Fields generated by a rotating disk

[lailola]

Homework Statement

We have an uniformly charged disk with total charge q, which is rotating around its axis with constant angular velocity w. Calculate electric and magnetic field in the axis and in the rotation plane. Calculate the radiated power in one cicle.

Homework Equations

Biot-Savart law.
v=wr

The Attempt at a Solution

I only know how to calculate the magnetic field in the center, using directly the biot-savart law:

$d\vec{B}=\frac{\mu}{4\pi}dq\frac{\vec{v}×\hat{r}}{r^2}$
$B=∫_{0}^{a} \frac{\mu}{4\pi}\sigma 2\pi r dr\frac{wr}{r^2}=\frac{\mu w \sigma a}{2}=\frac{\mu w q}{2\pi a}$

How can I calculate the rest?

Thank you.

 

[vela]

Your answer is wrong. For one thing, you should note your result is independent of where you are on the z-axis. That can't be right. Be a bit more careful about what each variable stands for in your integral.

 

[lailola]

Your answer is wrong. For one thing, you should note your result is independent of where you are on the z-axis. That can't be right. Be a bit more careful about what each variable stands for in your integral.

You're right, It's the field in the center.

 

[lailola]

I've already calculated the magnetic field along the axis. But, in the plane? And, what about the electric field? Any help?

thanks

 

[paranormal]

As a scientist, it is important to first clarify the context and assumptions of the problem. It seems that the disk in question is a non-magnetic, uniformly charged disk that is rotating around its axis with a constant angular velocity. Additionally, we can assume that the disk is infinitely thin and has a radius of a.

To calculate the electric and magnetic fields in the axis and rotation plane, we can use the laws of electromagnetism. The electric field at any point can be calculated using the Coulomb's law, which states that the electric field at a point is equal to the Coulomb constant times the charge of the disk divided by the square of the distance from the point to the disk.

In the axis, the distance from the point to the disk is equal to the radius of the disk, a. Therefore, the electric field in the axis can be calculated as:

E = k * q / a^2

Where k is the Coulomb constant.

In the rotation plane, the distance from the point to the disk can vary depending on the position. Therefore, we need to integrate over the disk to find the total electric field. This can be done using the Coulomb's law and the fact that the disk is uniformly charged. The result is:

E = k * q * (1 - cos θ) / 2πε_0 * a

Where θ is the angle between the point and the axis of rotation, and ε_0 is the permittivity of free space.

To calculate the magnetic field in the rotation plane, we can use the Biot-Savart law, as you have correctly done. However, we need to take into account the fact that the magnetic field is a vector quantity and therefore, we need to consider the direction of the magnetic field at different points on the disk. The result is:

B = μ_0 * w * q / (4π * a) * sin θ

Where μ_0 is the permeability of free space.

To calculate the radiated power in one cycle, we can use the Larmor formula, which states that the power radiated by a charged particle moving with an acceleration a is equal to:

P = q^2 * a^2 / (6π * ε_0 * c^3)

Where c is the speed of light. In this case, the acceleration of the charged particles on the disk is equal to w^