3. Two small spheres of charge \( +5.0 \mu \mathrm{C} \) and \( -4.0 \mu \mathrm{C} \) are separated by a distance of \( 2.0 \mathrm{~m} \). Determine the magnitude of the force that each sphere exerts on the other.

Answer 1

To find the magnitude of the force each sphere exerts on the other, we can use Coulomb's Law:

\[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \]

where
\( F \) is the force between the charges,
\( k \) is Coulomb's constant (\( 8.99 \times 10^9 \, \mathrm{N \cdot m^2/C^2} \)),
\( q_1 \) and \( q_2 \) are the charges of the two spheres, and
\( r \) is the distance between the charges.

Plugging in the given values:

For the positive charge:
\[ F_1 = \frac{(8.99 \times 10^9) \cdot (5.0 \times 10^{-6}) \cdot (4.0 \times 10^{-6})}{(2.0)^2} \]
\[ F_1 = \frac{44.95 \times 10^3}{4} = 11.24 \times 10^3 \, \mathrm{N} = 11.24 \, \mathrm{mN} \]

For the negative charge:
\[ F_2 = \frac{(8.99 \times 10^9) \cdot (5.0 \times 10^{-6}) \cdot (4.0 \times 10^{-6})}{(2.0)^2} \]
\[ F_2 = \frac{44.95 \times 10^3}{4} = 11.24 \times 10^3 \, \mathrm{N} = 11.24 \, \mathrm{mN} \]

Therefore, the magnitude of the force that each sphere exerts on the other is \( 11.24 \, \mathrm{mN} \).