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.
Torque depends on the orientation of the dipole in relation to the E field.
In a vector sense, torque=qL X E (Where X means cross product)
To find the net torque on the system of rod plus charges, we need to consider the torques caused by the electric field acting on each charge.
The torque on a point charge in an electric field is defined as the cross product of the position vector of the charge and the electric field:
τ = r x E
where τ is the torque, r is the position vector, and E is the electric field.
Let's assume that the positive charge +Q is located at the end of the rod with a distance of L/2 from the pivot point (the pin), and the negative charge -Q is located at the other end of the rod with a distance of -L/2 from the pivot point.
For the positive charge +Q, the torque τ1 on it is:
τ1 = (L/2) x E
Since the torque for a force about an axis is given by the product of the force and the perpendicular distance between the force and the axis, we can substitute the electric field force F1 for the charge +Q:
τ1 = (L/2) x F1
Similarly, for the negative charge -Q, the torque τ2 on it is:
τ2 = (-L/2) x E = -(L/2) x E
Again, we substitute the electric field force F2 for the charge -Q:
τ2 = (-L/2) x F2 = -(L/2) x F2
The total torque on the system is the sum of the torques on the positive and negative charges:
τ_net = τ1 + τ2 = (L/2) x F1 + -(L/2) x F2
Since the electric field E is uniform, the forces F1 and F2 on the charges are equal in magnitude and opposite in direction:
F1 = -F2 = qE
where q is the charge magnitude. Substituting this in, we get:
τ_net = (L/2) x (-qE) + -(L/2) x (qE)
τ_net = -(L/2)qE - (L/2)qE
τ_net = -qE(L/2 + L/2)
τ_net = -qEL
Finally, we can see that the net torque on the system of rod plus charges is -qEL in the direction opposite to the electric field E.