Problem 106 - A Complete Resource Book in Physics for JEE Main

Question

A dielectric slab of thickness \(d\) is inserted in a parallel plate capacitor whose negative plate is at \(x=0\) and positive plate is at \(x=3 d\). The slab is equidistant from the plates. The capacitor is given some charge. As \(x\) goes from 0 to \(3 d\), (A) the magnitude of the electric field remains the same. (B) the direction of the electric field remains the same. (C) the electric potential increases continuously. (D) the electric potential increases at first, then decreases and again increases.

Step-by-Step Solution

Answer

The correct answer is option (B), which states that the direction of the electric field remains the same.

Step 1: Identify the three regions in the capacitor

We will divide the capacitor into three regions: 1. Region I [0, d]: The space between the negative plate and the dielectric slab. 2. Dielectric Slab [d, 2d]: The dielectric slab region. 3. Region II [2d, 3d]: The space between the dielectric slab and the positive plate.

Step 2: Analyze the electric field in the regions

In a parallel plate capacitor, the electric field is uniform. However, when a dielectric slab is introduced, the electric field within the dielectric slab becomes weaker due to its dielectric property. Thus, we can summarize the electric field in the different regions as follows: 1. Region I [0, d]: E_1 is present between the negative plate and the dielectric slab and is directed towards the positive plate. 2. Dielectric Slab [d, 2d]: E_d is present within the dielectric slab and is also directed towards the positive plate, but its magnitude is smaller than E_1. 3. Region II [2d, 3d]: E_2 is present between the dielectric slab and the positive plate and has the same magnitude as E_1. Now, let's analyze the given options: (A) The magnitude of the electric field remains the same: As we can see, the magnitude of the electric field (E_1, E_d, E_2) is not the same in the different regions. Thus, option (A) is incorrect. (B) The direction of the electric field remains the same: In all three regions, the electric field is directed towards the positive plate. Thus, the direction remains the same throughout the capacitor. Option (B) is correct.

Step 3: Analyze the electric potential in the regions

Let's analyze the electric potential in the different regions: 1. Region I [0, d]: Electric potential decreases in the direction of the electric field E_1 from \(x=0\) to \(x=d\). 2. Dielectric Slab [d, 2d]: Electric potential still decreases in the direction of the electric field E_d from \(x=d\) to \(x=2d\), but at a slower rate due to the weaker electric field. 3. Region II [2d, 3d]: Electric potential continues to decrease in the direction of the electric field E_2 from \(x=2d\) to \(x=3d\). Now, let's analyze the given options: (C) The electric potential increases continuously: As we can see, the electric potential decreases continuously, not increases. Thus, option (C) is incorrect. (D) The electric potential increases at first, then decreases, and again increases: The electric potential continuously decreases as \(x\) goes from 0 to \(3d\). Thus, option (D) is also incorrect. In conclusion, the correct answer to the given exercise is option (B), which states that the direction of the electric field remains the same.