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Difficulty: HardElectrostatics and Electric Charges

A charged oil droplet of mass 3.2×10−15 kg3.2 \times 10^{-15}\text{ kg} remains stationary in a vacuum between two horizontal charged plates where there is a uniform vertical electric field of strength 2.0×104 N C−12.0 \times 10^4\text{ N C}^{-1}. Taking g=10 m s−2g = 10\text{ m s}^{-2} and the elementary charge e=1.6×10−19 Ce = 1.6 \times 10^{-19}\text{ C}, determine the number of excess electrons on the droplet.

Answer: 10

Answer

The number of excess electrons on the droplet is 10.
The droplet is in mechanical equilibrium under two equal and opposite forces: the downward gravitational force W=mgW = mg and the upward electric force Fe=qEF_e = qE. Setting qE=mgqE = mg gives q=mgE=1.6×10−18 Cq = \frac{mg}{E} = 1.6 \times 10^{-18}\text{ C}. By the quantization of charge (q=Neq = Ne), dividing this charge by the elementary charge e=1.6×10−19 Ce = 1.6 \times 10^{-19}\text{ C} gives exactly 10 excess electrons.

Step-by-Step Solution

1
Calculate the gravitational force (weight) acting on the droplet.
W=mg=(3.2×10−15 kg)×(10 m s−2)=3.2×10−14 NW = mg = (3.2 \times 10^{-15}\text{ kg}) \times (10\text{ m s}^{-2}) = 3.2 \times 10^{-14}\text{ N}.
For stationary equilibrium, weight provides the downward vertical force.
2
Apply the equilibrium condition to find the electric force.
Fe=W=3.2×10−14 NF_e = W = 3.2 \times 10^{-14}\text{ N}.
The net vertical force must be zero for the droplet to remain suspended.
3
Determine the charge qq using Fe=qEF_e = qE.
q=FeE=3.2×10−14 N2.0×104 N C−1=1.6×10−18 Cq = \frac{F_e}{E} = \frac{3.2 \times 10^{-14}\text{ N}}{2.0 \times 10^4\text{ N C}^{-1}} = 1.6 \times 10^{-18}\text{ C}.
Electric field strength relates force and charge.
4
Calculate the number of elementary charges using charge quantization q=Neq = Ne.
N=qe=1.6×10−18 C1.6×10−19 C=10N = \frac{q}{e} = \frac{1.6 \times 10^{-18}\text{ C}}{1.6 \times 10^{-19}\text{ C}} = 10.
Electric charge exists in discrete integer multiples of the elementary charge ee.

Key Concept

Equilibrium between electrostatic and gravitational forces combined with charge quantization.
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Topics

Electrostatics and Electric Charges

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