Before You Begin

Question #43691

Difficulty: 8/10 | 🟢 Free

The Question

A non-conducting solid sphere of radius $R$ is centered at the origin. It contains a radioactive isotope such that its volume charge density is given by $\rho(r, t) = \rho_0 \left(\frac{r}{R}\right) e^{-\lambda t}$ , where $r$ is the distance from the center, $t$ is the time, and $\lambda$ is the decay constant. A point charge $q$ is moved from the center of the sphere ( $r=0$ ) starting at $t=0$ to its surface ( $r=R$ ) along a radial line at a constant speed $v = \lambda R$ . What is the work done by the electric field on the point charge $q$ during this process?

Options

A.
$\frac{q \rho_0 R^2}{4\epsilon_0} \left( 2 - \frac{5}{e} \right)$
B.
$\frac{q \rho_0 R^2}{\epsilon_0} \left( \frac{1}{3} - \frac{1}{4e} \right)$
C.
$\frac{q \rho_0 R^2}{12\epsilon_0}$
D.
$\frac{q \rho_0 R^2}{12\epsilon_0 e}$

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

Radioactive Decay Law Dynamics|Coulombic Central Force|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Work-Energy Theorem in a time-varying electrostatic field.

A common lethal trap for students is assuming that the work done by an electric field is always equal to the negative change in potential energy ( $W = U_i - U_f$ ).

However, this is only true for static, conservative fields.

When the source charge distribution is time-dependent (due to radioactive decay), the electric potential $V(r,t)$ explicitly depends on time.

The correct approach requires integrating the instantaneous force over the path ( $W = \int \vec{F} \cdot d\vec{r}$ ).

Applying the static formula $U(0,0) - U(R, 1/\lambda)$ ignores the $\int (\partial U/\partial t) dt$ term and leads directly to the incorrect Option B.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Radioactive Decay Law Dynamics
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free