Question 167: Electrostats & Capacitance

Question

A tiny spherical oil drop carrying a net charge $$q$$ is balanced in still air with a vertical uniform electric field of strength $${{81\pi } \over 7} 	imes {10^5}\,\,V{m^{ - 1}}.$$ When the field is switched off, the drop is observed to fall with terminal velocity $$2 	imes {10^{ - 3}}\,\,m{s^{ - 1}}.$$ Given $$g = 9.8\,m\,{s^{ - 2}},$$ viscosity of the air $$ = 1.8 	imes {10^{ - 5}}\,\,Ns\,{m^{ - 2}}$$ and the density of coil $$=900$$ $$kg$$ $${m^{ - 3}},$$ the magnitude of $$q$$ is

Accepted Answer

The forces acting on the oil drop are its weight, buoyant force, and electrostatic force. The buoyant force on the oil drop is very small as compared to other two forces. Thus, the weight of the spherical oil drop is balanced by the electrostatic force
$$qE = mg$$
$$qE = {4 \over 3}\pi {r^3}ho g$$ ..... (i)
where, r = radius of oil drop, $$ho$$ = density of oil, q = charge on the oil drop
IN the absence of electric field,
IN equilibrium,
Viscous force on the drop = Weight of the drop
$$6\pi \eta r{v_T} = mg$$
$$6\pi \eta r{v_T} = {4 \over 3}\pi {r^3}ho g$$ ...... (ii)
where, v
T
= terminal velocity, $$\eta $$ = coefficient of viscosity of air
or $${r^2} = {{18 	imes \eta  	imes {v_T}} \over {4 	imes ho  	imes g}}$$
Substituting the given values, we get
$${r^2} = {{18 	imes 1.8 	imes {{10}^{ - 5}} 	imes 2 	imes {{10}^{ - 3}}} \over {4 	imes 900 	imes 9.8}}$$ or $$r = {3 \over 7} 	imes {10^{ - 5}}$$ m
From equation (i), we get $$q = {{4\pi {r^3}ho g} \over {3E}}$$
Substituting the given values, we get
$$q = {{4 	imes \pi  	imes {{\left( {{3 \over 7} 	imes {{10}^{ - 5}}} ight)}^3} 	imes 900 	imes 9.8} \over {3 	imes \left( {{{81\pi } \over 7} 	imes {{10}^5}} ight)}}$$
$$ = {{4 	imes \pi  	imes 7 	imes 3 	imes 3 	imes 3 	imes {{10}^{ - 15}} 	imes 900 	imes 9.8} \over {3 	imes 81\pi  	imes {{10}^5} 	imes 7 	imes 7 	imes 7}} = 8 	imes {10^{ - 19}}C$$

Answer

$$1.6 	imes {10^{ - 19}}C$$

Answer

$$3.2 	imes {10^{ - 19}}C$$

Answer

$$4.8 	imes {10^{ - 19}}C$$

Answer

$$8.0 	imes {10^{ - 19}}C$$

Electrostats & Capacitance

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