Contents

Section Summary

The crucial quantity in induction is magnetic flux $Φ Φ size 12{Φ} {}$ , defined to be $Φ = BA cos θ Φ = BA cos size 12{Φ= ital "BA""cos"θ} {}$ , where $B B size 12{B} {}$ is the magnetic field strength over an area $A A size 12{A} {}$ at an angle $θ θ size 12{θ} {}$ with the perpendicular to the area.
Units of magnetic flux $Φ Φ size 12{Φ} {}$ are $T \cdot m^{2} T \cdot m^{2} size 12{T cdot m rSup { size 8{2} } } {}$ .
Any change in magnetic flux $Φ Φ size 12{Φ} {}$ induces an emf—the process is defined to be electromagnetic induction.

Faraday’s law of induction states that the emfinduced by a change in magnetic flux is
$emf = - N \frac{Δ Φ}{Δ t} emf = - N \frac{Δ Φ}{Δ t} size 12{"emf"= - N { {ΔΦ} over {Δt} } } {}$
when flux changes by $Δ Φ Δ Φ size 12{ΔΦ} {}$ in a time $Δ t Δ t size 12{Δt} {}$ .
If emf is induced in a coil, $N N$ is its number of turns.
The minus sign means that the emf creates a current $I I size 12{I} {}$ and magnetic field $B B size 12{B} {}$ that oppose the change in flux $Δ Φ Δ Φ size 12{ΔΦ} {}$ —this opposition is known as Lenz’s law.

An emf induced by motion relative to a magnetic field $B B$ is called a motional emf and is given by
$emf = Bℓv (B, ℓ, and v perpendicular), emf = Bℓv size 12{"emf"=Bℓv} {}$
where $ℓ ℓ size 12{ℓ} {}$ is the length of the object moving at speed $v v size 12{v} {}$ relative to the field.

Current loops induced in moving conductors are called eddy currents.
They can create significant drag, called magnetic damping.

An electric generator rotates a coil in a magnetic field, inducing an emfgiven as a function of time by
$emf = NAB ω sin ωt, emf = NAB ω, size 12{"emf"= ital "NAB"ω"sin"ωt} {}$
where $A A size 12{A} {}$ is the area of an $N N size 12{N} {}$ -turn coil rotated at a constant angular velocity $ω ω size 12{ω} {}$ in a uniform magnetic field $B B size 12{B} {}$ .
The peak emf ${emf}_{0} {emf}_{0} size 12{"emf" rSub { size 8{0} } } {}$ of a generator is
${emf}_{0} = NAB ω . {emf}_{0} = NAB ω . size 12{"emf" rSub { size 8{0} } = ital "NAB"ω} {}$

Any rotating coil will have an induced emf—in motors, this is called back emf, since it opposes the emf input to the motor.

Transformers use induction to transform voltages from one value to another.
For a transformer, the voltages across the primary and secondary coils are related by
$\frac{V_{s}}{V_{p}} = \frac{N_{s}}{N_{p}}, \frac{V_{s}}{V_{p}} = \frac{N_{s}}{N_{p}}, size 12{ { {V rSub { size 8{s} } } over {V rSub { size 8{p} } } } = { {N rSub { size 8{s} } } over {N rSub { size 8{p} } } } } {}$
where $V_{p} V_{p} size 12{V rSub { size 8{p} } } {}$ and $V_{s} V_{s} size 12{V rSub { size 8{s} } } {}$ are the voltages across primary and secondary coils having $N_{p} N_{p} size 12{N rSub { size 8{p} } } {}$ and $N_{s} N_{s} size 12{N rSub { size 8{s} } } {}$ turns.
The currents $I_{p} I_{p} size 12{I rSub { size 8{p} } } {}$ and $I_{s} I_{s} size 12{I rSub { size 8{s} } } {}$ in the primary and secondary coils are related by $\frac{I_{s}}{I_{p}} = \frac{N_{p}}{N_{s}} \frac{I_{s}}{I_{p}} = \frac{N_{p}}{N_{s}} size 12{ { {I rSub { size 8{s} } } over {I rSub { size 8{p} } } } = { {N rSub { size 8{p} } } over {N rSub { size 8{s} } } } } {}$ .
A step-up transformer increases voltage and decreases current, whereas a step-down transformer decreases voltage and increases current.

Electrical safety systems and devices are employed to prevent thermal and shock hazards.
Circuit breakers and fuses interrupt excessive currents to prevent thermal hazards.
The three-wire system guards against thermal and shock hazards, utilizing live/hot, neutral, and earth/ground wires, and grounding the neutral wire and case of the appliance.
A ground fault interrupter (GFI) prevents shock by detecting the loss of current to unintentional paths.
An isolation transformer insulates the device being powered from the original source, also to prevent shock.
Many of these devices use induction to perform their basic function.

