Question

32. Two point charges having a total charge Q are separated by a distance x. Find the value of each charge so that the
force they exert on one another is maximum.​

Answer 1

Given:

• 2 charges having total charge Q are separated by distance x .

To find:

• Value of each charge for force to be maximum?

Calculation:

Let one charge be q and the other will be (Q-q).

• Now, the electrostatic force will be :

$F = \dfrac{kq(Q - q)}{ {x}^{2} }$

Now, 1st order derivation w.r.t q :

$\implies F = \dfrac{k(Q q- {q}^{2} )}{ {x}^{2} }$

$\implies \: \dfrac{dF}{dq} = \dfrac{k}{ {x}^{2} } \times \dfrac{d(Q q- {q}^{2})}{dq}$

$\implies \: \dfrac{dF}{dq} = \dfrac{k}{ {x}^{2}}( Q - 2q)$

For max value, dF/dq should be zero :

$\implies \: \dfrac{dF}{dq} = \dfrac{k}{ {x}^{2}}( Q - 2q) = 0$

$\implies \: \dfrac{k}{ {x}^{2}}( Q - 2q) = 0$

$\implies \: Q - 2q = 0$

$\implies \: q = \dfrac{Q }{2}$

So, the charges have to Q/2 and Q/2 for max force.