Magnetic Field due to a Long Straight Wire (Proof)

[FatPhysicsBoy]

Homework Statement

Whilst revising the Biot-Savart law, I came across the result:

for the magnetic field due to a long straight wire. I understand that this can be found by integrating the Biot-Savart equation over the length of the wire, however after attempting the integral myself and looking at the integration in my notes I find myself unable to understand how to come to the result.

Homework Equations

The Attempt at a Solution

Outlined above. Any help will be appreciated, thank you.

 

[Steely Dan]

Could you clarify what confuses you about the integral? The way to set it up would be as an integral over z, say, along the wire, with each wire element dz contributing to the field at P based on the distance to point P and the angle between the radius vector and the z-axis.

 

[FatPhysicsBoy]

This is the proof from my lecture notes:

I understand where the cross product has been replaced with dssinθ up to the integral of sinθ/r².

I also understand the first relationship r₂ = s² + R² as just being Pythagoras, however I do not understand it's necessity, also I do not understand the second relationship involving sinθ either.

So I suppose you could say I don't understand the relationships, and also why and how they are used.

 

[jtbell]

In order to do the integral, you have to write it in terms of a single variable. Here, the writer chose s as the variable of integration, so he has to eliminate $\theta$ and r in favor or s. R is a constant as far as the integral is concerned.

This is not the only way to do it. You can also do it by using $\theta$ as the integration variable, in which case you have to eliminate s and r. You should of course end up with the same final result.

You can also (at least in principle) use r as the integration variable. I've never done it that way in this particular example, but I've done something similar in other situations.

 

[FatPhysicsBoy]

In order to do the integral, you have to write it in terms of a single variable. Here, the writer chose s as the variable of integration, so he has to eliminate $\theta$ and r in favor or s. R is a constant as far as the integral is concerned.

This is not the only way to do it. You can also do it by using $\theta$ as the integration variable, in which case you have to eliminate s and r. You should of course end up with the same final result.

You can also (at least in principle) use r as the integration variable. I've never done it that way in this particular example, but I've done something similar in other situations.

Oh ok yeah I see, however I still do not understand this relationship then:

sin(∏-θ) = R/(s² + R²)^0.5

 

[Villyer]

Theta is the outside angle in that diagram, pi-theta makes it the inside angle so that you can use the simple definition (sin theta = opp / hyp) to remove the theta from the equation.

 

[FatPhysicsBoy]

Theta is the outside angle in that diagram, pi-theta makes it the inside angle so that you can use the simple definition (sin theta = opp / hyp) to remove the theta from the equation.

Thank you, I fully understand the derivation now!