Question

Two point charges \({{\rm{q}}_{\rm{1}}}\) and \({{\rm{q}}_{\rm{2}}}\) are \({\rm{3}}{\rm{.00 m}}\) apart, and their total charge is \({\rm{20 \mu C}}\). (a) If the force of repulsion between them is \(0.075{\rm{ N}}\), what are magnitudes of the two charges? (b) If one charge attracts the other with a force of \(0.525{\rm{ N}}\), what are the magnitudes of the two charges? Note that you may need to solve a quadratic equation to reach your answer.

Short answer

(a) The magnitude of one charge is \(15.0{\rm{ \mu C}}\) and the other is \(5.0{\rm{ \mu C}}\).

(b) The magnitude of one charge is \(35.0{\rm{ \mu C}}\) and the other is \( - 15.0{\rm{ \mu C}}\).

Step-by-step solution

Charge

An object is said to be charged when it has unequal number of electrons and protons.

Repulsive force

(a)

When two positive charges are separated by some distance in a system, they exert a repulsive force on each other. The net charge of the system is given as,

\({q_t} = {q_1} + {q_2}\)

Here, \({q_t}\) is the total charge in the system \(\left( {{q_t} = 20{\rm{ \mu C}}} \right)\), \({q_1}\) and \({q_2}\) are two positive charges.

The charge \({q_2}\) is given as,

\({q_2} = {q_t} - {q_1}\)

The force of repulsion between \({q_1}\) and \({q_2}\) is given as,

\[{F_r} = \frac{{K{q_1}{q_2}}}{{{r^2}}}\]

Here, \[{F_r}\] is the magnitude of repulsive force \(\left( {{F_r} = 0.075{\rm{ }}N} \right)\), K is the electrostatic force constant \(\left( {K = 9 \times {{10}^9}{\rm{ N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}} \right)\), \({q_1}\) and \({q_2}\) are two positive charges, and r is the separation between \({q_1}\) and \({q_2}\) \(\left( {r = 3.00{\rm{ m}}} \right)\).

From equation (1.1) and (1.2),

\[{F_r} = \frac{{K{q_1}\left( {{q_t} - {q_1}} \right)}}{{{r^2}}}\]

Substituting all known values,

\[\begin{aligned}\left( {0.075} \right) &= \frac{{\left( {9 \times {{10}^9}} \right) \times \left[ {{q_1} \times \left( {20 \times {{10}^{ - 6}}} \right) - q_1^2} \right]}}{{{{\left( 3 \right)}^2}}}\\7.5 \times {10^{ - 11}} &= {q_1} \times \left( {20 \times {{10}^{ - 6}}} \right) - q_1^2\\0 &= q_1^2 - {q_1} \times \left( {20 \times {{10}^{ - 6}}} \right) + 7.5 \times {10^{ - 11}}\end{aligned}\]

The roots of the following quadratic equation are \({q_1} = 15.0{\rm{ }}\mu C\) or \({q_1} = 5.0{\rm{ }}\mu C\).

If \({q_1} = 15.0{\rm{ }}\mu C\), then from equation (1.1) \({q_2} = 5.0{\rm{ }}\mu C\).

If \({q_1} = 5.0{\rm{ }}\mu C\), then from equation (1.1) \({q_2} = 15.0{\rm{ }}\mu C\).

Hence, magnitude of one charge is \(15.0{\rm{ }}\mu C\) and the other is \(5.0{\rm{ }}\mu C\).

Attractive force

(b)

When two opposite charges are separated by some distance in a system, they exert an attractive force on each other. The net charge of the system is given as,

\({q_t} = {q_1} - {q_2}\)

Here, \({q_t}\) is the total charge in the system \(\left( {{q_t} = 20{\rm{ }}\mu C} \right)\), \({q_1}\) is the positive charge and \({q_2}\) is the negative charge.

The charge \({q_2}\) is given as,

\[{q_2} = {q_1} - {q_t}\]

The force of attraction between \({q_1}\) and \({q_2}\) is given as,

\[{F_a} = \frac{{K{q_1}{q_2}}}{{{r^2}}}\]

Here, \[{F_a}\] is the magnitude of attractive force \(\left( {{F_a} = 0.525{\rm{ N}}} \right)\), K is the electrostatic force constant \(\left( {K = 9 \times {{10}^9}{\rm{ N}} \cdot {{\rm{m}}^{\rm{2}}}{\rm{/}}{{\rm{C}}^{\rm{2}}}} \right)\), \({q_1}\) is the positive charge, \({q_2}\) is the negative charge, and r is the separation between \({q_1}\) and \({q_2}\) \(\left( {r = 3.00{\rm{ m}}} \right)\).

From equation (1.3) and (1.4),

\[{F_a} = \frac{{K{q_1}\left( {{q_1} - {q_t}} \right)}}{{{r^2}}}\]

Substituting all known values,

\(\begin{aligned} 0.525 &= \frac{{\left( {9 \times {{10}^9}} \right) \times \left[ {q_1^2 - \left( {20 \times {{10}^{ - 6}}} \right){q_1}} \right]}}{{{{\left( {3.00} \right)}^2}}}\\5.25 \times {10^{ - 10}} &= q_1^2 - \left( {20 \times {{10}^{ - 6}}} \right){q_1}\\0 &= q_1^2 - \left( {20 \times {{10}^{ - 6}}} \right){q_1} - 5.25 \times {10^{ - 10}}\end{aligned}\)

The roots of the following quadratic equation are \({q_1} = 35.0{\rm{ \mu C}}\) or \({q_1} = - 15.0{\rm{ \mu C}}\).

If \({q_1} = 35.0{\rm{ \mu C}}\), then from equation (1.3) \({q_2} = - 15.0{\rm{ \mu C}}\).

If \({q_1} = - 15.0{\rm{ \mu C}}\), then from equation (1.3) \({q_2} = 35.0{\rm{ \mu C}}\).

Hence, the magnitude of one charge is \(35.0{\rm{ \mu C}}\) and the other is \( - 15.0{\rm{ \mu C}}\).