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Rivka Thorpe

Answered question

2021-05-09

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 a particles. For what value of q/Q will the electrostatic force between the two parts have 1/4 of the maximum possible value?

Answer & Explanation

oppturf

Skilled2021-05-11Added 94 answers

\frac{q}{Q} = 0.5

is not "the last step." This is the value of q that maximizes the force. To continue the problem:

q = \frac{Q}{2}

F_{max} = \frac{k {(\frac{Q}{2})}^{2}}{r^{2}} = \frac{k Q^{2}}{4 r^{2}}

F = \frac{1}{4} F_{max}

\frac{k q (Q - q)}{r^{2}} = \frac{1}{4} (\frac{k Q^{2}}{4 r^{2}})

Q q - q^{2} = \frac{Q^{2}}{16}

16 \frac{q}{Q} - 16 {(\frac{q}{Q})}^{2} - 1 = 0

This is a quadratic in q/Q, the solution of which is (using the positive root):

\frac{q}{Q} = \frac{1 + \sqrt{3}}{2} \approx 1.336

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