Question 144: Electrostatics & Capacitance

Question

Two metal plates
(A, B)
are kept horizontally with separation of
$ \left( \frac{12}{\pi} \right) $ cm
, with plate A on the top.
An atomizer jet sprays oil (density
1.5 g/cm
3
) droplets of radius
1 mm
horizontally. All oil droplets carry a charge
5 nC
. The potentials
V
A
and
V
B
are required on plates A and B respectively in order to ensure the droplets do not descend. The values of
V
A
and
V
B
are ______.
(Neglect the air resistance to the droplets and take
g = 10 m/s
2
)

Accepted Answer

For the droplet to
not descend
, the electric force on it must balance its weight.
So, we use
$$ qE = mg $$
where:
$q = 5\,	ext{nC} = 5 	imes 10^{-9}\,	ext{C}$
$m = ho 	imes 	ext{volume}$
Now calculate the mass of one oil droplet.
Given:
density of oil $= 1.5\,	ext{g/cm}^3 = 1500\,	ext{kg/m}^3$
radius $r = 1\,	ext{mm} = 10^{-3}\,	ext{m}$
Volume of droplet:
$$ V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (10^{-3})^3 = \frac{4}{3}\pi 	imes 10^{-9}\,	ext{m}^3 $$
Mass:
$$ m = ho V = 1500 	imes \frac{4}{3}\pi 	imes 10^{-9} $$
$$ m = 2000\pi 	imes 10^{-9} = 2\pi 	imes 10^{-6}\,	ext{kg} $$
Weight:
$$ mg = 2\pi 	imes 10^{-6} 	imes 10 = 2\pi 	imes 10^{-5}\,	ext{N} $$
Now from equilibrium:
$$ qE = mg $$
$$ E = \frac{mg}{q} = \frac{2\pi 	imes 10^{-5}}{5	imes 10^{-9}} $$
$$ E = \frac{2\pi}{5} 	imes 10^4 = 4000\pi\,	ext{V/m} $$
Distance between plates:
$$ d = \frac{12}{\pi}\,	ext{cm} = \frac{12}{\pi}	imes 10^{-2}\,	ext{m} $$
Potential difference needed:
$$ |V_A - V_B| = Ed $$
$$ |V_A - V_B| = 4000\pi 	imes \frac{12}{\pi}	imes 10^{-2} $$
$$ |V_A - V_B| = 480\,	ext{V} $$
Now decide which plate should be at higher potential.
The droplet is positively charged, so electric force is in the direction of electric field.
To balance weight, electric force must act
upward
.
Electric field goes from higher potential to lower potential, so for upward electric field:
lower plate $B$ must be at higher potential
upper plate $A$ must be at lower potential
So we need
$$ V_B - V_A = 480\,	ext{V} $$
Among the options, this is satisfied by
$$ V_A = 100\,	ext{V}, \quad V_B = 580\,	ext{V} $$
Hence, the correct answer is:
$$ \boxed{V_A = 100\,	ext{V}, \quad V_B = 580\,	ext{V}} $$
So,
Option A
is correct.

Answer

100 V and 580 V

Answer

580 V and 100 V

Answer

60 V and 400 V

Answer

0 V and -200 V

Electrostatics & Capacitance

Choose the correct answer: