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Chapter 16: Problem 34

Flux \(\phi\) (in weber) in a closed circuit of resistance \(10 \Omega\) varies with time \(t\) (in seconds) according to the equation \(\phi=6 t^{2}-5 t+1\). The magnitude of the induced current in the circuit at \(t=0.25 \mathrm{~s}\) is (A) \(0.2 \mathrm{~A}\) (B) \(0.6 \mathrm{~A}\) (C) \(0.8 \mathrm{~A}\) (D) \(1.2 \mathrm{~A}\)

Short Answer

Expert verified
The magnitude of the induced current in the circuit at \(t=0.25s\) is \(0.2 A\), which corresponds to option (A).

Step by step solution

01

Calculate the derivative of magnetic flux with respect to time.

Using the given equation for magnetic flux, \(\phi=6 t^{2}-5 t+1\), we can find the rate of change of magnetic flux with respect to time by taking the derivative with respect to time: \(\phi' = \frac{d\phi}{dt} = \frac{d(6t^2 - 5t + 1)}{dt}\) Applying the power rule, we get: \(\phi' = 12t - 5\)
02

Find the rate of change of magnetic flux at time t=0.25s

Substitute the given time, \(t=0.25s\), into the derivative equation: \(\phi'(0.25)=12(0.25)-5\) Evaluating this expression, we get: \(\phi'(0.25)=-2\)
03

Calculate the induced EMF using Faraday's law

Faraday's law states that the induced EMF is equal to the negative rate of change of magnetic flux: EMF = \(-\phi'\) Substituting the value we found for \(\phi'(0.25)\): EMF = \(-(-2)\) EMF = \(2 V\)
04

Calculate the induced current using Ohm's law

Ohm's law relates the current, resistance, and voltage by the following equation: \(I = \frac{EMF}{R}\) We are given that the resistance R = 10 惟. Substituting the EMF value and the resistance into the equation: \(I = \frac{2}{10}\) Simplifying, we get: \(I = 0.2 A\) The induced current in the circuit at \(t=0.25s\) is 0.2 A, which corresponds to option (A).

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