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Faraday's Law

CSCA Faraday's Law study guide organized around the publicly available CSCA syllabus. Practice Physics questions on aicsca.com.

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Topic: Electromagnetism > Electromagnetic Induction > Faraday's Law

1. Core Formulas

Faraday's Law solves the problem of "**Magnitude of Induced EMF**".

(1) General Formula (Flux Change)

Applies to all induction phenomena, calculates **Average EMF**.

$$ \mathcal{E} = N \frac{\Delta \Phi}{\Delta t} $$

* $\mathcal{E}$: Induced EMF (V).

* $N$: Number of turns (**Don't forget to multiply by N!**).

* $\frac{\Delta \Phi}{\Delta t}$: Rate of change of flux.

**Tip**: For magnitude calculations, ignore the negative sign. Use Lenz's Law for direction.

(2) Cutting Formula (Motional EMF)

Applies to conductors cutting magnetic lines, calculates **Instantaneous EMF**.

$$ \mathcal{E} = B L v \sin\theta $$

* **$L$**: **Effective Length** (m).

* *Curved wire*: Straight line distance between ends.

* **$v$**: Relative velocity (m/s).

* **$\theta$**: Angle between $v$ and $B$.

* Most common ($v \perp B, L \perp B, v \perp L$): **$\mathcal{E} = BLv$**.

(3) Rotational Cutting Formula (Extension)

A rod rotating around one end in a perpendicular field.

$$ \mathcal{E} = \frac{1}{2} B L^2 \omega $$

* $\omega$: Angular velocity (rad/s).

* *Derivation*: Using average velocity $\bar{v} = \frac{0 + \omega L}{2}$ in $BL\bar{v}$.

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2. Three Models for $\Delta \Phi$

Determined by $\Phi = B S \cos\theta$:

1. **B changes, S constant**:

$$ \mathcal{E} = N S \frac{\Delta B}{\Delta t} $$

* *Graph Problem*: $\frac{\Delta B}{\Delta t}$ is the **Slope** of the B-t graph.

2. **S changes, B constant**:

$$ \mathcal{E} = N B \frac{\Delta S}{\Delta t} $$

3. **$\theta$ changes (Rotation)** (AC Generator):

Produces Sine wave. Instantaneous $\mathcal{E} = NBS\omega \sin(\omega t)$.

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3. Typical Examples

**Example 1: B-t Graph (Slope Method)**

**Problem**: Coil $N=100$, Area $S=0.1 m^2$. B decreases uniformly from 2T to 0T in 2s. Find EMF.

**Solution**:

1. **Model**: S constant, B changes.

2. **Slope**: $\frac{\Delta B}{\Delta t} = 1 T/s$ (magnitude).

3. **Calc**:

$$ \mathcal{E} = N S \left| \frac{\Delta B}{\Delta t} \right| = 100 \times 0.1 \times 1 = 10 V $$

**Example 2: Rotating Coil (Average vs Max)**

**Problem**: A coil rotates from "Neutral Plane" ($\perp$ B) by $90^\circ$ to Parallel position.

(1) Find Average EMF $\bar{\mathcal{E}}$.

(2) Find Instantaneous EMF at the end $\mathcal{E}_{max}$.

**Solution**:

* **Average**: Use Faraday's Law.

$\Delta \Phi = BS - 0 = BS$. Time $\Delta t = \frac{\pi/2}{\omega}$.

$$ \bar{\mathcal{E}} = N \frac{BS}{\pi/2 / \omega} = \frac{2NBS\omega}{\pi} $$

* **Instantaneous** (Max): Use $NBS\omega$.

At parallel, cutting speed is max, $\mathcal{E}_{max} = NBS\omega$.

* **Note**: Distinguish between Average and Instantaneous requests!

**Example 3: Rotational Cutting (Rod)**

**Problem**: Rod $L=1m$, fixed at one end, rotates at $\omega=2 rad/s$ in $B=0.5T$. Find Voltage.

**Solution**:

$$ \mathcal{E} = \frac{1}{2} B L^2 \omega = 0.5 \times 0.5 \times 1^2 \times 2 = 0.5 V $$

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4. CSCA Common Pitfalls

1. **Units**: Convert $cm^2$ to $10^{-4} m^2$ immediately!

2. **Turns N**: Look for "coil" vs "single loop". Don't miss multiplying by $N$.

3. **Effective Area S**: If coil radius $r >$ Magnetic field radius $R$, use the **smaller area** ($\pi R^2$) for flux calculation.