Two metal plates (A, B) are kept horizontally with separation of $$\frac{12}{\pi}$$ 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$$^2$$)

Density of oil $$\rho = 1.5 \,\text{g cm}^{-3} = 1.5 \times 10^{3}\,\text{kg m}^{-3}$$

Radius of droplet $$r = 1\,\text{mm} = 1 \times 10^{-3}\,\text{m}$$

Volume of a sphere $$V = \frac{4}{3}\pi r^{3} = \frac{4}{3}\pi(1 \times 10^{-3})^{3} = \frac{4}{3}\pi \times 10^{-9}\,\text{m}^{3}$$

Mass of one droplet
$$m = \rho V = 1.5 \times 10^{3} \times \frac{4}{3}\pi \times 10^{-9} = 2\pi \times 10^{-6}\,\text{kg}$$

Weight acting downward
$$W = mg = 2\pi \times 10^{-6} \times 10 = 2\pi \times 10^{-5}\,\text{N} \approx 6.28 \times 10^{-5}\,\text{N}$$

Charge on each droplet $$q = 5\,\text{nC} = 5 \times 10^{-9}\,\text{C}$$

For the droplet to float, the upward electric force must equal its weight:
$$qE = mg \quad\Longrightarrow\quad E = \frac{mg}{q} = \frac{6.28 \times 10^{-5}}{5 \times 10^{-9}} = 1.26 \times 10^{4}\,\text{V m}^{-1}$$

Separation between the plates
$$d = \frac{12}{\pi}\,\text{cm} = \frac{12}{\pi} \times 10^{-2}\,\text{m} = \frac{0.12}{\pi}\,\text{m} \approx 3.82 \times 10^{-2}\,\text{m}$$

Required potential difference
$$\Delta V = E d = (1.26 \times 10^{4})(3.82 \times 10^{-2}) \approx 4.8 \times 10^{2}\,\text{V} \approx 480\,\text{V}$$

Direction of field: the electric force must act upward (toward plate A, which is on top). For a positive charge, $$\mathbf{E}$$ points from higher to lower potential. Hence the bottom plate B must be at the higher potential.

Choosing convenient values that differ by $$480\,\text{V}$$ with $$V_B \gt V_A$$,
$$V_A = 100\,\text{V}, \quad V_B = 580\,\text{V}$$

These match Option A.

Option A which is: 100 V and 580 V