A 2.4·10⁻¹⁵ kg oil drop is in equilibrium between plates (d = 1 cm, U = 500 V). Find the charge of the drop and the number of electrons — Physics problems | PhysLabo
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Charge of an oil drop suspended between capacitor plates

Medium 10 grade Electricity / Electrostatics
Problem statement

An oil drop of mass $m = 2.4 \cdot 10^{-15}$ kg is in equilibrium between the horizontal plates of a parallel-plate capacitor. The distance between the plates is $d = 1$ cm, the voltage is $U = 500$ V, and the upper plate is positively charged (see the figure). Find the charge of the drop, determine its sign and the number of excess or missing electrons. The electron charge is $e = 1.6 \cdot 10^{-19}$ C, $g = 10$ m/s².

An oil drop between horizontal capacitor plates: distance d, voltage U, the upper plate is positive
Given
  • $m = 2.4 \cdot 10^{-15}$ kg
  • $d = 1$ cm $= 0.01$ m
  • $U = 500$ V
  • $e = 1.6 \cdot 10^{-19}$ C, $g = 10$ m/s²
  • $q = ?$, $N = ?$
Hints
Hint 1 / 2
The drop is in equilibrium: the electric force $qE$ balances $mg$. The field strength in a capacitor is $E = U/d$.
Hint 2 / 2
The field points from the positive plate to the negative one (downward), while the force must point upward — use this to find the sign of the charge.
Solution

The drop is at rest, so gravity is balanced by the electric force. The field between the plates is uniform, $E = U/d$, and points from the positive plate to the negative one — downward. The electric force must point up, that is, against the field, so the charge of the drop is negative.

The electric force qE balances gravity mg
The electric force qE balances gravity mg

1. Equilibrium condition:

$$qE = mg \;\Rightarrow\; |q| = \frac{mg}{E} = \frac{mgd}{U}$$

2. Substitute the numbers:

$$|q| = \frac{2.4 \cdot 10^{-15} \cdot 10 \cdot 0.01}{500} = 4.8 \cdot 10^{-19} \text{ C}$$

3. The number of electrons:

$$N = \frac{|q|}{e} = \frac{4.8 \cdot 10^{-19}}{1.6 \cdot 10^{-19}} = 3$$

The drop has an excess of three electrons.

Answer

$q = -4.8 \cdot 10^{-19}$ C (negative); an excess of three electrons

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