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 $Δ I_{1} / Δ t Δ I_{1} / Δ t size 12{ΔI rSub { size 8{1} } /Δt} {}$ in one induces an emf ${emf}_{2} {emf}_{2} size 12{"emf" rSub { size 8{2} } } {}$ in the second:
${emf}_{2} = - M \frac{Δ I_{1}}{Δ t}, {emf}_{2} = - M \frac{Δ I_{1}}{Δ t}, size 12{"emf" rSub { size 8{2} } = - M { {ΔI rSub { size 8{1} } } over {Δt} } } {}$
where $M M$ 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 $Δ I_{2} / Δ t Δ I_{2} / Δ t size 12{ΔI rSub { size 8{2} } /Δt} {}$ through the second device induces an emf ${emf}_{1} {emf}_{1} size 12{"emf" rSub { size 8{1} } } {}$ in the first:
${emf}_{1} = - M \frac{Δ I_{2}}{Δ t}, {emf}_{1} = - M \frac{Δ I_{2}}{Δ t}, size 12{"emf" rSub { size 8{1} } = - M { {ΔI rSub { size 8{2} } } over {Δt} } } {}$
where $M M$ is the same mutual inductance as in the reverse process.
Current changes in a device induce an emf in the device itself.
Self-inductance 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
$emf = - L \frac{Δ I}{Δ t}, emf = - L \frac{Δ I}{Δ t}, size 12{"emf"= - L { {ΔI} over {Δt} } } {}$
where $L L size 12{L} {}$ is the self-inductance of the inductor, and $Δ I / Δ t Δ I / Δ t size 12{ΔI/Δt} {}$ 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 $1 H = 1 Ω \cdot s 1 H = 1 Ω \cdot s size 12{1`H=1` %OMEGA cdot s} {}$ .
The self-inductance $L L size 12{L} {}$ of an inductor is proportional to how much flux changes with current. For an $N N size 12{N} {}$ -turn inductor,
$L = N \frac{Δ Φ}{Δ I} . L = N \frac{Δ Φ}{Δ I} . size 12{L=N { {ΔΦ} over {ΔI} } } {}$
The self-inductance of a solenoid is
$L = \frac{μ_{0} N^{2} A}{ℓ} (solenoid), L = \frac{μ_{0} N^{2} A}{ℓ} (solenoid), size 12{L= { {μ rSub { size 8{0} } N rSup { size 8{2} } A} over {ℓ} } } {}$
where $N N size 12{N} {}$ is its number of turns in the solenoid, $A A size 12{A} {}$ is its cross-sectional area, $ℓ ℓ size 12{ℓ} {}$ is its length, and $μ_{0} = 4π \times 10^{−7} T \cdot m/A μ_{0} = 4π \times 10^{−7} size 12{μ rSub { size 8{0} } =4π times "10" rSup { size 8{"-7"} } `T cdot "m/A"} {}$ is the permeability of free space.
The energy stored in an inductor $E_{ind} E_{ind} size 12{E rSub { size 8{"ind"} } } {}$ is
$E_{ind} = \frac{1}{2} {LI}^{2} . E_{ind} = \frac{1}{2} {LI}^{2} . size 12{E rSub { size 8{"ind"} } = { {1} over {2} } ital "LI" rSup { size 8{2} } } {}$

When a series connection of a resistor and an inductor—an RL circuit—is connected to a voltage source, the time variation of the current is
$I = I_{0} (1 - e^{- t / τ}) (turning on). I = I_{0} (1 - e^{- t / τ}) (turning on). size 12{I=I rSub { size 8{0} } $ 1 - e rSup { size 8{ - t/τ} } $ } {}$
where $I_{0} = V / R I_{0} = V / R size 12{I rSub { size 8{0} } =V/R} {}$ is the final current.
The characteristic time constant $τ τ size 12{τ} {}$ is $τ = \frac{L}{R} τ = \frac{L}{R} size 12{τ= { {L} over {R} } } {}$ , where $L L$ is the inductance and $R R$ is the resistance.
In the first time constant $τ τ size 12{τ} {}$ , the current rises from zero to $0.632 I_{0} 0.632 I_{0} size 12{0 "." "632"I rSub { size 8{0} } } {}$ , and 0.632 of the remainder in every subsequent time interval $τ τ size 12{τ} {}$ .
When the inductor is shorted through a resistor, current decreases as
$I = I_{0} e^{- t / τ} (turning off). I = I_{0} e^{- t / τ} (turning off). size 12{I=I rSub { size 8{0} } e rSup { size 8{ - t/τ} } } {}$
Here $I_{0} I_{0} size 12{I rSub { size 8{0} } } {}$ is the initial current.
Current falls to $0.368 I_{0} 0.368 I_{0} size 12{0 "." "368"I rSub { size 8{0} } } {}$ in the first time interval $τ τ size 12{τ} {}$ , and 0.368 of the remainder toward zero in each subsequent time $τ τ size 12{τ} {}$ .

