A small plastic ball with a mass of 6.80 10-3 kg and with a charge of +0.161 µC is suspended from an insulating thread and hangs between the plates of a capacitor (see the drawing). The ball is in equilibrium, with the thread making an angle of 30.0° with respect to the vertical. The area of each plate is 0.0155 m2. What is the magnitude of the charge on each plate?
m=6.8•10⁻³ kg, q=0.161•10⁻⁶C, A=0.0155 m², α=30°, Q=?
Let’s find the projections of the forces acting on the plastic ball:
x-projections: Tsinα =F ……. (1)
y-projections: Tcosα = mg ……(2)
Divide (1) by (2)
Tsinα /Tcosα=F/mgá
tanα =F/mg…………(3)
F=qE=qσ/ε₀=q•Q/A ε₀ …(4)
Substitute (4) in (3)
tanα = qQ/A• ε₀•m•gá
Q= A•ε₀•mg• tanα/q=
=0.0155•8.85•10⁻¹²•6.8•10⁻³•9.8•0.577/0.161•10⁻⁶=
= 3.28•10⁻⁸ C
To find the magnitude of the charge on each plate, we need to use the equilibrium conditions of the ball hanging between the plates of the capacitor.
First, let's analyze the forces acting on the ball. The gravitational force pulling the ball down can be represented as:
F_gravity = m * g
where m is the mass of the ball and g is the acceleration due to gravity (approximately 9.8 m/s^2).
The electrostatic force pulling the ball towards the positive plate of the capacitor can be expressed as:
F_electric = q * E
where q is the charge on the ball and E is the electric field between the plates.
Since the ball is in equilibrium, the net force acting on the ball must be zero. Therefore, we can write:
F_electric - F_gravity = 0
Now, let's calculate the electric field between the plates. The electric field (E) between parallel plates of a capacitor is given by:
E = V / d
where V is the voltage across the plates and d is the distance between the plates.
However, in this case, we are given the area (A) of each plate and the value of E can be rewritten as:
E = σ / ε0
where σ is the surface charge density on each plate and ε0 is the permittivity of free space.
To find the surface charge density σ, we can use the formula:
σ = Q / A
where Q is the charge on each plate and A is the area of each plate.
Using this information, we can set up an equation to find Q:
q * E = m * g
(σ / ε0) * A = m * g
Q / (ε0 * A) = m * g
Q = (m * g * ε0 * A) /E
Substituting the given values into this equation, we can find the magnitude of the charge on each plate.