A small sperical insulator of mass 8.00X10^-2 kg and charge +0.600 mC is hung by a thin wire of negligible mass. A charge of -0.900 mC is held 0.150 m away from the spere and directly to the right of it, so the wire makes an angle [theta] with the vertical. Find (a) the angle [theta] and (b) the tension in the wire.
Let T be the tension in the wire.
A vertical force balance of the insulator says:
T sin theta = M g
A horizontal force balance says :
T cos theta = k Q1*Q2/R^2
which is Coulomb's law. k is the Coulomb constant and Q1 and Q2 are the two charges.
Solve the two equations in the two unknowns, theta and T.
Hint: divide one equation by the other to get tan theta and eliminiate T.
To solve this problem, we can apply the principle of electrostatic equilibrium, which states that the net electrostatic force on a charged object is zero when it is in equilibrium.
Let's break down the problem and find the forces acting on the insulator:
(a) Angle θ:
The insulator experiences two forces: the gravitational force (Fg) and the electrostatic force (Fe) due to the charge -0.900 mC. These forces can be resolved into components along the wire and perpendicular to the wire.
1. Gravitational force:
The gravitational force acts vertically downward and can be expressed as:
Fg = mg
2. Electrostatic force:
The electrostatic force acts horizontally due to the repulsion between charges. The magnitude of the electrostatic force can be calculated using Coulomb's law:
Fe = (k * |q1 * q2|) / r^2
where k is the electrostatic constant (9 * 10^9 N.m^2/C^2), q1 and q2 are the charges (+0.600 mC and -0.900 mC), and r is the distance between the charges (0.150 m).
Now, let's proceed to find the angle θ:
First, calculate the magnitude of the gravitational force:
Fg = (8.00 * 10^-2 kg) * (9.8 m/s^2) = 7.84 * 10^-1 N
Next, calculate the magnitude of the electrostatic force:
Fe = (9 * 10^9 N.m^2/C^2) * (0.600 * 10^-3 C) * (0.900 * 10^-3 C) / (0.150 m)^2 = 2.16 N
Using the magnitudes of these forces, we can find the angle formed by the wire and the vertical direction:
tan θ = (Fe / Fg)
θ = tan^(-1)(Fe / Fg)
Substituting the values:
θ = tan^(-1)(2.16 / 7.84 * 10^-1)
(b) Tension in the wire:
To find the tension in the wire, we need to consider the vertical component of the electrostatic force. This vertical component balances the vertical component of the gravitational force, resulting in tension in the wire.
The vertical component of the electrostatic force is given by:
Fe_vertical = Fe * sin(θ)
The tension in the wire can be calculated by equating the vertical forces:
Tension = Fe_vertical + Fg
Substituting the values and using the calculated angle, we can find the tension in the wire.