Question

What is the charge on the ball bearing?
A small glass bead has been charged to +20 nC. A tiny ball bearing 1.0 cm above the bead feels a 0.018 N downward electric force. What is the charge on the ball bearing?

Answer 1

The magnitude of electric force between two charges is directly proportional to the product of the charges and inversely proportional to the distance between two charges.

$$ F=K \frac{\left|q_{1}\right|\left|q_{2}\right|}{r^{2}} $$

Here, \(q_{1}\) is the amount of first charge, \(q_{2}\) is the amount of the second charge, \(r\) is the

separation between two charges and \(K\) is electric constant.

Solve for \(q_{2}\)

$$ \left|q_{2}\right|=\frac{F r^{2}}{K\left|q_{1}\right|} \quad \ldots \ldots $$

Convert the unit for electric charge from \(\mathrm{nC}\) to \(\mathrm{C}\).

$$ \begin{aligned} q &=+20 \mathrm{nC} \\ &=(+20 \mathrm{nC})\left(\frac{1.0 \times 10^{-9} \mathrm{C}}{1.0 \mathrm{nC}}\right) \\ &=+20 \times 10^{-9} \mathrm{C} \end{aligned} $$

Convert the unit for distance from \(\mathrm{cm}\) to \(\mathrm{m}\).

$$ r=1.0 \mathrm{~cm} $$

$$ \begin{array}{l} =(1.0 \mathrm{~cm})\left(\frac{1.0 \times 10^{-2} \mathrm{~m}}{1.0 \mathrm{~cm}}\right) \\ =1.0 \times 10^{-2} \mathrm{~m} \end{array} $$

Substitute \(+20 \times 10^{-9} \mathrm{C}\) for \(q_{1}, 1.0 \times 10^{-2} \mathrm{~m}\) for \(r\) and \(9.0 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}\) for \(K\) in equation (1),

solve for \(F\).

$$ \begin{aligned} \left|q_{2}\right| &=\frac{F r^{2}}{K\left|q_{1}\right|} \\ &=\frac{(0.018 \mathrm{~N})\left(1.0 \times 10^{-2} \mathrm{~m}\right)^{2}}{\left(9.0 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}\right)\left(20 \times 10^{-9} \mathrm{C}\right)} \\ &=10.0 \times 10^{-9} \mathrm{C} \\ &=\left(10.0 \times 10^{-9} \mathrm{C}\right)\left(\frac{1.0 \mathrm{nC}}{1.0 \times 10^{-9} \mathrm{C}}\right) \\ =& 10.0 \mathrm{nC} \end{aligned} $$

As the force on the metal ball is downward and towards the glass bead, sign of charge on the

metal ball and glass bead must be opposite. Therefore, the charge on the metal ball should be negative.

Therefore, the charge on the metal bead is \(-10.0 \mathrm{nC}\).