Problem 17 - A Complete Resource Book in Physics for JEE Main

Question

A charge \(Q\) is divided into two parts of magnitude \(q\) and \(Q-q\). If the coulomb repulsion between them when they are separated at some distance is to be maximum, the ratio of \(\frac{Q}{q}\) should be (A) 2 (B) \(1 / 2\) (C) 4 (D) \(1 / 4\)

Step-by-Step Solution

Answer

The ratio of $\frac{Q}{q}$ should be 2 for the Coulomb repulsion between the charges to be maximum. The answer is (A) 2.

Step 1: Write down the expression for Coulomb repulsion

First, we need to write down the expression for Coulomb repulsion (F) between the two charges (q and Q - q) separated by some distance (r). The general expression for Coulomb repulsion is given by: \[F = k \frac{q_1 \cdot q_2}{r^2}\] In our case, \(q_1 = q\) and \(q_2 = Q - q\). Substituting these values we get: \[F = k \frac{q(Q - q)}{r^2}\]

Step 2: Differentiate the expression with respect to q

In this step, we are going to differentiate the above expression for F with respect to q. This will help us find the maximum value of F. We have: \[F = k \frac{q(Q - q)}{r^2}\] Differentiating with respect to q (we can treat Q as a constant while differentiating): \[\frac{dF}{dq} = k \frac{Q - 2q}{r^2}\]

Step 3: Set the derivative equal to zero and solve for q

To find the maximum value for F, we need to set its derivative equal to zero and solve for the unknown variable, q. So: \[\frac{dF}{dq} = 0\] \[k \frac{Q - 2q}{r^2} = 0\] The above equation implies that: \[Q - 2q = 0\] Solving for q: \[q = \frac{Q}{2}\]

Step 4: Find the ratio of Q/q

Finally, we need to find the ratio of Q/q: \[\frac{Q}{q} = \frac{Q}{\frac{Q}{2}} = 2\] So the ratio of Q/q should be 2 for the Coulomb repulsion between the charges to be maximum. The answer is (A) 2.