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Biot-Savart for current density in a volume

  • Thread starter henrybrent
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  • #1
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Homework Statement


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

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

Homework Equations



[itex] d\vec{B} = \frac{\mu_0\vec{j}\times\vec{r}}{4\pi r^3} dV[/itex]

The Attempt at a Solution



If I integrate over the whole volume, that will give me B, but it wont be correct for a moving point charge?

[itex] \vec{B} = \frac{\mu_0}{4\pi} \int \frac{\vec{j}\times\vec{r}}{ r^3} dV[/itex]
 

Answers and Replies

  • #2
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Why not?
You have to think about how to get a current from a point particle with an "infinite" charge density.
 
  • #3
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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 ?
 
  • #4
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What is your distribution of j? You'll need it to evaluate your integral.
 
  • #5
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What is your distribution of j? You'll need it to evaluate your integral.
I'm not sure, but;

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

Am I on the right lines?
 
  • #6
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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.
 
  • #7
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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.


BSlaw1.jpg



BsLaw.jpg


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?
 
  • #8
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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?
 
  • #9
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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 ?
 
Last edited:
  • #10
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No, whatever I/A is supposed to mean.
 
  • #11
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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.
 
  • #12
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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 ..."?
 

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