====== IX. Electromagnetic induction ======
===== - Faraday's law =====
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Magnetic flux is defined as

$$ \Phi_m = \int_S \vec{B} \cdot d\vec{A} = \int_S \vec{B} \cdot \hat{n} dA $$

where $\hat{n}$ is the normal vector to the area $\vec{A}$. Unit is weber, Wb. 1 Wb = 1 T m$^2$.

Michael Faraday observed the following phenomena.

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Faraday's law says that the induced emf (electromotive force, voltage) in a circuit

$$ \varepsilon = -\frac{d\Phi_m}{dt} $$

that is the emf is the negative of the rate of change of the magnetic flux through the circuit.

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If there are multiple turns like above then

$$ \varepsilon = -N \frac{d\Phi_m}{dt}. $$

===== - Lenz's law =====
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//The direction of the induced emf drives current around a wire loop to always oppose the change in magnetic flux that causes the emf.//
===== - Motional emf =====
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Magnetic flux varies with field, area and the angle between field and area. Variation of any one of these will induce an emf.

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The area of MNOP $A=lx$ and the flux $\Phi=BA=Blx$. The induced emf

$$ \varepsilon = -\frac{d\Phi}{dt} = Bl\frac{dx}{dt} = Blv $$

and the current induced in the circuit

$$ I = \frac{\varepsilon}{R} = \frac{Blv}{R}. $$

From the energy point of view, $\vec{F}_a$ produces power $P=F_a v$:

$$ F_a v = IlBv = \frac{Blv}{R} Blv = \frac{l^2 B^2 v^2}{R}. $$

The power dissipated

$$ P = I^2 R = \frac{l^2 B^2 v^2}{R}. $$

The produced and dissipated power are equal.

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===== - Induced electric field =====
Faraday's law can rewritten in terms of the induced electric field.

$$ \varepsilon = \oint \vec{E} \cdot d\vec{l} = - \frac{d\Phi_m}{dt}. $$

Note that for the electrostatic case $\oint \vec{E} \cdot d\vec{l}=0$.

===== - Electric generators and back emf =====
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Calculate the emf induced when the call is rotated from $\theta=0^\circ$ to $90^\circ$:

$$ \varepsilon = -\frac{d\Phi}{dt} = NBA \sin\theta \frac{d\theta}{dt}. $$

Here $A=\pi r^2$, $N=200$, $r=5$ cm, $B=0.8$ T, $d\theta=\pi/2$ and $dt=15$ ms.

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In the above diagram, emf is produced on both side of the loop, total emf

$$ \varepsilon = 2Blv\sin\theta = 2Blv\sin(\omega t) $$

where $\omega$ is the angular velocity. So $v=r\omega$ and $r=w/2$ if $w$ is the width of the loop. So

$$ \varepsilon =  2Bl \frac{w}{2}\omega \sin(\omega t) = (lw)B\omega \sin(\omega t) = AB\omega \sin(\omega t). $$

If there are $N$ loops just multiply by $N$. We can write

$$ \varepsilon = \varepsilon_0 \sin(\omega t) $$

where $\varepsilon_0 = NBA\omega$ as you see below.

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The frequency of the oscillation of this alternating current is $f=\omega/(2\pi)$ and the period $T=2\pi/omega$.

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