Problem Statement
A circular coil of $N = 70$ turns and radius $R = 5\,\text{cm}$ is in a field that changes by $\Delta B = 1.0\,\text{T}$ in $\Delta t = 0.1\,\text{s}$. The plane is perpendicular to the field. Find the induced emf.
Given Information
- $N = 70$ turns
- $R = 0.05\,\text{m}$, area $A = \pi R^2 = 0.00785\,\text{m}^2$
- $\Delta B = 1.0\,\text{T}$
- $\Delta t = 0.1\,\text{s}$
Physical Concepts & Formulas
Faraday’s law: $\mathcal{E} = -N\,d\Phi/dt = -NA\,dB/dt$. The magnitude of induced emf equals $N$ times the rate of flux change through one turn. Lenz’s law gives the direction opposing the change.
- $|\mathcal{E}| = NA\,|dB/dt|$ — Faraday’s law for N-turn coil
Step-by-Step Solution
Step 1 — Area of coil: $$A = \pi R^2 = \pi\times(0.05)^2 = 0.00785\,\text{m}^2$$
Step 2 — Rate of field change: $$dB/dt = 1.0/0.1 = 10.00\,\text{T s}^{-1}$$
Step 3 — Induced emf: $$|\mathcal{E}| = NA\cdot|dB/dt| = 70\times0.00785\times10.00 = 5.4978\,\text{V}$$
Worked Calculation
$$|\mathcal{E}| = 70\times0.00785\times10.00 = 5.4978\,\text{V}$$
Answer
$$\boxed{\mathcal{E} = 5.4978\,\text{V} \approx 5497.79\,\text{mV}}$$
The 5497.79 mV emf is induced by the changing flux. More turns, larger area, or faster field change all increase the emf — the design parameters of transformers and generators.
Physical Interpretation
The 5497.79 mV emf is induced by the changing flux. More turns, larger area, or faster field change all increase the emf — the design parameters of transformers and generators.
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