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.
[tex]F = k_e\frac{Q_1Q_2}{r^2}[/tex], 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
[tex]F = k_e\frac{Q_1(Q_t - Q_1)}{r^2}[/tex]
Simplify to obtain,
[tex]F = \frac{k_e}{r^2} (Q_tQ_1 - Q_1^2)[/tex]
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,
[tex]\frac{dF}{dQ_1} = \frac{k_e}{r^2} (Q_t - 2Q_1)[/tex]
Equate the differential to 0, to obtain the value of Q1 for which F is the maximum.
[tex]\frac{k_e}{r^2} (Q_t - 2Q_1) = 0\\Q_t - 2Q_1 = 0\\2Q_1 = Q_t\\Q1 = \frac{Q_t}{2}[/tex]
It follows that
[tex]Q_2 = \frac{Q_t}{2}[/tex].
