A spherical shell of radius R carries a uniform surface charge density (charge per unit area) \sigma. The center of the sphere is at the origin and the shell rotates with angular velocity \omega (in rad/sec) around the z-axis (z=0 at the origin). Seen from below, the sphere rotates clockwise. (See the figure below)
a) (a) Calculate the magnitude of the total current (in A) carried by the rotating sphere for the following values of sigma,omega and R:
sigma = 5 times 10^{-4},C/m^2, omega = 4 rad/sec and R = 1 m
b)(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 sigma, omega , z and R:
sigma = 5 times 10^{-4} {C m}^{-2}, \omega = 4 rad/sec , z= 2.1 m and R =1m
a) I = Sigma*A*omega
b) B = mu*I/(2*(z-R))
Thanks!! But how to calculate A in the first part??
Wrong formula!!!
I could not get it right with that formula too.
Anonymous, is there chance that you could give us a calculated example with the above numbers?
We could trasfer it into ours, thanks.
Please step by step answer, I cannot figure it out too!
Yes, please explain the way guys!
Anonymous and Phy please help!
a) I = Sigma*A*f
f = omega/(2pi)
b) B = mu*I/(2*(z-R))
Anonymous, the formula is wrong. Did you get the answer right by the way?
If somebody got the answer could they provide it step by step with numbers though?
thanks
To find the magnitude of the total current carried by the rotating sphere, we need to first calculate the total charge on the sphere.
a) Recall that the surface charge density (\sigma) is defined as the charge per unit area. Therefore, the total charge (Q) on the sphere can be calculated as:
Q = \sigma * A
where A is the surface area of the sphere. The surface area of a sphere can be found using the formula:
A = 4 * \pi * R^2
where R is the radius of the sphere.
In this case, we are given that sigma = 5 * 10^-4 C/m^2, omega = 4 rad/sec, and R = 1 m. Using these values, we can calculate the surface area and the total charge:
A = 4 * \pi * (1)^2 = 4 * \pi m^2
Q = (5 * 10^-4 C/m^2) * (4 * \pi m^2) = 2 * 10^-3 \pi C
Next, to find the magnitude of the total current (I) carried by the rotating sphere, we need to consider the rotation of the sphere. As the sphere rotates with angular velocity omega, the charges on its surface will move, creating a current. The current can be calculated as:
I = Q * \omega
where Q is the total charge on the sphere and \omega is the angular velocity.
In this case, we are given that omega = 4 rad/sec and Q = 2 * 10^-3 \pi C. Plugging in these values, we can calculate the magnitude of the total current:
I = (2 * 10^-3 \pi C) * (4 rad/sec) = 8 * 10^-3 \pi A
Therefore, the magnitude of the total current carried by the rotating sphere is 8 * 10^-3 \pi A.
b) To calculate the magnitude of the magnetic field (B(z)) generated by the circular current at point P on the z-axis, we can use Ampere's Law. According to Ampere's Law, the magnitude of the magnetic field at any point near a current-carrying conductor can be found using the formula:
B(z) = \frac{\mu_0}{2\pi} * \frac{I}{r}
where B(z) is the magnetic field at point P, \mu_0 is the magnetic constant (\mu_0 = 4 * \pi * 10^-7 T*m/A), I is the current, and r is the distance between the point and the axis of rotation.
In this case, we are given that sigma = 5 * 10^-4 C/m^2, omega = 4 rad/sec, R = 1 m, and z = 2.1 m. Using these values, we can calculate the magnetic field:
First, we know that the current I is the same as the magnitude of the total current calculated in part (a), which is 8 * 10^-3 \pi A.
Next, we need to calculate the distance r between point P and the axis of rotation. In this case, r is the distance between point P and the origin (center of the sphere), which is 2.1 m.
Plugging in the values, we can calculate the magnetic field:
B(z) = \frac{4 * \pi * 10^-7 T*m/A}{2\pi} * \frac{8 * 10^-3 \pi A}{2.1 m}
Simplifying the equation, we get:
B(z) = \frac{2 * 10^-7}{2.1} T
Therefore, the magnitude of the magnetic field at point P on the z-axis is approximately 9.52 * 10^-8 T.