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  • Calculate the inductance of an inductor.
  • Calculate the energy stored in an inductor.
  • Calculate the emf generated in an inductor.


Induction is the process in which an emf is induced by changing magnetic flux. Many examples have been discussed so far, some more effective than others. Transformers, for example, are designed to be particularly effective at inducing a desired voltage and current with very little loss of energy to other forms. Is there a useful physical quantity related to how “effective” a given device is? The answer is yes, and that physical quantity is called inductance    .

Mutual inductance is the effect of Faraday’s law of induction for one device upon another, such as the primary coil in transmitting energy to the secondary in a transformer. See [link] , where simple coils induce emfs in one another.

The figure shows two coils coil one, of five turns and coil two, of four turns are kept adjacent to each other. The magnetic field lines of strength B are shown to pass through the two coils. Coil one is shown to be connected to an A C source. The changing current in the coil one is given as I one in clock wise direction. Coil two is connected to a galvanometer. A change in current in coil one is shown to induce an e m f in coil two.The induced e m f in coil two is measured as a deflection in galvanometer.
These coils can induce emfs in one another like an inefficient transformer. Their mutual inductance M indicates the effectiveness of the coupling between them. Here a change in current in coil 1 is seen to induce an emf in coil 2. (Note that " E 2 size 12{E rSub { size 8{2} } } {} induced" represents the induced emf in coil 2.)

In the many cases where the geometry of the devices is fixed, flux is changed by varying current. We therefore concentrate on the rate of change of current, Δ I t size 12{ΔI} {} , as the cause of induction. A change in the current I 1 size 12{I rSub { size 8{1} } } {} in one device, coil 1 in the figure, induces an emf 2 size 12{"emf" rSub { size 8{2} } } {} in the other. We express this in equation form as

emf 2 = M Δ I 1 Δ t , size 12{"emf" rSub { size 8{2} } = - M { {ΔI rSub { size 8{1} } } over {Δt} } } {}

where M size 12{M} {} is defined to be the mutual inductance between the two devices. The minus sign is an expression of Lenz’s law. The larger the mutual inductance M size 12{M} {} , the more effective the coupling. For example, the coils in [link] have a small M size 12{M} {} compared with the transformer coils in [link] . Units for M size 12{M} {} are ( V s ) /A = Ω s size 12{ \( V cdot s \) "/A"= %OMEGA cdot s} {} , which is named a henry    (H), after Joseph Henry. That is, 1 H = 1 Ω s size 12{1`H=1` %OMEGA cdot s} {} .

Nature is symmetric here. If we change the current I 2 size 12{I rSub { size 8{2} } } {} in coil 2, we induce an emf 1 size 12{"emf" rSub { size 8{1} } } {} in coil 1, which is given by

emf 1 = M Δ I 2 Δ t , size 12{"emf" rSub { size 8{1} } = - M { {ΔI rSub { size 8{2} } } over {Δt} } } {}

where M size 12{M} {} is the same as for the reverse process. Transformers run backward with the same effectiveness, or mutual inductance M size 12{M} {} .

A large mutual inductance M size 12{M} {} may or may not be desirable. We want a transformer to have a large mutual inductance. But an appliance, such as an electric clothes dryer, can induce a dangerous emf on its case if the mutual inductance between its coils and the case is large. One way to reduce mutual inductance M size 12{M} {} is to counterwind coils to cancel the magnetic field produced. (See [link] .)

The figure describes the heating coils of electric clothes dryer that are counter wound on a cylindrical core. There magnetic fields cancel each other.
The heating coils of an electric clothes dryer can be counter-wound so that their magnetic fields cancel one another, greatly reducing the mutual inductance with the case of the dryer.

Self-inductance , the effect of Faraday’s law of induction of a device on itself, also exists. When, for example, current through a coil is increased, the magnetic field and flux also increase, inducing a counter emf, as required by Lenz’s law. Conversely, if the current is decreased, an emf is induced that opposes the decrease. Most devices have a fixed geometry, and so the change in flux is due entirely to the change in current Δ I size 12{ΔI} {} through the device. The induced emf is related to the physical geometry of the device and the rate of change of current. It is given by

Questions & Answers

Calculate the work done by an 85.0-kg man who pushes a crate 4.00 m up along a ramp that makes an angle of 20.0º20.0º with the horizontal. (See [link] .) He exerts a force of 500 N on the crate parallel to the ramp and moves at a constant speed. Be certain to include the work he does on the crate an
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