and the magnetic flux lines pass through the coil itself. Let $\Phi_B$ be the magnetic flux linked with each turn of the coil due to the current $I$ in the coil. If the coil has $N$ turns, the total magnetic flux linkage is $N\Phi_B$.

It is found experimentally that the total magnetic flux linkage $N\Phi_B$ is directly proportional to the current $I$ in the coil:

$ N\Phi_B \propto I $

Introducing a constant of proportionality, we write:

$ N\Phi_B = L I $

Where $L$ is a constant called the self-inductance (or simply inductance) of the coil. Its value depends on the geometry of the coil (size, shape, number of turns, turn density) and the properties of the medium within the coil (e.g., air core or ferromagnetic core).

From this definition, the self-inductance $L$ is given by:

$ L = \frac{N\Phi_B}{I} $

where $N\Phi_B$ is the total flux linkage and $I$ is the current causing this flux.

If the current $I$ in the coil changes with time, the magnetic flux linked with the coil also changes. According to Faraday's Law, an EMF ($\mathcal{E}$) is induced in the coil itself:

$ \mathcal{E} = -\frac{d(N\Phi_B)}{dt} $

Substituting $N\Phi_B = LI$:

$ \mathcal{E} = -\frac{d}{dt}(LI) $

If $L$ is constant (for fixed geometry and medium, not saturated ferromagnetic core):

$ \mathcal{E} = -L \frac{dI}{dt} $

This equation gives the self-induced EMF (back EMF) in a coil when the current through it changes at the rate $dI/dt$. The negative sign signifies that the induced EMF opposes the change in current (Lenz's Law). If the current is increasing ($dI/dt > 0$), the induced EMF is negative, opposing the increase. If the current is decreasing ($dI/dt < 0$), the induced EMF is positive, opposing the decrease.

The SI unit of self-inductance is also the henry (H). One henry is the self-inductance of a coil if a change of 1 ampere per second in the current through it induces an EMF of 1 volt in the coil.

Inductors are electrical components designed to have a specific self-inductance. They are often coils of wire and are used in circuits to oppose changes in current, store energy in a magnetic field, and in filtering and tuning applications (like in AC circuits, radio receivers).

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Self-Inductance of a Long Solenoid

Let's calculate the self-inductance of a long air-cored solenoid. Assume the solenoid has $N$ turns, length $L$, and cross-sectional area $A$. The number of turns per unit length is $n = N/L$. If a current $I$ flows through it, the magnetic field inside is approximately uniform and given by $B = \mu_0 n I = \mu_0 \frac{N}{L} I$.

The magnetic flux through each turn is $\Phi_B = BA = \left(\mu_0 \frac{N}{L} I\right) A$.

The total flux linkage with the coil is $N\Phi_B$:

Total flux linkage $= N \left(\mu_0 \frac{N}{L} I A\right) = \frac{\mu_0 N^2 A}{L} I $

Using the definition of self-inductance $LI = N\Phi_B$, we have:

$ L I = \frac{\mu_0 N^2 A}{L} I $

So, the self-inductance of a long solenoid is:

$ L = \frac{\mu_0 N^2 A}{L} $

This can also be written in terms of turns per unit length $n=N/L$, so $N=nL$:

$ L = \frac{\mu_0 (nL)^2 A}{L} = \mu_0 n^2 L A $

This formula shows that the self-inductance of a solenoid depends on its geometry ($N, L, A$) and the permeability of the core material ($\mu_0$ for air/vacuum). If a core of relative permeability $\mu_r$ is inserted, $\mu_0$ is replaced by $\mu = \mu_0 \mu_r$.

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