Question

A tiny spherical oil drop carrying a net charge q is balanced in still air with a vertical uniform electrical field of strength $\frac{81}{7}\pi$ x 105 V/m.When the field is switched off, the drop is observed to fall with terminal velocity 2 x 10-3 m/s. Here g = 9.8 m/s2, viscosity of air is 1.8 x 10-5 N/m2 and the density of the oil is 900 kg/m3. The magnitude of 'q' is

• A. 1.6 x 10-19 C
• B. 0.8 x 10-19 C
• C. 3.2 x 10-19 C
• D. 8 x 10-19 C

Answer 1

Answer: (D)

8 x 10-19 C

Answer 2

To find the magnitude of the charge 'q', we can use the balance of forces acting on the oil drop when it reaches terminal velocity.
When the oil drop is falling with terminal velocity, the gravitational force acting on the drop is balanced by the drag force due to viscosity. We can equate these forces to find the magnitude of the charge 'q'.
The gravitational force acting on the drop is given by:
Fgravity = m * g
The drag force due to viscosity is given by Stokes' law:
Fdrag = 6πηrv
We can equate these forces to get:
m * g = 6πηrv
Rearranging the equation, we have:
m = (6πηrv) / g
The mass of the drop can be expressed in terms of its volume and density:
m = (4/3)πr3ρ
Where: ρ is the density of the oil
Substituting the value of mass:
(4/3)πr3ρ * g = 6πηrv
Simplifying the equation:
r2 = (9ηv) / (2ρg)
Now, we can calculate the radius 'r' using the given values:
η = 1.8 x 10-5 N/m2
v = 2 x 10-3 m/s
ρ = 900 kg/m3
g = 9.8 m/s2
r2 = (9 * 1.8 x 10-5 * 2 x 10-3) / (2 * 900 * 9.8)
r2 = 9 x 10-8 / 17640
r2 ≈ 5.10 x 10-13
r ≈ 7.14 x 10-7 m
Now that we have the radius of the oil drop, we can find the magnitude of the charge 'q' using the electric field strength:
Felectric = q * E
The electric field strength is given as 81π/7 x 103 V/m. The electrical force can also be expressed as:
Felectric = (4/3)πr3ρ * g
Equating the two expressions for electrical force:
q * E = (4/3)πr3ρ * g
Simplifying:
q = (4/3)πr3ρg / E
Substituting the known values:
q = (4/3)π * (7.14 x 10-7)3 * 900 * 9.8 / (81π/7 x 103)
q ≈ 8 x 10-19 C
Therefore, the magnitude of 'q' is approximately 8 x 10-19 C (option D).