A spherical shell of radius carries a uniform surface charge density (charge per unit area) . The center of the sphere is at the origin and the shell rotates with angular velocity (in rad/sec) around the -axis ( at the origin). Seen from below, the sphere rotates clockwise. (See the figure below)
(a) Calculate the magnitude of the total current (in A) carried by the rotating sphere for the following values of , and :
8 , 5 and
(b) Calculate the magnitude of the magnetic field (in T) that is generated by the circular current of the rotating shell at a point P on the -axis for the following values of , , and :
8 , 5 , 2.3 and
(a) To calculate the magnitude of the total current carried by the rotating sphere, we need to find the total charge on the sphere and the time it takes for one complete rotation.
The total charge on the sphere can be calculated by multiplying the surface charge density (σ) by the surface area of the sphere (4πR²). In this case, the radius of the sphere does not matter because we are only interested in the surface charge.
Total charge (Q) = σ * surface area = σ * 4πR²
The time it takes for the rotating sphere to complete one full revolution is given by the period (T) which is equal to 2π/ω, where ω is the angular velocity.
T = 2π/ω
The total current (I) is equal to the total charge divided by the time:
I = Q / T = Q / (2π/ω)
Substituting the given values:
R = 8
σ = 5
ω = (empty value)
The magnitude of the total current can be calculated using the formula above.
(b) To calculate the magnetic field generated by the circular current of the rotating shell at point P on the z-axis, we use Ampere's law. Ampere's law states that the magnetic field (B) at a point due to a circular current is proportional to the current (I) and inversely proportional to the distance (r) from the center of the circular current.
B = (μ₀ / 2π) * (I / r)
Where:
μ₀ is the permeability of free space (4π x 10^-7 T·m/A)
I is the current
r is the distance from the center of the circular current
Substituting the given values:
I = (empty value)
r = 2.3 (distance from the center of the circular current to point P)
The magnitude of the magnetic field can be calculated using the formula above.
To calculate the magnitude of the total current carried by the rotating sphere, we need to determine the total charge on the sphere and the time it takes to complete one rotation.
(a) The total charge on the sphere can be calculated by multiplying the surface charge density by the surface area of the sphere. The formula for the surface area of a sphere is given by 4πr^2, where r is the radius of the sphere. So, the total charge is Q = σ * 4πr^2.
Next, we need to find the time it takes for the sphere to complete one rotation. The time period (T) of a rotation is given by the formula T = 2π/ω, where ω is the angular velocity.
Finally, the magnitude of the total current (I) is given by I = Q/T.
Therefore, the formula to calculate the magnitude of the total current is:
I = σ * 4πr^2 / (2π/ω)
For the given values of σ = 8 C/m^2, r = 5 m, and ω = 3 rad/s, we can substitute these values into the formula to calculate the magnitude of the total current.
I = 8 * 4π * 5^2 / (2π/3)
I = 8 * 4 * 25 / (6)
I = 16 A
Therefore, the magnitude of the total current carried by the rotating sphere is 16 A for the given values.
(b) To calculate the magnitude of the magnetic field generated by the circular current of the rotating shell at a point P on the z-axis, we can use Ampere's Law. Ampere's Law states that the magnetic field (B) at a distance r from a long, straight current-carrying wire is given by B = μ0 * I / (2π * r), where μ0 is the permeability of free space, I is the current, and r is the distance from the wire.
In this case, the rotating sphere acts as a circular current-carrying wire. Therefore, the magnetic field at point P can be calculated using the above formula by considering the radius of the sphere (r') as the distance from the wire.
The formula to calculate the magnetic field is:
B = μ0 * I / (2π * r')
For the given values of μ0 = 4π * 10^-7 T*m/A, I = 8 A, r' = 2.3 m, we can substitute these values into the formula to calculate the magnitude of the magnetic field.
B = (4π * 10^-7) * 8 / (2π * 2.3)
B = 1.73 * 10^-6 T
Therefore, the magnitude of the magnetic field at point P is 1.73 * 10^-6 T for the given values.