# Current of Spinning sphere of constant voltage

• Z90E532
In summary, the conversation discusses finding the magnetic dipole moment of a spinning sphere with given parameters. The correct approach involves finding the charge density on the sphere using the known voltage and then using this to calculate the current density. Dimensional analysis can also be used to determine where a factor of the radius is missing in the final equation.
Z90E532

## Homework Statement

Find the magnetic dipole moment of a spinning sphere of voltage ##V## and radius ##R## with angular frequency ##\omega##.

## The Attempt at a Solution

To find the dipole moment, we need to do ##I \int d \vec{a}##, which would be ##I 4 \pi R^2 \hat{r}##, but I need to find the current knowing only the voltage and angular velocity. I'm not sure how to proceed.

Edit: Thought about this again and noticed a mistake. What I need to do is integrate to add up all the current rings on the sphere. So I want to be able to do something like (the sphere is spinning along the z axis):

$$\vec{m} = \sigma R^3 \omega \int ^{2 \pi} _{0} \int ^{\pi} _{0} \sin^3 \theta d \theta \phi \hat{z}$$ .

Where ##I = \sigma R \omega \sin \theta## and ##\vec{A} = R^2 \sin^2 theta##

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Your expression for the dipole moment is the dipole moment of a conductor carrying a current I. You have a more general current density. It is also unclear what you mean by ##\hat r##.

Hint: You know the potential so you should be able to find the charge density on the sphere. From the charge density and rotation of the sphere, you should be able to deduce the current density.

Orodruin said:
Your expression for the dipole moment is the dipole moment of a conductor carrying a current I. You have a more general current density. It is also unclear what you mean by ##\hat r##.

Hint: You know the potential so you should be able to find the charge density on the sphere. From the charge density and rotation of the sphere, you should be able to deduce the current density.
I corrected my mistake I think, but I still am not sure how to find the surface charge density from only the voltage. I think my brain stopped working.

Z90E532 said:
I corrected my mistake I think, but I still am not sure how to find the surface charge density from only the voltage. I think my brain stopped working.
Did you perhaps do electrostatics before magnetostatics?

Orodruin said:
Did you perhaps do electrostatics before magnetostatics?

Yeah, I think I was just having a brain malfunction... it's pretty late here.

I think this is the right approach: For a surface charged sphere: ##\vec{E} = \frac{R^2 \sigma}{\epsilon _{0}r^2}## then we have ##V = - \int ^{R} _{\infty} \frac{R^2 \sigma}{\epsilon _{0}r^2} dr =\frac{R^2 \sigma}{\epsilon _{0}R}## which gives ##\sigma = \frac{V \epsilon _{0}}{R}##.

Edit: Just heard from a friend this is the correct solution, although my answer is missing a factor of R somehow. I have ##I = \sigma \omega \sin \theta R## and ## da = \pi R^2 \sin ^2 \theta ##, putting them together I get ##\vec{m} = \sigma \omega R^3 \int ^{\pi} _{0} \sin ^3 \theta d\theta##

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Z90E532 said:
although my answer is missing a factor of R somehow

What is the width of the circular loop between ##\theta## and ##\theta + d\theta##?

Edit: Dimensional analysis is also a powerful tool to figure out where a factor of ##R## is missing ...

## 1. What is a spinning sphere of constant voltage?

A spinning sphere of constant voltage is a scientific concept that involves the flow of electric charge on a spherical object that is rotating at a constant speed. This phenomenon is often used in experiments and studies to better understand the behavior of electricity and electrical currents.

## 2. How does a spinning sphere of constant voltage work?

In a spinning sphere of constant voltage, the electrical charge is distributed evenly across the surface of the sphere. As the sphere rotates, the charge moves and creates an electric current. This current is affected by factors such as the speed of rotation, the voltage applied, and the resistance of the material the sphere is made of.

## 3. What are the applications of a spinning sphere of constant voltage?

A spinning sphere of constant voltage has various applications in the field of electricity and magnetism. It is used in experiments to study the effects of electric fields on rotating objects, as well as in devices such as generators and motors.

## 4. How is a spinning sphere of constant voltage different from a regular electric current?

A spinning sphere of constant voltage differs from a regular electric current in that it involves a rotating object and the charge is distributed evenly across the surface instead of flowing through a wire. This can result in different behaviors and effects, making it a useful tool for scientific research.

## 5. What are the advantages of using a spinning sphere of constant voltage in experiments?

Using a spinning sphere of constant voltage in experiments allows scientists to manipulate and control the electric current in a more precise and controlled manner. It also provides a unique perspective on the behavior of electricity and can lead to new discoveries and advancements in the field.

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