14. III A small plastic sphere with a charge of -8.0nC is near another small plastic sphere with a charge of -10.0nC If the spheres repel one another with a force of magnitude 5.6times 10^-4 N, what is the distance between the spheres?

To solve this problem, we can use Coulomb's law, which states that the force between two charged objects is proportional to the product of their charges and inversely proportional to the square of the distance between them.<br /><br />The formula for Coulomb's law is:<br /><br />$F = \frac{k \cdot |q_1| \cdot |q_2|}{r^2}$<br /><br />where:<br />- $F$ is the force between the charges (in Newtons)<br />- $k$ is the Coulomb constant ($8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2$)<br />- $q_1$ and $q_2$ are the charges of the two objects (in Coulombs)<br />- $r$ is the distance between the charges (in meters)<br /><br />Given information:<br />- Charge of the first sphere, $q_1 = -8.0 \, \text{nC} = -8.0 \times 10^{-9} \, \text{C}$<br />- Charge of the second sphere, $q_2 = -10.0 \, \text{nC} = -10.0 \times 10^{-9} \, \text{C}$<br />- Force between the spheres, $F = 5.6 \times 10^{-4} \, \text{N}$<br /><br />We need to find the distance between the spheres, $r$.<br /><br />Rearranging the Coulomb's law formula to solve for $r$, we get:<br /><br />$r = \sqrt{\frac{k \cdot |q_1| \cdot |q_2|}{F}}$<br /><br />Substituting the given values:<br /><br />$r = \sqrt{\frac{(8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2) \cdot (8.0 \times 10^{-9} \, \text{C}) \cdot (10.0 \times 10^{-9} \, \text{C})}{5.6 \times 10^{-4} \, \text{N}}}$<br /><br />$r = \sqrt{\frac{(8.99 \times 10^9) \cdot (8.0 \times 10^{-9}) \cdot (10.0 \times 10^{-9})}{5.6 \times 10^{-4}}}$<br /><br />$r = \sqrt{\frac{7.192 \times 10}}{5.6 \times 10^{-4}}}$<br /><br />$r = \sqrt{1.2857 \times 10^{-3}}$<br /><br />$r \approx 0.036 \, \text{m}$<br /><br />Therefore, the distance between the spheres is approximately $0.036 \, \text{m}$ or $3.6 \, \text{cm}$.