Question

Two positive charges are fixed a distance apart.the sun of their charges is Qt.what charge must each have in order to maximise the electric force between them

Answer 1

Answer:

Both charges must have the same charge, Qt/2.

Explanation:

Let the two charges have charge Q1 and Q2, respectively.

Use Coulombs's Law to find an expression for the force between the two charges.

$F = k_e\frac{Q_1Q_2}{r^2}$, where

Ke is Coulomb's contant and

r is the distance between the charges.

We know from the question that

Q1 + Q2 = Qt

So,

Q2 = Qt - Q1

$F = k_e\frac{Q_1(Q_t - Q_1)}{r^2}$

Simplify to obtain,

$F = \frac{k_e}{r^2} (Q_tQ_1 - Q_1^2)$

In order to find the value of Q1 for which F is the maximum, we will use the optimization technique of calculus.

Differentiate F with respect to Q1,

$\frac{dF}{dQ_1} = \frac{k_e}{r^2} (Q_t - 2Q_1)$

Equate the differential to 0, to obtain the value of Q1 for which F is the maximum.

$\frac{k_e}{r^2} (Q_t - 2Q_1) = 0\\Q_t - 2Q_1 = 0\\2Q_1 = Q_t\\Q1 = \frac{Q_t}{2}$

It follows that

$Q_2 = \frac{Q_t}{2}$.