Before You Begin

Question #42912

Difficulty: 8/10 | 🟢 Free

The Question

A spherically symmetric charge distribution in free space has a volume charge density given by $\rho(r) = \frac{A}{r} e^{-\alpha r}$ for $r > 0$ , where $r$ is the radial distance from the origin, and $A$ and $\alpha$ are positive constants. Which of the following expressions correctly represents the magnitude of the electric field $E(r)$ at a distance $r$ from the origin?

Options

A.
$\frac{A}{\alpha^2 \epsilon_0 r^2} \left[ 1 - (1 + \alpha r) e^{-\alpha r} \right]$
B.
$\frac{A}{\alpha \epsilon_0 r^2} \left( 1 - e^{-\alpha r} \right)$
C.
$\frac{A}{\alpha^2 \epsilon_0 r^2} \left[ 1 + (\alpha r - 1) e^{-\alpha r} \right]$
D.
$\frac{A}{\alpha \epsilon_0 r} e^{-\alpha r}$

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

Continuous Charge Distribution|Dimensional Consistency and Transcendental Functions|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of Gauss's Law alongside dimensional analysis for transcendental charge distributions.

A common trap is incorrectly formulating the volume integral by dropping the $r^2$ term from the spherical volume element $d\tau = 4\pi r^2 dr$ , leading to an easier but physically meaningless integral (Option B).

Furthermore, students often fail to extract the implicit dimensional constraints from the transcendental function ( $[\alpha] = L^{-1}$ and $[A] = Q L^{-2}$ ).

By weaponizing dimensional homogeneity, the problem allows astute students to immediately eliminate options that violate $[E] = Q/(\epsilon_0 L^2)$ , while punishing those who blindly manipulate formulas without verifying the physicality of the resulting field, such as its finite limit at the origin.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Continuous Charge Distribution
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free