Of the charge Q initially on a tiny sphere, a portion q is to be transferred to a second, nearby sphere. Both spheres can be treated as particles and are fixed with a certain separation. For what value of q/Q will the electrostatic force between the two spheres be maximized?

#### Solution:

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The first particle will have a charge of q, and the secondÂ particle will have a charge of Q-q. (This is because a portion of q is transferred from the particle that had a charge of Q).

Let r represent the distance between the two particles.

Let us now write Coulomb’s law.

Substitute the values of charge into our equation.

To find the value which maximizes the electrostatic force, we must first take the derivative of our equation and set it to zero.

F(q)=k\dfrac{(q)(Q-q)}{r^2}

F'=\dfrac{k}{r^2}(Q-2q)

(Remember k and r are both constants)

Set it equal to zero to find the maximum.

0=Q-2q

We can now find the ratio.

\dfrac{q}{Q}=0.5

#### Final Answer: