Physics II: Chapter 10

Reference: Beginning Physics II

Chapter 10: INDUCTANCE



Self-Inductance, Mutual Inductance, Inductor, Energy in an Inductor, Energy Density in Space, Transformers



For details on the following concepts, please consult Chapter 10.

Self-inductance arises from the flux that a current circuit produces within its own area. It is distinguished from mutual inductance. Self-inductance depends only on the geometry of the circuit. It connects the flux with the current as follows,

The unit for inductance is Wb/A, which is given the name henry. Practical circuits have inductance much smaller than one henry, more in the range of millihenries. The main use of the concept of inductance will be in circuits where the current changes, thus causing a proportional change in flux. This changing flux induces an EMF:

The self-inductance, L, of the solenoid is: L = μ0 n2Ad; and the inductance per unit length is:
L/d = μ0 n2A.

The self-inductance, L, of the solenoid is: L = μ0 N2A/2πr. If the toroid is filled with material of permeability μ, then: L = μ N2A/2πr.

Whenever one has two circuits near each other, it will be possible for a current which exists in one circuit to produce flux through the second circuit. If φ12 is the flux in circuit 2 caused by a current I1, in circuit 1, Then, the mutual inductance, M12, connects these two quantities, as follows,

The exact value of M12 is determined by the geometrical relationship between the two circuits. The change in current in circuit 1 changes the flux in circuit 2 proportionally, which produces an EMF in circuit 2, given by

There is only one mutual inductance for the two circuits meaning M12 = M21. We can measure the mutual inductance by measuring the induced EMF produced in one circuit by a known rate of change in current in the other circuit.

Coil on Solenoid

Coil on Toroid

Coil Near Long Wire

Coil at Center of Loop

Any circuit element that generates an inductance when current flows through it (e.g. a coil, a solenoid, a toroid) is called an inductor. An inductor has the property that it produces a back EMF if the current is changing, but does nothing if the current is steady. While an inductor will not affect a DC circuit once a current has been established it will be of great importance during the time that the current is being turned on or off.

In order to increase the current in the inductor, an external driving voltage must be imposed on the circuit to overcome the back EMF, and this voltage will do work against the resisting EMF. The voltage will continue to do work until the current reaches its final value, at which time the current is no longer changing and no back EMF is being produced. During the time that the current is building up from zero to its final value, however, work must be done on the inductor. The work, or energy stored in the inductor is,

This result is similar to the case of storing energy in a capacitor by virtue of the charge that we have placed on the plates of the capacitor. There the energy stored = (1/2)Q2/C.

In terms of the magnetic field that have been set up in space, we have,

At any point in space, where there is a magnetic field, a certain amount of energy is stored. This energy equals the energy density times the volume of space being considered. The same general consideration holds for electric fields as well and indeed the electric field energy density is given by (1/2) ε0E2. In other words, wherever electric or magnetic fields exist in space, energy is being stored in the form of these fields

The total energy density at any point in space is the sum of the electric and the magnetic field energy densities. Since the units for energy density are the same irrespective of their source, this offers a means of comparing the relative magnitudes of electric and magnetic fields. Electric and magnetic fields with the same energy density can be considered to be comparable to each other. The electric and the magnetic fields associated with electromagnetic waves have equal energy densities. These considerations lend credence to the idea that these fields are real physical quantities that actually exist in space and are not merely mathematical contrivances that make it easier to calculate the forces exerted by the electric and magnetic interactions.

We can induce EMFs in one circuit by changing the current in another circuit. This forms the basis of the transformer, which is used to transform voltage in one circuit into a different voltage in a second circuit. All the magnetic flux established by the first winding, called the primary coil, passes through the turns of the other winding, called the secondary coil. In order to get large fluxes, it is useful to place ferromagnetic material within the solenoid that has a large permeability, such as iron. The figure below shows a typical transformer:

Here, the primary winding, with N1 turns, is wound on one side of the rectangular ring, and the secondary winding, with N2 turns, is wound on the other side of the ring. This is a typical transformer. If one changes the voltage in the primary circuit, the current in the primary circuit will change, and therefore the flux. For a perfect transformer, the flux through one turn of the secondary is the same as the flux through one turn of the primary. Therefore, the total EMF developed in each winding will depend on the number of turns in that circuit.

A transformer is useful only with currents that are changing, as with AC. In that case, it is possible to use a transformer to convert a voltage applied to the primary circuit into a larger or smaller voltage in the secondary circuit. This ability to easily convert (transform) voltages in AC, which is much more difficult for DC, is the main reason why AC is the primary source of power throughout the world.


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