For uniform field and flat surface: $\Phi_B = BA\cos\theta$

Units: 1 weber (Wb) = 1 T·m²

Faraday's Law

Induced EMF: $\mathcal{E} = -\frac{d\Phi_B}{dt}$

Negative sign is Lenz's law: induced current opposes change in flux.

For N-turn coil: $\mathcal{E} = -N\frac{d\Phi_B}{dt}$

Ways to Change Flux

1. Change B: $\Phi_B = BA\cos\theta$, vary $B$
2. Change A: Move wire to change loop area
3. Change θ: Rotate loop in field

Motional EMF

Rod of length $L$ moving with velocity $v$ perpendicular to field $B$:

Magnetic force on charges: $F = qvB$

EMF: $\mathcal{E} = BLv$

From Faraday's law: $\mathcal{E} = -\frac{d(BA)}{dt} = -B\frac{dA}{dt} = -BL\frac{dx}{dt} = -BLv$

(Magnitude $BLv$; sign from Lenz's law)

Induced Electric Field

Changing magnetic flux creates electric field:

$\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$

This $\vec{E}$ is non-conservative! (Circulation not zero)

For cylindrical symmetry: $E(2\pi r) = -\frac{d\Phi_B}{dt}$

Eddy Currents

Induced currents in bulk conductor:

• Flow in loops (eddies)
• Dissipate energy as heat
• Create magnetic braking

Applications:

• Metal detectors
• Magnetic braking
• Induction heating

Generators

Rotating coil in magnetic field:

$\Phi_B = BA\cos(\omega t)$

$\mathcal{E} = -\frac{d\Phi_B}{dt} = BA\omega\sin(\omega t)$

$\mathcal{E} = \mathcal{E}_0\sin(\omega t)$

where $\mathcal{E}_0 = NBA\omega$ (N turns, area A).

Betatron

Accelerates electrons in circular path:

Condition for stable orbit: $B_{orbit} = \frac{1}{2}B_{avg}$

where $B_{avg}$ is average field inside orbit.

Lenz's Law

Induced current creates magnetic field that opposes the change in flux.

• Flux increasing: induced $\vec{B}$ opposes it
• Flux decreasing: induced $\vec{B}$ tries to maintain it

Energy conservation: Work required to change flux against induced current.

Change in flux: $\Delta \Phi_B = A \Delta B = A(B_1 - B_0)$ $\Delta \Phi_B = (0.0201)(0.8 - 0.2) = 0.0121 \text{ Wb}$

Faraday's law: $|\mathcal{E}| = \left|\frac{\Delta \Phi_B}{\Delta t}\right| = \frac{0.0121}{0.5}$

$|\mathcal{E}| = \boxed{0.0241 \text{ V} = 24.1 \text{ mV}}$

(b) Induced current:

$I = \frac{|\mathcal{E}|}{R} = \frac{0.0241}{2.0}$

$I = \boxed{0.0121 \text{ A} = 12.1 \text{ mA}}$

Direction (Lenz's law):

• B is increasing into page
• Induced current opposes this change
• Induced B must point out of page
• By right-hand rule: current flows counterclockwise (viewed from front)

(c) Total charge:

$Q = \int I \, dt = \int \frac{|\mathcal{E}|}{R} dt = \frac{1}{R} \int |\mathcal{E}| \, dt$

$Q = \frac{\Delta \Phi_B}{R} = \frac{0.0121}{2.0}$

$Q = \boxed{6.03 \times 10^{-3} \text{ C} = 6.03 \text{ mC}}$