Chapter 16, Problem 11P

(a)

To determine

To Find: The charge of each to maximize the electric force between them.

Answer to Problem 11P

  $\frac{Q_T}{2}$ , $\frac{Q_T}{2}$

Explanation of Solution

Given:

Total Charge $=Q_T$

Formula used:

Electric force can be obtained by:

  $F=\frac{k{q}_1{q}_2}{r^2}$

Here, k is the Coulomb’s constant, $q_1$ and $q_2$ are the charges and r is the distance between them.

Calculation:

Let the charge of 1 particle be $q_1=q$ , then charge of the second particle would be $q_2=Q_T-q$ . Electric force between the two charges would be:

  $F=\frac{kq\left(Q_T-q\right)}{r^2}$

To find the charges for which the electric force would be maximum, find the extremum. Differentiate with respect to charge q.

  $\begin{array}{l}\frac{dF}{dq}=\frac{d}{dq}\left\{\frac{kq\left(Q_T-q\right)}{r^2}\right\}=0\\ \frac{dF}{dq}=\frac{k{Q}_T-2q}{r^2}=0\\ q=\frac{Q_T}{2}\end{array}$

Conclusion:

Thus, when both the particles have the same amount of charge i.e. $\frac{Q_T}{2}$ .

(b)

To determine

To Find: The charge of each to minimize the electric force between them.

Answer to Problem 11P

  $Q_T$ ,0

Explanation of Solution

Given:

Total Charge $=Q_T$

Formula used:

Electric force can be obtained by:

  $F=\frac{k{q}_1{q}_2}{r^2}$

Here, k is the Coulomb’s constant, $q_1$ and $q_2$ are the charges and r is the distance between them.

Calculation:

Let the charge of 1 particle be $q_1=q$ , then charge of the second particle would be $q_2=Q_T-q$ . Electric force between the two charges would be:

  $F=\frac{kq\left(Q_T-q\right)}{r^2}$

The electric force will be minimum when one of charge would be zero. So, there will be zero electric force.

  $\begin{array}{l}{q}_1=0\\ q_2=Q_T\end{array}$

Conclusion:

Thus, the electric force would be minimum when one of the particles has zero charge.