A long hollow non-conducting cylinder of radius 0.060 m and length 0.70 m carries a uniform charge per unit area of 4.0 C/m^2 on its surface. Beginning from rest, an externally applied torque causes the cylinder to rotate at constant acceleration of 40 rad/s^2 about the cylinder axis. Find the net power entering the interior volume of the cylinder from the surrounding electromagnetic fields at the instant the angular velocity reaches 200 rad/s.
4.6 μW
To find the net power entering the interior volume of the cylinder, we need to calculate the electromagnetic work done on the cylinder as it rotates from rest to an angular velocity of 200 rad/s.
The net power is given by the equation:
Net Power = Torque * Angular velocity
To find the torque, we can use the equation:
Torque = Moment of inertia * Angular acceleration
The moment of inertia, in this case, can be calculated as:
Moment of inertia = (1/2) * mass * radius^2
Since the cylinder is hollow, we need to find the mass of the cylinder first. The charge per unit area on the surface of the cylinder allows us to determine the charge enclosed within its surface.
The charge enclosed can be calculated as:
Charge = Surface charge density * Surface area
Surface area = 2π * radius * length
Once we know the charge enclosed, we can calculate the mass of the cylinder using the formula:
Mass = Charge / Electric field
To find the electric field, we can use Gauss's Law for a cylinder, which states:
Electric field = 2 * π * k * surface charge density
Where k is the Coulomb constant.
Now that we have the mass and moment of inertia of the cylinder, we can calculate the torque. Given the constant angular acceleration of 40 rad/s^2, we can now find the torque using the following equation:
Torque = Moment of inertia * Angular acceleration
Finally, we can find the net power by multiplying the torque with the angular velocity:
Net Power = Torque * Angular velocity
Substituting the known values, we can calculate the net power entering the interior volume of the cylinder.