Problem Statement
A circular coil of $N = 50$ turns and radius $R = 20\,\text{cm}$ is in a field that changes by $\Delta B = 2.0\,\text{T}$ in $\Delta t = 0.4\,\text{s}$. The plane is perpendicular to the field. Find the induced emf.
Given Information
- $N = 50$ turns
- $R = 0.20\,\text{m}$, area $A = \pi R^2 = 0.12566\,\text{m}^2$
- $\Delta B = 2.0\,\text{T}$
- $\Delta t = 0.4\,\text{s}$
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: $$A = \pi R^2 = \pi\times(0.20)^2 = 0.12566\,\text{m}^2$$
Step 2 — Rate of field change: $$dB/dt = 2.0/0.4 = 5.00\,\text{T s}^{-1}$$
Step 3 — Induced emf: $$|\mathcal{E}| = NA\cdot|dB/dt| = 50\times0.12566\times5.00 = 31.4159\,\text{V}$$
Worked Calculation
$$|\mathcal{E}| = 50\times0.12566\times5.00 = 31.4159\,\text{V}$$
Answer
$$\boxed{\mathcal{E} = 31.4159\,\text{V} \approx 31415.93\,\text{mV}}$$
The 31415.93 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 31415.93 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.
The magnitude of this result is physically reasonable and consistent with the expected order of magnitude for this class of problem. Comparing with standard values from physical tables confirms we are in the correct range.
This problem illustrates a fundamental principle that appears throughout physics: small changes in one parameter can lead to measurable, predictable changes in the observable quantity. Understanding this relationship is key to experimental design.
Note that the result depends on the square (or square root) of the key variable — this nonlinear dependence is characteristic of many physics phenomena and means that doubling the parameter does not simply double the result. Students should always check dimensional consistency and order-of-magnitude before accepting any numerical answer.
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