A long, thin rod (length = 4.0 m) lies along the x axis, with its midpoint at the origin. In a vacuum, a +5.0 µC point charge is fixed to one end of the rod, while a -5.0 µC point charge is fixed to the other end. Everywhere in the x, y plane there is a constant external electric field (magnitude = 5.00 103 N/C) that is perpendicular to the rod. With respect to the z axis, find the magnitude of the net torque applied to the rod.
To find the magnitude of the net torque applied to the rod with respect to the z-axis, we need to use the formula for electric torque:
τ = r x F
where:
τ is the torque
r is the position vector from the point of rotation to the point where the force is applied
F is the force vector
In this case, the force acting on each charge along the rod is the external electric field E. However, because the electric field is perpendicular to the rod, the force will be parallel to the z-axis, resulting in a torque.
To determine the torque on each charge, we can use the formula:
τ = q * r * E
where:
q is the charge
r is the distance from the charge to the point of rotation
E is the magnitude of the electric field
Since the charges are fixed at the ends of the rod, the distance r is equal to half the length of the rod (2.0 m). Substituting the values:
τ = (5.0 µC) * (2.0 m) * (5.00 × 10^3 N/C)
= 50 × 10^-6 C * 2.0 m * 5.00 × 10^3 N/C
= 500 × 10^-6 N*m
Since there are two charges on the rod, the total torque on the rod is the sum of the torques on each charge:
τ_net = 2 * τ
= 2 * (500 × 10^-6 N*m)
= 1000 × 10^-6 N*m
Therefore, the magnitude of the net torque applied to the rod with respect to the z-axis is 1000 × 10^-6 N*m.