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Chapter 13: Problem 17

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) \(1 / 4\)

Short Answer

Expert verified
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 by step solution

01

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: \[F = k \frac{q(Q - q)}{r^2}\]
02

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): \[\frac{dF}{dq} = k \frac{Q - 2q}{r^2}\]
03

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: \[q = \frac{Q}{2}\]
04

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.

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