Biot-Savart for current density in a volume

[henrybrent]

Homework Statement

The Biot-Savart law for a current density j in a volume V is: $d\vec{B} = \frac{\mu_0\vec{j}\times\vec{r}}{4\pi r^3} dV$

Derive the formula for the magnetic field created by a single point-like particle carrying charge $q$ moving with velocity $\vec{v}$. Explain and justify all important steps.

Homework Equations

$d\vec{B} = \frac{\mu_0\vec{j}\times\vec{r}}{4\pi r^3} dV$

The Attempt at a Solution

If I integrate over the whole volume, that will give me B, but it won't be correct for a moving point charge?

$\vec{B} = \frac{\mu_0}{4\pi} \int \frac{\vec{j}\times\vec{r}}{ r^3} dV$

 

[mfb]

Why not?
You have to think about how to get a current from a point particle with an "infinite" charge density.

 

[henrybrent]

Why not?
You have to think about how to get a current from a point particle with an "infinite" charge density.

Sorry, can you elaborate ?

 

[mfb]

What is your distribution of j? You'll need it to evaluate your integral.

 

[henrybrent]

What is your distribution of j? You'll need it to evaluate your integral.

I'm not sure, but;

$\vec{B} = \frac{\mu_0}{4\pi} \int \frac{\vec{j}\times\vec{r}}{ r^3} dV$ then,
$\vec{B} = \frac{\mu_0}{4\pi} \int \frac{\vec{v}\times\hat{r}}{ r^2} dq$

Am I on the right lines?

 

[mfb]

j and v are not the same. They have completely different units, for example, and v does not depend on the position. Also, your equation is now independent of q, which clearly shows something is wrong.

 

[henrybrent]

j and v are not the same. They have completely different units, for example, and v does not depend on the position. Also, your equation is now independent of q, which clearly shows something is wrong.

These are all my lecture notes on the Biot-Savart Law.

That's literally all I have to go on, my lecturer barely went over it at all. I have some textbooks as well, but they just provide me with what I wrote earlier, which you said was wrong?

 

[mfb]

That's literally all I have to go on, my lecturer barely went over it at all. I have some textbooks as well, but they just provide me with what I wrote earlier, which you said was wrong?

I don't see any j=v there.

Let's start with an easier example: you have a ball of total charge Q with radius R and a uniform charge density. Its center is currently at the origin of the coordinate system, and it is moving in x-direction with a speed v. What is the current density for every point in space?

 

[henrybrent]

I don't see any j=v there.

Let's start with an easier example: you have a ball of total charge Q with radius R and a uniform charge density. Its center is currently at the origin of the coordinate system, and it is moving in x-direction with a speed v. What is the current density for every point in space?

I/A ?

 

[mfb]

No, whatever I/A is supposed to mean.

 

[henrybrent]

No, whatever I/A is supposed to mean.

Sigh, then I don't know. I really don't. I know you can't tell me either.

Those pictures I have included are everything I have been taught (taught is an exceptionally strong word to use) regarding Biot-Savart law, our lecturer did not even derive them as he said you can't 'technically' derive them.

I have said this in the many other threads I've had to create, I'm not a physics student, I just take a module in Electromagnetism. I am not (necessarily..) asking to be spoon fed, but it's safe assume I know next to nothing regarding the Biot-Savart Law.

This forum is just another resource I've had to find because my textbooks assume the reader already has a sound base in Physics/maths - which I do not. Alas, I cannot keep using this an excuse.

 

[mfb]

The current subproblem has nothing to do with Biot-Savart.

What do you know about current densities?
Can you find something like "current density is ... per ... and ..."?