Electromagnetic Induction

Magnetic flux through a surface is given by:

$\Phi_B = \vec{B} \cdot \vec{A} = BA \cos\theta$

• $\vec{B}$: Magnetic field vector (Tesla)
• $\vec{A}$: Area vector normal to the surface (m²)
• $\theta$: Angle between $\vec{B}$ and $\vec{A}$

Special Cases:

• $\theta = 0^\circ \Rightarrow \Phi_B = BA$ (maximum flux)
• $\theta = 90^\circ \Rightarrow \Phi_B = 0$ (no flux)

SI Unit: Weber (Wb), where $1\, \text{Wb} = 1\, \text{T} \cdot \text{m}^2$

Law: The induced emf in a closed loop equals the negative rate of change of magnetic flux through the loop.

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

For a coil of N turns:

$\varepsilon = -N \frac{d(BA\cos\theta)}{dt}$

Lenz's Law

The negative sign shows that the induced emf always opposes the change in magnetic flux, preserving energy conservation:

• Change in B ↑ → Induced B opposes increase (B ↓)
• Change in area or angle also generates emf

For a straight conductor of length $l$ moving with velocity $v$ perpendicular to magnetic field $B$:

$\varepsilon = Blv \quad \text{(only if motion is perpendicular)}$

This can be derived from the Lorentz force:

$F = q(\vec{v} \times \vec{B}) \Rightarrow \varepsilon = \frac{F \cdot l}{q} = Blv$

Let a coil of N turns and area A rotate in a uniform magnetic field B with angular velocity $\omega$. Magnetic flux at time $t$:

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

Then induced emf:

$\varepsilon = -\frac{d\Phi_B}{dt} = \omega NBA \sin(\omega t)$

Peak emf:

$\varepsilon_0 = \omega NBA$

This forms the basis of AC generation.

When two coils are near each other, changing current in one induces emf in the other.

$\varepsilon_2 = -M \frac{dI_1}{dt}, \quad \varepsilon_1 = -M \frac{dI_2}{dt}$

• $M$: Mutual inductance (H)
• $M = \frac{N_2 \Phi_{21}}{I_1}$

Applications: Transformers, wireless charging, induction coils

Changing current in a coil induces emf in the same coil:

$\varepsilon = -L \frac{dI}{dt}$

• $L$: Self inductance (H), defined as $L = \frac{N \Phi_B}{I}$

Energy stored in an inductor:

$U = \frac{1}{2} L I^2$

When a conductor moves in a magnetic field, loops of current are induced inside the material, called eddy currents.

• Causes energy loss as heat (e.g., transformer cores)
• Reduced by laminating cores

Example 1: Induced emf in a Solenoid

Given a solenoid of length $l$, turn density $n$, changing current rate $\frac{dI}{dt}$, and coil inside it:

$B = \mu_0 n I \Rightarrow \varepsilon = -N \mu_0 n \pi r^2 \frac{dI}{dt}$

Example 2: Copper Loop and Resistivity

$\varepsilon = IR = I \cdot \frac{\rho l}{A}, \quad \frac{dB}{dt} = \frac{\varepsilon}{\text{loop area}}$

Example 3: Rotating Coil

Given N = 8000, A = 100 cm², f = 600 rpm, B = 0.05 T

$\omega = 2\pi f = 62.8 \, \text{rad/s}, \quad \varepsilon_0 = \omega NBA = 251\, V$

• Electric Generators: Convert mechanical to electrical energy
• Transformers: Transfer energy between circuits
• Induction Cookers: Use eddy currents for heating
• Magnetic Brakes: Use eddy currents to slow down motion
• Wireless Charging: Mutual induction between coils