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Question #43313

Difficulty: 8/10 | 🟢 Free

The Question

A long cylindrical region of radius $R \text{ (in m)}$ contains a time-varying magnetic field directed parallel to the axis of the cylinder. The magnetic field is non-uniform and its magnitude is given by $B(r,t) = B_0 \left( 1 - \frac{r^2}{R^2} \right) \beta t$ for $0 \le r \le R$ , and $B(r,t) = 0 \text{ T}$ for $r > R$ . Here, $r \text{ (in m)}$ is the radial distance from the axis, $t \text{ (in s)}$ is time, and $B_0 \text{ (in T)}$ and $\beta \text{ (in s}^{-1}\text{)}$ are positive constants. What is the ratio of the magnitude of the induced electric field at a radial distance $r_1 = R/2$ to the magnitude of the induced electric field at a radial distance $r_2 = 2R$ ?

Options

A.
1.75
B.
1.50
C.
1.00
D.
2.00

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Induced Electric Fields|Flux and Surface Geometry|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply Faraday's Law in integral form to a region with a spatially non-uniform, time-varying magnetic field.

The primary trap is the misapplication of the formula $E = \frac{r}{2} \frac{dB}{dt}$ , which is only valid for spatially uniform magnetic fields.

If a student incorrectly uses the local magnetic field value at $r = R/2$ (which is $\frac{3}{4} B_0 \beta t$ ), they will calculate an induced field of $\frac{3}{16} B_0 \beta R$ , leading to the incorrect ratio of 1.50.

Another common trap is assuming the global average magnetic field ( $B_0/2$ ) applies uniformly across all sub-regions, which yields a ratio of 1.00.

The correct approach requires recognizing that the induced EMF depends strictly on the area integral of the magnetic field enclosed by the specific Amperian loop.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Induced Electric Fields
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free