For inductors in AC circuits, we find that when a sinusoidal voltage is applied to an inductor, the voltage leads the current by one-fourth of a cycle, or by a $90º 90º$ phase angle.
The opposition of an inductor to a change in current is expressed as a type of AC resistance.
Ohm’s law for an inductor is
$I = \frac{V}{X_{L}}, I = \frac{V}{X_{L}}, size 12{I= { {V} over {X rSub { size 8{L} } } } } {}$
where $V V size 12{V} {}$ is the rms voltage across the inductor.
$X_{L} X_{L} size 12{X rSub { size 8{L} } } {}$ is defined to be the inductive reactance, given by
$X_{L} = 2π fL, X_{L} = 2π fL, size 12{X rSub { size 8{L} } =2π ital "fL"} {}$
with $f f size 12{f} {}$ the frequency of the AC voltage source in hertz.
Inductive reactance $X_{L} X_{L} size 12{X rSub { size 8{L} } } {}$ has units of ohms and is greatest at high frequencies.
For capacitors, we find that when a sinusoidal voltage is applied to a capacitor, the voltage follows the current by one-fourth of a cycle, or by a $90º 90º$ phase angle.
Since a capacitor can stop current when fully charged, it limits current and offers another form of AC resistance; Ohm’s law for a capacitor is
$I = \frac{V}{X_{C}}, I = \frac{V}{X_{C}}, size 12{I= { {V} over {X rSub { size 8{C} } } } } {}$
where $V V size 12{V} {}$ is the rms voltage across the capacitor.
$X_{C} X_{C} size 12{X rSub { size 8{C} } } {}$ is defined to be the capacitive reactance, given by
$X_{C} = \frac{1}{2π fC} . X_{C} = \frac{1}{2π fC} . size 12{X rSub { size 8{C} } = { {1} over {2π ital "fC"} } } {}$
$X_{C} X_{C} size 12{X rSub { size 8{C} } } {}$ has units of ohms and is greatest at low frequencies.

The AC analogy to resistance is impedance $Z Z$ , the combined effect of resistors, inductors, and capacitors, defined by the AC version of Ohm’s law:
$I_{0} = \frac{V_{0}}{Z} or I_{rms} = \frac{V_{rms}}{Z}, I_{0} = \frac{V_{0}}{Z} size 12{I rSub { size 8{0} } = { {V rSub { size 8{0} } } over {Z} } " or "I rSub { size 8{ ital "rms"} } = { {V rSub { size 8{ ital "rms"} } } over {Z} } ,} {}$
where $I_{0} I_{0} size 12{I rSub { size 8{0} } } {}$ is the peak current and $V_{0} V_{0} size 12{V rSub { size 8{0} } } {}$ is the peak source voltage.
Impedance has units of ohms and is given by $Z = \sqrt{R^{2} + (X_{L} - X_{C})^{2}} Z = \sqrt{R^{2} + (X_{L} - X_{C})^{2}} size 12{Z= sqrt {R rSup { size 8{2} } + $ X rSub { size 8{L} } - X rSub { size 8{C} } $ rSup { size 8{2} } } } {}$ .
The resonant frequency $f_{0} f_{0} size 12{f rSub { size 8{0} } } {}$ , at which $X_{L} = X_{C} X_{L} = X_{C} size 12{X rSub { size 8{L} } =X rSub { size 8{C} } } {}$ , is
$f_{0} = \frac{1}{2π \sqrt{LC}} . f_{0} = \frac{1}{2π \sqrt{LC}} . size 12{f rSub { size 8{0} } = { {1} over {2π sqrt { ital "LC"} } } } {}$
In an AC circuit, there is a phase angle $ϕ ϕ size 12{ϕ} {}$ between source voltage $V V size 12{V} {}$ and the current $I I size 12{I} {}$ , which can be found from
$cos ϕ = \frac{R}{Z}, cos, size 12{"cos"ϕ= { {R} over {Z} } } {}$
$ϕ = 0º ϕ = 0º size 12{ϕ=0 rSup { size 8{ circ } } } {}$ for a purely resistive circuit or an RLC circuit at resonance.
The average power delivered to an RLC circuit is affected by the phase angle and is given by
$P_{ave} = I_{rms} V_{rms} cos ϕ, P_{ave} = I_{rms} V_{rms}, size 12{P rSub { size 8{"ave"} } =I rSub { size 8{"rms"} } V rSub { size 8{"rms"} } "cos"ϕ} {}$
$cos ϕ cos size 12{"cos"ϕ} {}$ is called the power factor, which ranges from 0 to 1.