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.
To find the net power entering the interior volume of the cylinder, we need to find the rate at which work is being done by the electromagnetic fields on the cylinder.
The power is given by the equation:
P = τα
where P is the power, τ is the torque, and α is the angular acceleration.
Step 1: Calculate the torque:
The torque on a rotating object is given by the equation:
τ = I α
where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
For a hollow cylinder rotating about its axis, the moment of inertia is given by:
I = MR²
where M is the mass of the cylinder and R is the radius.
The mass of the cylinder can be calculated using the equation:
M = (surface charge density) × (area)
where (surface charge density) is the uniform charge per unit area and (area) is the surface area of the cylinder.
The surface area of the cylinder is given by:
A = 2πRL
where R is the radius and L is the length.
Step 2: Calculate the mass of the cylinder:
M = (surface charge density) × (area)
= (4.0 C/m²) × (2π × 0.06 m × 0.70 m)
= 2.646 kg
Step 3: Calculate the moment of inertia:
I = MR²
= (2.646 kg) × (0.06 m)²
= 0.009001 kg·m²
Step 4: Calculate the torque:
τ = I α
= (0.009001 kg·m²) × (40 rad/s²)
= 0.36004 N·m
Step 5: Calculate the power:
P = τα
= (0.36004 N·m) × (200 rad/s)
= 72.008 W
Therefore, 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 is 72.008 W.
To find the net power entering the interior volume of the cylinder, we need to consider the work done by electromagnetic fields to accelerate the cylinder and the work done by the cylinder's interior to overcome the electromagnetic forces. The net power can be calculated using the equation:
Net Power = Torque * Angular Velocity
First, let's find the torque acting on the cylinder. Since the torque causes the cylinder to rotate at a constant acceleration, we can use the equation:
Torque = Moment of Inertia * Angular Acceleration
The moment of inertia of a cylinder rotating about its central axis is given by:
Moment of Inertia (I) = (1/2) * Mass * Radius^2
To find the mass of the cylinder, we need the surface charge density and the length of the cylinder. The surface charge density is given as 4.0 C/m^2.
Surface Charge Density = Charge / Area
Charge = Surface Charge Density * Area
The area of the cylinder's surface is given by:
Area = 2 * π * Radius * Length
Substituting these values, we can find the charge:
Charge = (4.0 C/m^2) * (2 * π * 0.060 m * 0.70 m)
Now, we can calculate the mass using the equation:
Mass = Charge / Electric Field
The electric field inside a hollow cylinder is zero. Therefore, the mass of the cylinder is zero.
As the mass of the cylinder is zero, the torque acting on it is also zero. Therefore, the net power entering the interior volume of the cylinder from the surrounding electromagnetic fields is zero.