Magnetic Fields from Current Carrying Wires

[gibberingmouther]

I've had a lot of problems that involve a segment of current carrying wire, for example when you have a square loop of wire.

I have a formula for "long" wires that is B = μ0 * I/(2 * π * d).

Can I use this for shorter wire segments, and if not, what formula can I use?

 

[kuruman]

It depends on where you want to find the field relative to the segment.

On edit: See here for a derivation.

 

[Cryo]

A universal way to get the magnetic vector potential ($\mathbf{A}$), given the current density ($\mathbf{J}$) in magnetostatics is:

$\mathbf{A}(\mathbf{r})=\frac{\mu_0}{4\pi}\int d^3 r' \frac{\mathbf{J}(\mathbf{r}')}{\left|\mathbf{r}-\mathbf{r}'\right|}$

Where integration is over the whole space. To get the magnetic field you simply take the curl

$\mathbf{B}(\mathbf{r})=\boldsymbol{\nabla}\times\mathbf{A}(\mathbf{r})=-\frac{\mu_0}{4\pi}\int d^3 r' \mathbf{J}(\mathbf{r}')\times\boldsymbol{\nabla}\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}=\frac{\mu_0}{4\pi}\int d^3 r' \mathbf{J}(\mathbf{r}')\times\frac{\left(\mathbf{r}-\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|^3}$

Now define $\mathbf{R}=\mathbf{r}-\mathbf{r}'$ and integrate over the corross-section of the wire, assuming the wire is thin (comapred to $R$). This will convert the volume integral into integral along the wire: $\int d^3 r' \mathbf{J}(\mathbf{r}')\to\int dl I(l) \mathbf{\hat{l}}$ where $I$ is current (i.e. current density over the whole cross-section of the wire) and $\mathbf{\hat{l}}$ is parallel to the wire (along the direction of current flow).

Thus:

$\mathbf{B}(\mathbf{r})=\frac{\mu_0}{4\pi}\int dl I(l) \boldsymbol{\hat{l}}\times\frac{\mathbf{\hat{R}}}{R^2}$

Now $\mathbf{R}$ points from the section of the wire (at position $l$) towards the observer. This is the Biot-Savart law. It works in all cases, including shorter wires, your formula its special case.