### Electrolytically Charged Sphere

Part of an apparatus known as a torsion balance, the spheres are electrically charged and the electrostatic force between them is measured by calculating the equilibrium distance between the two charged spheres.

#### Can You Apply It?

- When the spheres in the system are initially charged, the force between them when they are brought near one another is:

\begin{align*}F=(4 \varepsilon \pi a^2)\frac{V_1V_2}{r^2}\end{align*}

where \begin{align*}a\end{align*} is the radius of each of the spheres, \begin{align*}V_1\end{align*} and \begin{align*}V_2\end{align*} are the respective voltages and \begin{align*}r\end{align*} is the distance between the center of the spheres. One of the spheres is attached to a torsion spring that is calibrated to measure the to angle of deflection from equilibrium when a torque \begin{align*}\tau\end{align*} is applied. The spring is tightened to bring the charged sphere back to equilibrium, at which point the angle is measured. For small displacements the torque is related to the angle by

\begin{align*}\tau=k\theta\end{align*}

- The torque is a result of the force that is applied to the sphere over a distance \begin{align*}l\end{align*}, from the axis of rotation. This force takes the form of

\begin{align*}F=\frac{k}{l}\theta\end{align*}

#### Show What You Know

- What relationship exists between the angle of deflection and the voltage applied to the spheres?
- As the distance between the charged spheres increases, what do you expect to happen to the angle of deflection?
- What assumption is being made about the torque equation? (Hint: think about the angle between the force and the lever arm.)