(a) Two point charges totaling\({\bf{8}}.{\bf{00}}{\rm{ }}{\bf{\mu C}}\)exert a repulsive force of\({\bf{0}}.{\bf{150}}{\rm{ }}{\bf{N}}\)on one another when separated by\({\bf{0}}.{\bf{500}}{\rm{ }}{\bf{m}}\). What is the charge on each? (b) What is the charge on each if the force is attractive?

Charge is a fundamental property of matter. An object is said to be charged when it has unequal number of electrons and protons.

The electrostatic force between two charges separated by some distance is given as,

\(F = \frac{{KQq}}{{{d^2}}}\)

Here,\(K\)is the electrostatic force constant\(\left( {K = 9 \times {{10}^9}{\rm{ N}} \cdot {{\rm{m}}^2}/{{\rm{C}}^2}} \right)\), and\(d\)is the separation between charges\(Q\)and\(q\).

(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\(\left( {{q_t} = 8.00{\rm{ }}\mu {\rm{C}}} \right)\),\({q_1}\)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\(\left( {{F_r} = 0.150{\rm{ N}}} \right)\),\(K\)is the electrostatic force constant\(\left( {K = 9 \times {{10}^9}{\rm{ N}} \cdot {{\rm{m}}^2}/{{\rm{C}}^2}} \right)\),\({q_1}\)and\({q_2}\)are two positive charges, and\(r\)is the separation between\({q_1}\)and\({q_2}\)\(\left( {r = 0.500{\rm{ m}}} \right)\).

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,

\(\begin{array}{c}\left( {0.150} \right) = \frac{{\left( {9 \times {{10}^9}} \right) \times \left[ {{q_1} \times \left( {8 \times {{10}^{ - 6}}} \right) - q_1^2} \right]}}{{{{\left( {0.500} \right)}^2}}}\\4.17 \times {10^{ - 12}} = {q_1} \times \left( {8 \times {{10}^{ - 6}}} \right) - q_1^2\\0 = q_1^2 - {q_1} \times \left( {8 \times {{10}^{ - 6}}} \right) + 4.17 \times {10^{ - 12}}\end{array}\)

The roots of the following quadratic equation are\({q_1} = 7.44{\rm{ }}\mu {\rm{C}}\)or\({q_1} = 0.56{\rm{ }}\mu C\).

If\({q_1} = 7.44{\rm{ }}\mu {\rm{C}}\), then from equation (1.1)\({q_2} = 0.56{\rm{ }}\mu {\rm{C}}\).

If\({q_1} = 0.56{\rm{ }}\mu C\), then from equation (1.1)\({q_2} = 7.44{\rm{ }}\mu {\rm{C}}\).

Hence, for repulsive force, one charge is\(7.44{\rm{ }}\mu {\rm{C}}\)whereas the other charge is\(0.56{\rm{ }}\mu {\rm{C}}\).

(b)

When two opposite charges are separated by some distance in a system, they exert a 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\(\left( {{q_t} = 8.00{\rm{ }}\mu {\rm{C}}} \right)\),\({q_1}\) 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\(\left( {{F_a} = 0.150{\rm{ N}}} \right)\),\(K\)is the electrostatic force constant\(\left( {K = 9 \times {{10}^9}{\rm{ N}} \cdot {{\rm{m}}^2}/{{\rm{C}}^2}} \right)\),\({q_1}\) is the positive charge,\({q_2}\)is the negative charge, and\(r\)is the separation between\({q_1}\)and\({q_2}\)\(\left( {r = 0.500{\rm{ m}}} \right)\).

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,

\(\begin{array}{c}0.150 = \frac{{\left( {9 \times {{10}^9}} \right) \times \left[ {q_1^2 - \left( {8 \times {{10}^{ - 6}}} \right){q_1}} \right]}}{{{{\left( {0.500} \right)}^2}}}\\4.17 \times {10^{ - 12}} = q_1^2 - \left( {8 \times {{10}^{ - 6}}} \right){q_1}\\0 = q_1^2 - \left( {8 \times {{10}^{ - 6}}} \right){q_1} - 4.17 \times {10^{ - 12}}\end{array}\)

The roots of the following quadratic equation are\({q_1} = 8.49{\rm{ }}\mu {\rm{C}}\)or\({q_1} = - 0.49{\rm{ }}\mu {\rm{C}}\).

If\({q_1} = 8.49{\rm{ }}\mu {\rm{C}}\), then from equation (1.3)\({q_2} = - 0.49{\rm{ }}\mu {\rm{C}}\).

If\({q_1} = - 0.49{\rm{ }}\mu {\rm{C}}\), then from equation (1.3)\({q_2} = 8.49{\rm{ }}\mu {\rm{C}}\).

Hence, for attractive force, one charge is\(8.49{\rm{ }}\mu {\rm{C}}\)whereas the other charge is\( - 0.49{\rm{ }}\mu {\rm{C}}\).