A spherical shell of radius R carries a uniform surface charge density (charge per unit area) σ. The center of the sphere is at the origin and the shell rotates with angular velocity ω (in rad/sec) around the z-axis (z=0 at the origin)

(a) Calculate the magnitude of the total current (in A) carried by the rotating sphere for the following values of σ, ω and R:
σ= 6 ×10−4C m−2, ω= 7 rad⋅s−1 and R=1m

(b) Calculate the magnitude of the magnetic field B(z) (in T) that is generated by the circular current of the rotating shell at a point P on the z-axis for the following values of σ, ω , z and R:
σ= 6 ×10−4C m−2, ω= 7 rad⋅s−1, z= 2.1 m and R=1m

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Tut tut, your supposed to do your own MIT exam questions, not ask other to do them for you.

To find the magnitude of the total current carried by the rotating sphere, you can use the formula for the current on a surface:

I = σ * A * v,

where I is the current, σ is the surface charge density, A is the area, and v is the velocity.

(a) Given: σ = 6 × 10^−4 C/m^2, ω = 7 rad/s, R = 1 m.

The area of the spherical shell can be calculated using the formula:

A = 4πR^2,

where A is the area and R is the radius.

Plugging in the values:

A = 4π * (1)^2
= 4π m^2.

The velocity of a point on the surface of the shell can be given by:

v = ω * r,

where v is the velocity, ω is the angular velocity, and r is the distance from the axis of rotation.

Since the shell is rotating around the z-axis, the distance from the origin to the z-axis is z.

Plugging in the values:

v = ω * z
= 7 rad/s * 1 m
= 7 m/s.

Now, substitute the values of σ, A, and v into the current formula:

I = σ * A * v
= (6 × 10^−4 C/m^2) * (4π m^2) * (7 m/s)
= 1.68 × 10^−2 C/s
= 1.68 × 10^−2 A.

Therefore, the magnitude of the total current carried by the rotating sphere is 1.68 × 10^−2 A.

(b) Given: σ = 6 × 10^−4 C/m^2, ω = 7 rad/s, z = 2.1 m, and R = 1 m.

To find the magnetic field B(z) generated by the circular current of the rotating shell at point P on the z-axis, you can use the formula for the magnetic field produced by a current-carrying wire:

B(z) = (μ₀ * I * R^2) / (2 * z^3),

where B(z) is the magnetic field, μ₀ is the permeability of free space, I is the current, R is the radius, and z is the distance from the wire.

Plugging in the values:

B(z) = (μ₀ * I * R^2) / (2 * z^3)
= (4π × 10^−7 T⋅m/A) * (1.68 × 10^−2 A) * (1 m)^2 / (2 * (2.1 m)^3)
= 4.01 × 10^−10 T.

Therefore, the magnitude of the magnetic field B(z) generated by the circular current of the rotating shell at point P on the z-axis is 4.01 × 10^−10 T.

To find the magnitude of the total current carried by the rotating sphere (part a) and the magnitude of the magnetic field generated by the circular current at a point on the z-axis (part b), we can use the formulas related to rotating charged objects.

(a) To calculate the magnitude of the total current carried by the rotating sphere (I), we need to find the total charge on the sphere and divide it by the time period. The charge on the sphere is given by the product of the surface charge density (σ) and the surface area of the sphere.

1. Calculate the surface area of the sphere:
The surface area of a spherical shell is given by 4πR², where R is the radius of the sphere.
In this case, R = 1m, so the surface area is 4π(1)² = 4πm².

2. Calculate the total charge on the sphere:
The total charge on the sphere is given by the product of the surface charge density (σ) and the surface area of the sphere.
In this case, σ = 6 × 10^-4 C/m², so the total charge is (6 × 10^-4 C/m²) × (4πm²) = 2.4π × 10^-3 C.

3. Calculate the time period:
The time period (T) is the time taken for one complete rotation, and it is given by the formula T = 2π/ω, where ω is the angular velocity.
In this case, ω = 7 rad/s, so the time period is T = (2π)/(7 rad/s) = 2π/7 s.

4. Calculate the magnitude of the total current:
The magnitude of the total current (I) is given by the formula I = Q/T, where Q is the total charge on the sphere and T is the time period.
In this case, I = (2.4π × 10^-3 C) / (2π/7 s) = 7 × 10^-3 A.

Therefore, the magnitude of the total current carried by the rotating sphere is 7 × 10^-3 A.

(b) To calculate the magnitude of the magnetic field generated by the circular current at a point P on the z-axis (B(z)), we can use the formula for the magnetic field at the center of a circular loop.

1. Calculate the current in the circular loop:
The current in the circular loop is the same as the magnitude of the total current calculated in part a, which is 7 × 10^-3 A.

2. Calculate the distance from the center of the loop to point P:
In this case, the distance from the center of the loop to point P is given as z = 2.1 m.

3. Use the formula for the magnetic field at the center of a circular loop:
The formula for the magnetic field at the center of a circular loop is given by B = (μ₀ I)/(2R), where μ₀ is the permeability of free space, I is the current in the loop, and R is the radius of the loop.

In this case, μ₀ is a constant equal to 4π × 10^-7 T·m/A, I = 7 × 10^-3 A, and R = 1 m.

Substituting the values into the formula, we get B(z) = (4π × 10^-7 T·m/A) × (7 × 10^-3 A) / (2 × 1 m) = 14π × 10^-10 T.

Therefore, the magnitude of the magnetic field B(z) generated by the circular current of the rotating shell at point P on the z-axis is 14π × 10^-10 T.