Problem 7 - A Complete Resource Book in Physics for JEE Main

Question

The electric field intensity at a point at a distance \(2 \mathrm{~m}\) from a charge \(q\) is \(E\). The amount of work done in bringing a charge of 2 coulomb from infinity to this point will be (A) \(2 E \mathrm{~J}\) (B) \(4 E \mathrm{~J}\) (C) \(\frac{E}{2} \mathrm{~J}\) (D) \(\frac{E}{4} \mathrm{~J}\)

Step-by-Step Solution

Answer

The work done in bringing a charge of 2 coulombs from infinity to a point at a distance 2 meters from a charge q is (B) \(4 E \mathrm{~J}\). This is calculated by first finding the charge q using the electric field formula, then calculating the electric potential, and finally computing the work done using the test charge and the electric potential.

Step 1: Find the charge q

To find the charge q, we use the formula for the electric field: \(E = \frac{k * q}{r^2}\), where E is the electric field intensity, k is Coulomb's constant (\(8.9875 * 10^9 \frac{Nm^2}{C^2}\)), and r is the distance from the charge (2 meters). Rearranging this equation to solve for q, we get: \(q = \frac{E * r^2}{k}\)

Step 2: Calculate the electric potential

To calculate the electric potential (V), we use the formula: \(V = \frac{k * q}{r}\) Substitute the value of q obtained in step 1: \(V = \frac{k * E * r^2}{k * r}\) The k terms cancel out, and we get: \(V = E * r\) Since r is 2 meters, the electric potential is: \(V = 2 * E\)

Step 3: Compute the work done

Finally, we can calculate the amount of work done (W) to bring a 2 coulomb charge from infinity to a point 2 meters from charge q. The formula for calculating the work done is: \(W = q₀ * V\) Substitute the value of V obtained in step 2 and q₀ = 2 C: \(W = 2 * (2 * E)\) Therefore, the work done is: \(W = 4 * E\) The correct answer is (B) \(4 E \mathrm{~J}\).