How Do You Calculate the Magnetic Field Above a Current-Carrying Loop?

[skrat]

Homework Statement

Calculate the magnetic field above a loop shown in picture with radius $R$ and current $I$.

Homework Equations

The Attempt at a Solution

Firstly, curved part:

$\vec{H}(\vec{r})=\frac{I}{4\pi }\int \frac{d\vec{{r}'}\times (\vec{r}-\vec{{r}'})}{|\vec{r}-\vec{{r}'}|^3}cos\varphi =\frac{I}{4\pi }\int \frac{Rd\varphi \hat{e}_{\varphi }\times (\vec{r}-\vec{{r}'})}{(R^2+z^2)^{3/2}}\frac{R}{\sqrt{R^2+z^2}}$

Where $\hat{e}_{\varphi }\times (\vec{r}-\vec{{r}'})=|\hat{e}_{\varphi }||\vec{r}-\vec{{r}'}|sin\varphi \hat{n}=\sqrt{R^2+z^2}\hat{n}$ and finally:

$\vec{H}(\vec{r})=\frac{I}{4\pi }\int_{\pi }^{2\pi }\frac{R^2\hat{n}d\varphi }{(R^2+z^2)^{3/2}}=\frac{IR^2}{4(R^2+z^2)^{3/2}}\hat{n}$.

For the straight line I have some troubles with the integral...

$\vec{H}(\vec{r})=\frac{I}{4\pi }\int \frac{d\vec{{r}'}\times (\vec{r}-\vec{{r}'})}{|\vec{r}-\vec{{r}'}|^3}$

Now $d\vec{{r}'}\times (\vec{r}-\vec{{r}'})=d\vec{{r}'}\times \vec{r}$ since $d\vec{{r}'}$ and $\vec{{r}'}$ are parallel. Also $d\vec{{r}'}=dx\hat{e}_x$.

$\vec{H}(\vec{r})=\frac{I}{4\pi }\int_{-R}^{R}\frac{dx(\hat{e}_x\times \vec{r})}{((\vec{r}-\hat{e}_xx)(\vec{r}-\hat{e}_xx))^{3/2}}=\frac{I(\hat{e}_x\times \vec{r})}{4\pi }\int_{-R}^{R}\frac{dx}{(r^2-2\hat{e}_x\vec{r} x+x^2)^{3/2}}$

Now what? Is this even ok?

 

[Simon Bridge]

Presumably you've already done it, or seen it done, for an infinite straight wire?

 

[skrat]

Aha, ok, I see it now...

$\hat{e}_x\vec{r}=0$ since they are always perpendicular. (My question here: What if they werent?)

In that case, things simplify a lot and the integral over the straight line should be $\vec{H}(\vec{r})=\frac{IR}{2\pi r\sqrt{R^2+r^2}}\hat{e}_y$, where $\vec{r}$ is the distance from the center of the straight line to point $T$.

 

[Simon Bridge]

When you don't have easy symmetry, the calculation can get arbitrarily difficult.
In many cases the integral has to be solved numerically.

That is why you will only see simple geometries at this stage.
Also consider what happens to the field close to one corner of the D ;)

 

[paranormal]

I would suggest that the approach taken in the attempt at a solution is not completely correct. It is important to first clearly define the problem and the variables involved. In this case, the problem states that the loop has a radius R and carries a current I. The magnetic field above the loop is to be calculated at a point with coordinates (x,y,z).

The correct approach would be to use the Biot-Savart law, which states that the magnetic field at a point due to a current element is given by:

d𝐵 = (𝜇₀/4𝜋) * (𝑑𝐼 * 𝑑𝐿 * sin𝜃)/𝑟²

where 𝜇₀ is the permeability of free space, 𝑑𝐼 is the current element, 𝑑𝐿 is the length of the current element, 𝜃 is the angle between the current element and the line joining the current element and the point, and 𝑟 is the distance between the current element and the point.

Using this law, the magnetic field at a point above the loop can be calculated by considering the contributions from all the current elements that make up the loop. The integral can be set up as follows:

𝐵 = ∫(𝜇₀/4𝜋) * (𝑑𝐼 * 𝑑𝐿 * sin𝜃)/𝑟²

where the limits of integration are from 0 to 2𝜋 for the angle and from 0 to R for the radius of the loop.

Once the integral is set up, it can be solved using appropriate mathematical techniques. It is important to note that the magnetic field will have different components in the x, y, and z directions and will depend on the coordinates of the point above the loop.

In conclusion, as a scientist, I would suggest that a more systematic and rigorous approach should be taken to solve the problem and to clearly define the variables and equations being used.