



Section summary
 Inductance is the property of a device that tells how effectively it induces an emf in another device.
 Mutual inductance is the effect of two devices in inducing emfs in each other.
 A change in current
$\mathrm{\Delta}{I}_{1}/\mathrm{\Delta}t$ in one induces an emf
${\text{emf}}_{2}$ in the second:
${\text{emf}}_{2}=M\frac{\mathrm{\Delta}{I}_{1}}{\mathrm{\Delta}t}\text{,}$
where
$M$ is defined to be the mutual inductance between the two devices, and the minus sign is due to Lenz’s law.
 Symmetrically, a change in current
$\mathrm{\Delta}{I}_{2}/\mathrm{\Delta}t$ through the second device induces an emf
${\text{emf}}_{1}$ in the first:
${\text{emf}}_{1}=M\frac{\mathrm{\Delta}{I}_{2}}{\mathrm{\Delta}t}\text{,}$
where
$M$ is the same mutual inductance as in the reverse process.
 Current changes in a device induce an emf in the device itself.
 Selfinductance is the effect of the device inducing emf in itself.
 The device is called an inductor, and the emf
induced in it by a change in current through it is
$\text{emf}=L\frac{\mathrm{\Delta}I}{\mathrm{\Delta}t}\text{,}$
where
$L$ is the selfinductance of the inductor, and
$\mathrm{\Delta}I/\mathrm{\Delta}t$ is the rate of change of current through it. The minus sign indicates that emf opposes the change in current, as required by Lenz’s law.
 The unit of self and mutual inductance is the henry (H), where
$\mathrm{1\; H}=1\; \Omega \cdot \text{s}$ .
 The selfinductance
$L$ of an inductor is proportional to how much flux changes with current. For an
$N$ turn inductor,
$L=N\frac{\mathrm{\Delta}\Phi}{\mathrm{\Delta}I}\text{.}$
 The selfinductance of a solenoid is
$L=\frac{{\mu}_{0}{N}^{2}A}{\ell}\text{(solenoid),}$
where
$N$ is its number of turns in the solenoid,
$A$ is its crosssectional area,
$\ell $ is its length, and
${\text{\mu}}_{0}=\mathrm{4\pi}\times {\text{10}}^{\text{\u22127}}\phantom{\rule{0.25em}{0ex}}\text{T}\cdot \text{m/A}\phantom{\rule{0.10em}{0ex}}$ is the permeability of free space.
 The energy stored in an inductor
${E}_{\text{ind}}$ is
${E}_{\text{ind}}=\frac{1}{2}{\text{LI}}^{2}\text{.}$
Conceptual questions
Problems&Exercises
Two coils are placed close together in a physics lab to demonstrate Faraday’s law of induction. A current of 5.00 A in one is switched off in 1.00 ms, inducing a 9.00 V emf in the other. What is their mutual inductance?
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If two coils placed next to one another have a mutual inductance of 5.00 mH, what voltage is induced in one when the 2.00 A current in the other is switched off in 30.0 ms?
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Starting with
${\text{emf}}_{2}=M\frac{\mathrm{\Delta}{I}_{1}}{\mathrm{\Delta}t}$ , show that the units of inductance are
$(\text{V}\cdot \text{s})\text{/A}=\Omega \cdot \text{s}$ .
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Camera flashes charge a capacitor to high voltage by switching the current through an inductor on and off rapidly. In what time must the 0.100 A current through a 2.00 mH inductor be switched on or off to induce a 500 V emf?
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A large research solenoid has a selfinductance of 25.0 H. (a) What induced emf opposes shutting it off when 100 A of current through it is switched off in 80.0 ms? (b) How much energy is stored in the inductor at full current? (c) At what rate in watts must energy be dissipated to switch the current off in 80.0 ms? (d) In view of the answer to the last part, is it surprising that shutting it down this quickly is difficult?
(a) 31.3 kV
(b) 125 kJ
(c) 1.56 MW
(d) No, it is not surprising since this power is very high.
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(a) Calculate the selfinductance of a 50.0 cm long, 10.0 cm diameter solenoid having 1000 loops. (b) How much energy is stored in this inductor when 20.0 A of current flows through it? (c) How fast can it be turned off if the induced emf cannot exceed 3.00 V?
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A precision laboratory resistor is made of a coil of wire 1.50 cm in diameter and 4.00 cm long, and it has 500 turns. (a) What is its selfinductance? (b) What average emf is induced if the 12.0 A current through it is turned on in 5.00 ms (onefourth of a cycle for 50 Hz AC)? (c) What is its inductance if it is shortened to half its length and counterwound (two layers of 250 turns in opposite directions)?
(a) 1.39 mH
(b) 3.33 V
(c) Zero
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The heating coils in a hair dryer are 0.800 cm in diameter, have a combined length of 1.00 m, and a total of 400 turns. (a) What is their total selfinductance assuming they act like a single solenoid? (b) How much energy is stored in them when 6.00 A flows? (c) What average emf opposes shutting them off if this is done in 5.00 ms (onefourth of a cycle for 50 Hz AC)?
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When the 20.0 A current through an inductor is turned off in 1.50 ms, an 800 V emf is induced, opposing the change. What is the value of the selfinductance?
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Integrated Concepts
A very large, superconducting solenoid such as one used in MRI scans, stores 1.00 MJ of energy in its magnetic field when 100 A flows. (a) Find its selfinductance. (b) If the coils “go normal,” they gain resistance and start to dissipate thermal energy. What temperature increase is produced if all the stored energy goes into heating the 1000 kg magnet, given its average specific heat is
$\text{200 J/kg\xb7\xbaC}$ ?
(a) 200 H
(b)
$\text{5.00\xbaC}$
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Unreasonable Results
A 25.0 H inductor has 100 A of current turned off in 1.00 ms. (a) What voltage is induced to oppose this? (b) What is unreasonable about this result? (c) Which assumption or premise is responsible?
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Source:
OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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