Question

Two charged spheres are held a distance, r, apart. One sphere has a charge of +4µC, and the other sphere has a
charge of +8µC. Compare the force of the +4µC sphere on the +8µC sphere with the force of the +8µC sphere
on the +4µC sphere.

Answer 1

Two charged spheres are held a distance, $r$, apart. One sphere has a charge of $+4\mu C$, and the other sphere has a charge of $+8\mu C$. Compare the force of the $+4\mu C$ sphere on the $+8\mu C$ sphere with the force of the $+8\mu C$ sphere on the $+4\mu C$ sphere.

Coulomb's law states that the force $F$ on a charge, $q_{1}$ at position $r_{1}$, experiencing an electric field due to the presence of another charge, $q_{2}$ at position $r_{2}$ in vacuum is:

$$\vec{F} = \frac{q_{1} q_{2}}{4 \pi \varepsilon_{0}} \frac{\widehat{r_{12}}}{\widehat{r_{12}}^{2}}$$

where $\overrightarrow{r_{12}} = \overrightarrow{r_1} -\overrightarrow{r_2}$ and $\varepsilon_0$ is the electric constant. This is simply the scalar definition of Coulomb's law with the direction given by the unit vector, $\widehat{\vec{r}_{12}}$, parallel with the line from charge $q_{2}$ to charge $q_{1}$.

The Coulomb's law equation provides an accurate description of the force between two objects whenever the objects act as point charges. A charged conducting sphere interacts with other charged objects as though all of its charge were located at its center. While the charge is uniformly spread across the surface of the sphere, the center of charge can be considered to be the center of the sphere. The sphere acts as a point charge with its excess charge located at its center.

Magnitude of force in our case:

$$F = \frac{8 \mu C * 4 \mu C}{4 \pi \varepsilon_{0} r^{2}}$$

Therefore, the forces are force pairs of each other so they will always be equal in magnitude and opposite in direction.

Answer: equal in magnitude and opposite in direction.