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Question #42652

Difficulty: 8/10 | 🟢 Free

The Question

A solid non-conducting sphere of radius RR has a uniform volume charge density ρ\rho. A small electric dipole with dipole moment p⃗=p0j^\vec{p} = p_0 \hat{j} is placed at a point AA inside the sphere. The position vector of point AA from the center of the sphere is r⃗=R2i^\vec{r} = \frac{R}{2} \hat{i}, where i^,j^,k^\hat{i}, \hat{j}, \hat{k} are standard Cartesian unit vectors. What is the net electric force F⃗\vec{F} and the net torque τ⃗\vec{\tau} acting on the dipole? (Assume the presence of the dipole does not alter the charge distribution of the sphere, and ε0\varepsilon_0 is the permittivity of free space).

Options

  • A.

    F⃗=ρp03ε0j^,τ⃗=−ρp0R6ε0k^\vec{F} = \frac{\rho p_0}{3\varepsilon_0} \hat{j}, \quad \vec{\tau} = -\frac{\rho p_0 R}{6\varepsilon_0} \hat{k}

  • B.

    F⃗=ρp03ε0i^,τ⃗=ρp0R6ε0k^\vec{F} = \frac{\rho p_0}{3\varepsilon_0} \hat{i}, \quad \vec{\tau} = \frac{\rho p_0 R}{6\varepsilon_0} \hat{k}

  • C.

    F⃗=0⃗,τ⃗=−ρp0R6ε0k^\vec{F} = \vec{0}, \quad \vec{\tau} = -\frac{\rho p_0 R}{6\varepsilon_0} \hat{k}

  • D.

    F⃗=ρp06ε0j^,τ⃗=−ρp0R3ε0k^\vec{F} = \frac{\rho p_0}{6\varepsilon_0} \hat{j}, \quad \vec{\tau} = -\frac{\rho p_0 R}{3\varepsilon_0} \hat{k}

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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

Non-conducting Charged Sphere|Dipole in Uniform Field|Vector Decomposition Ambiguity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between the local value of an electric field and its spatial gradient when evaluating dipole interactions.

A common trap is assuming that because the electric field is purely radial, the force on the dipole must also be radial (or zero, if they incorrectly treat the field as locally uniform).

Students often misapply the 1D force formula F=pdEdrF = p \frac{dE}{dr} along the radial direction, yielding an incorrect force vector.

By requiring the full vector operator F⃗=∇(p⃗⋅E⃗)\vec{F} = \nabla(\vec{p} \cdot \vec{E}), the problem exposes the vector decomposition ambiguity, proving that the force on a dipole inside a uniform charge density is constant and strictly parallel to the dipole moment itself, regardless of its position.

Quick Facts About This Question

Question Information

SubjectPhysics
ChapterElectric Charges and Fields - Class 12
ConceptNon-conducting Charged Sphere
Difficulty8
Cognitive Load7
Target Solve Time160 sec
AccessFree