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.
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 \( 11.24 \, \mathrm{mN} \).