Before You Begin

Question #42810

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor has plates of area $A$ separated by a distance $d$ (in m). The space between the plates is filled with a non-uniform dielectric whose dielectric constant varies as $\kappa(x) = 1 + \frac{x}{d}$ , where $x$ is the distance from the positive plate ( $0 \le x \le d$ ). A narrow evacuated tunnel exists along the x-axis through the dielectric, which does not affect the macroscopic electric field distribution. The capacitor is connected to a battery that maintains a constant potential difference $V_0$ (in V) across the plates. A particle of mass $m$ (in kg) and positive charge $+q$ (in C) is released from rest at the positive plate ( $x=0$ ). Neglecting gravity and fringe effects, what is the speed of the particle when it reaches the midpoint between the plates ( $x = d/2$ )?

Options

A.
$\sqrt{ \frac{2 q V_0}{m} \frac{\ln(3/2)}{\ln 2} } \text{ m/s}$
B.
$\sqrt{ \frac{2 q V_0}{m} \ln(3/2) } \text{ m/s}$
C.
$\sqrt{ \frac{q V_0}{m} } \text{ m/s}$
D.
$\sqrt{ \frac{5 q V_0}{6 m} } \text{ m/s}$

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

Non-Uniform Dielectric Filling|Non-Uniform Acceleration Constraint|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to navigate non-uniform electric fields within varying dielectrics and exposes the common trap of misapplying constant acceleration kinematics ( $v^2 = u^2 + 2as$ ).

Students frequently make one of two fatal errors: they either assume the electric field is a uniform $E = V_0/d$ despite the dielectric, or they incorrectly assume $E(x) = V_0 / (d \cdot \kappa(x))$ without normalizing the field integral to the total potential difference $V_0$ .

By forcing the dielectric constant to vary spatially as $\kappa(x) = 1 + x/d$ , the electric field becomes non-linear.

To solve this, students must first use Gauss's Law for dielectrics to establish that the displacement field $D$ is constant, integrate $E(x)$ to find the correct boundary value constant, and finally apply the Work-Energy Theorem by integrating the spatially varying force over the particle's path.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Non-Uniform Dielectric Filling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free