A sphere of radius r =5cm and volume charge density ρ=14μC/m^3 spins at ω=200rad/s

. What is the magnetic field at the center of the sphere in Tesla?

You have a solid sphere. this requires a lot of integration. Here is an example of just a spherical radius, examine it, then adapt it for an inegration thru dr.

https://ocw.mit.edu/courses/physics/8-07-electromagnetism-ii-fall-2012/exams/MIT8_07F12_quizsol2.pdf

Why did the charged sphere go to therapy? It had some serious spin issues!

Okay, let's calculate the magnetic field at the center of the sphere. We can use the formula for the magnetic field due to a spinning charged sphere:

B = (μ₀/4π) * (2/3) * (ω * q) / r³

Where:
B is the magnetic field
μ₀ is the permeability of free space (4π * 10⁻⁷ T·m/A)
ω is the angular velocity (200 rad/s)
q is the total charge of the sphere (volume charge density * volume)
r is the radius of the sphere (5 cm or 0.05 m)

First, let's calculate the total charge of the sphere:

Volume = (4/3) * π * r³
q = ρ * Volume

q = 14 μC/m³ * ((4/3) * π * (0.05 m)³)

q ≈ 46.74 μC

Now, we can substitute the values in the formula:

B = (4π * 10⁻⁷ T·m/A / 4π) * (2/3) * (200 rad/s * 46.74 μC) / (0.05 m)³

B ≈ 6.37 × 10⁻⁷ T (Tesla)

So, the magnetic field at the center of the sphere is approximately 6.37 × 10⁻⁷ T. Keep in mind this calculation is a bit lighthearted, just like my jokes!

To find the magnetic field at the center of a spinning charged sphere, we can make use of the Biot-Savart Law.

The Biot-Savart Law states that the magnetic field at any point due to a small segment of wire carrying current is directly proportional to the product of the current, the length of the segment, and the sine of the angle between the segment and the direction of the magnetic field. In this case, we can consider a small segment of the spinning sphere as current carrying.

The formula for the magnetic field due to a current element at the center of a sphere is given by:

dB = (μ₀ * I * ω * r² * sinθ) / (2 * R³)

Where:
dB is the magnetic field at the center of the sphere (in Tesla)
μ₀ is the permeability of free space (4π × 10^⁻7 T·m/A)
I is the total current running through the sphere (in Ampere)
ω is the angular velocity of the spinning sphere (in rad/s)
r is the radius of the sphere (in meters)
R is the distance from the segment to the center of the sphere (in meters)
θ is the angle between the segment and the direction of the magnetic field (in radians)

Given:
Radius r = 5 cm = 0.05 m
Volume charge density ρ = 14 μC/m³ = 14 × 10⁻⁶ C/m³
Angular velocity ω = 200 rad/s

First, we need to find the total charge Q within the sphere using the volume charge density:
Q = ρ * V
= (14 × 10⁻⁶ C/m³) * (4/3 * π * (0.05 m)³)
≈ 0.058 C

Next, we can calculate the total current I flowing through the sphere:
I = Q / T
= 0.058 C / (2π / ω)
≈ 0.058 C / (2π / 200 rad/s)
≈ 58 A

Now, we can find the magnetic field at the center of the sphere using the Biot-Savart Law:

dB = (4π × 10^⁻7 T·m/A) * (58 A) * (200 rad/s) * (0.05 m)² / (2 * 0.05 m)³
= (4π × 10^⁻7 T·m/A) * (58 A) * (200 rad/s) / (2 * 0.05 m)
≈ 0.0072 T

Therefore, the magnetic field at the center of the sphere is approximately 0.0072 Tesla (T).

To find the magnetic field at the center of a spinning charged sphere, you can use Ampere's circuital law.

Ampere's law states that the line integral of the magnetic field around a closed loop is equal to the product of the current passing through the loop and the permeability of free space (μ0).

In this case, since the sphere is spinning, a current is induced due to the motion of the charges. This induced current produces a magnetic field.

The formula for the magnetic field (B) at the center of a spinning charged sphere is given by:

B = (μ0/4π) * (2π * r^2 * ρ * ω)

Where:
- B is the magnetic field
- μ0 is the permeability of free space, approximately equal to 4π x 10^(-7) T·m/A
- r is the radius of the sphere
- ρ is the volume charge density
- ω is the angular velocity of the spinning sphere

Now let's substitute the given values into the formula:

r = 5 cm = 0.05 m
ρ = 14 μC/m^3 = 14 x 10^(-6) C/m^3
ω = 200 rad/s

Plugging these values in:

B = (4π x 10^(-7) T·m/A) * (2π * (0.05 m)^2 * (14 x 10^(-6) C/m^3) * 200 rad/s)

Calculating this expression will give you the magnetic field at the center of the sphere in Tesla.