A 0.7 um-diameter droplet of oil, having a charge of +e, is suspended in midair between two horizontal plates of a parallel-plate capacitor. The upward electric force on the droplet is exactly balanced by the downward force of gravity. The oil has a density of 860 (kg/m^3), and the capacitor plates are 5.0 mm apart. What must the potential difference be to hold the droplet in equilibrium?
To find the potential difference required to hold the droplet in equilibrium, we need to consider the balance of forces acting on the droplet.
Let's break down the forces involved:
1. Force of gravity (Fg): The force pulling the droplet downwards is given by the equation Fg = m*g, where m is the mass of the droplet and g is the acceleration due to gravity (9.8 m/s^2).
2. Electric force (Fe): The upward electric force on the droplet is given by the equation Fe = q*E, where q is the charge on the droplet and E is the electric field between the capacitor plates.
For the droplet to remain suspended in midair, the electric force must equal the force of gravity:
Fe = Fg
Let's calculate these forces step by step:
1. Mass (m) of the droplet:
The volume of a spherical droplet is given by the equation V = (4/3)*π*r^3, where r is the radius of the droplet. Given the diameter as 0.7 um, the radius can be calculated as r = 0.35 um = 0.35*10^-6 m.
The mass of the droplet can be calculated using the density (ρ) and the volume (V) as m = ρ*V.
Substituting the values, we have:
m = 860 kg/m^3 * ((4/3) * π * (0.35*10^-6 m)^3)
2. Force of gravity (Fg):
Fg = m * g
3. Electric force (Fe):
The electric field between the capacitor plates (E) is given by the equation E = V/d, where V is the potential difference and d is the distance between the plates.
The charge on the droplet (q) is given as +e, where e is the elementary charge.
Let's substitute the values and solve for the potential difference:
Fe = Fg
q * E = m * g
e * E = m * g
V/d = (m * g) / e (Substituting E = V/d)
V = (m * g * d) / (e) (Rearranging the equation)
Substituting the calculated values, we have:
V = [(m * g * d) / (e)] = [(860 kg/m^3 * ((4/3) * π * (0.35*10^-6 m)^3) * 9.8 m/s^2 * 5.0 mm] / e
Computing this expression will give us the potential difference required to hold the droplet in equilibrium.