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Question #43077

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical region of space $x^2 + y^2 < R_s^2$ contains a uniform time-varying magnetic field $\vec{B} = \alpha t \hat{k}$ (where $\alpha > 0$ is a constant). Outside this region, the magnetic field is zero. A non-conducting solid sphere of radius $R$ (where $R \ll R_s$ ) has a uniform volume charge density and a total charge $Q$ . The sphere is centered at coordinates $(d, 0, 0)$ such that $d + R < R_s$ . Find the interaction electric energy $U_{int} = \int \epsilon_0 (\vec{E}_{stat} \cdot \vec{E}_{ind}) dV$ stored within the hemisphere $y > 0$ of the charged sphere, where $\vec{E}_{stat}$ is the static electric field of the sphere and $\vec{E}_{ind}$ is the induced electric field.

Options

A.
$-\frac{Q \alpha d R}{32}$
B.
$+\frac{Q \alpha d R}{32}$
C.
$-\frac{Q \alpha d R}{16}$
D.
$0$

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

Transformer Flux Linkage|Energy Density Locality|Non-conducting Charged Sphere

Why This Question Is Tricky

Common Mistakes

This question tests the principle of energy density superposition and the correct reference frame for induced electric fields.

A common trap is assuming the induced electric field is locally centered around the charged object itself (which would incorrectly yield zero interaction energy everywhere).

Students must recognize that the induced E-field is globally determined by the magnetic field's axis of symmetry.

Furthermore, while the total interaction energy over the entire sphere vanishes due to symmetry (leading many to hastily guess 0), the local energy density cross-term $\epsilon_0 \vec{E}_{stat} \cdot \vec{E}_{ind}$ is non-zero and redistributes the electric energy, yielding a net negative interaction energy in the upper hemisphere.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Transformer Flux Linkage
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free