Two point charges, +Q and -Q of mass m, are placed on the ends of a massless rod of length L, which is fixed to a table by a pin through its center.
If the apparatus is then subjected to a uniform electric field E parallel to the table and perpendicular to the rod, find the net torque on the system of rod plus charges.
Express your answer in terms of some or all of the variables Q, m, L, and E.
To find the net torque on the system of rod plus charges, we need to consider the torque due to both the electric field and the gravitational field acting on the charges.
1. Torque due to the electric field:
The torque due to the electric field is given by the formula:
τ_e = Q * E * d
where Q is the charge on each point charge, E is the magnitude of the electric field, and d is the perpendicular distance between the center of the rod and the line along which the electric field acts.
Since the rod is of length L and the pin is at the center, the distance d between the center and the line of action of the electric field is L/2.
Therefore, the torque due to the electric field can be written as:
τ_e = Q * E * (L/2)
2. Torque due to the gravitational field:
The torque due to the gravitational field is given by the formula:
τ_g = (m * g * h) * sin(θ)
where m is the mass of each point charge, g is the acceleration due to gravity, h is the perpendicular distance between the center of the rod and the line of action of the gravitational force, and θ is the angle between the rod and the vertical direction.
Since the charges are at the ends of the rod, the distance h between the center and the line of action of the gravitational force is L/2 as well.
Also, since the rod is fixed to the table, it will be horizontal, so the angle θ between the rod and the vertical direction is 90 degrees, and sin(θ) = 1.
Therefore, the torque due to the gravitational field can be written as:
τ_g = (m * g * (L/2)) * 1
Combining these two torques, the net torque on the system can be calculated as:
τ_net = τ_e + τ_g
= Q * E * (L/2) + (m * g * (L/2))
So, the net torque on the system of rod plus charges is given by:
τ_net = Q * E * (L/2) + (m * g * (L/2))
This expression provides the net torque in terms of the given variables Q, m, L, and E.