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Question #43397

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The Question

A straight wire of finite length $2L$ is placed along the $z$ -axis from $z = -L$ to $z = L$ . It carries a time-varying current $I(t) = \alpha t$ , where $\alpha$ is a positive constant. Assuming the quasi-static approximation is valid and that the electric field vanishes at infinity, what is the magnitude of the non-conservative induced electric field (ignoring any electrostatic contributions from charge accumulation at the ends) at a point $P$ located on the perpendicular bisector of the wire at a distance $r$ from its center?

Options

A.
$\frac{\mu_0 \alpha}{2\pi} \ln\left( \frac{\sqrt{r^2+L^2}+L}{r} \right)$
B.
$\frac{\mu_0 \alpha}{2\pi} \ln\left( \frac{\sqrt{r^2+L^2}}{r} \right)$
C.
$\frac{\mu_0 \alpha}{2\pi} \ln\left( \frac{L}{r} \right)$
D.
$\frac{\mu_0 \alpha}{4\pi} \ln\left( \frac{\sqrt{r^2+L^2}+L}{r} \right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Infinite Wire vs. Finite Wire|Induced Electric Fields|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

The problem tests the application of Faraday's Law in integral form for a dynamic induction scenario.

A common trap is assuming the magnetic field of an infinite wire ( $B = \mu_0 I / 2\pi r$ ) can be used.

However, integrating the infinite wire field to infinity yields a divergent flux, making the induced electric field undefined.

Students must recognize that the finite length of the wire is physically necessary to compute a finite flux linkage.

They must apply the Biot-Savart law for a finite wire and integrate the non-uniform, time-varying flux over a semi-infinite strip to correctly determine the induced electric field.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Infinite Wire vs. Finite Wire
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free