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Question #42661

Difficulty: 8/10 | 🟢 Free

The Question

A uniform time-varying magnetic field $\vec{B} = \alpha t \hat{k}$ exists in a cylindrical region $x^2 + y^2 \le R^2$ , where $\alpha$ is a positive constant in $\text{T/s}$ , $t$ is time in $\text{s}$ , and $R$ is the radius in $\text{m}$ . Outside this region, the magnetic field is zero. Consider two points $P(R, 0, 0)$ and $Q(0, R, 0)$ with coordinates given in meters. Let $W_1$ be the work done (in Joules) by the induced electric field on a point charge $+q$ (in Coulombs) as it is moved from $P$ to $Q$ along the path $P \to O \to Q$ (where $O$ is the origin). Let $W_2$ be the work done by the induced electric field on the same charge as it is moved from $P$ to $Q$ along the straight line segment connecting them. Which of the following gives the correct values for $W_1$ and $W_2$ ?

Options

A.
$W_1 = 0$ , $W_2 = -\frac{q \alpha R^2}{2}$
B.
$W_1 = -\frac{q \alpha R^2}{2}$ , $W_2 = -\frac{q \alpha R^2}{2}$
C.
$W_1 = 0$ , $W_2 = -\frac{\pi q \alpha R^2}{4}$
D.
$W_1 = \frac{q \alpha R^2}{2}$ , $W_2 = 0$

Check Your Answer

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

Potential Gradient Direction|Induced Electric Fields|Vector Calculus (Curl)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between conservative electrostatic fields and non-conservative induced electric fields.

A common trap is to incorrectly apply the concept of a scalar potential gradient ( $\vec{E} = -\nabla V$ ) to a time-varying magnetic flux scenario, leading students to assume that the work done is path-independent ( $W_1 = W_2$ ).

By forcing the evaluation of the line integral along two distinct paths, the problem highlights that $\nabla \times \vec{E} \neq 0$ .

Students who memorize formulas without understanding the Maxwellian framework will likely choose the trap option assuming path independence, failing to recognize that the induced EMF strictly depends on the area swept or enclosed by the specific path taken.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Potential Gradient Direction
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free