# How to calculate the B field at the center of a torus with a large current carrying diameter?

I know the B field at the center axis of a torus (closed current carrying loop of wire) can be simplified to: My question is what if the wire carrying the current has a very wide diameter - how can this wire diameter be accounted for when defining its B field (assuming current creating the field is flowing evenly throughout this thick wire)? In the equation above, the radius value R pertains only to the radius of the entire torus loop, not the small radius inside the loop defining the loop's thickness.

I only need to calculate the B field at the center of this thick loop of wire, but if there was an equation to find the B field at any point along the axis that would be helpful to know as well (similar to the equation on hyperphysics: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/curloo.html)

## 1 Answer

The expression you look for is given by Bergeman, Erez and Metcalf, Phys. Rev. A 35, 1535 (1987) and for a current ring of radius $R$ and current $I$ centered at the position $(z=A, r=0)$ reads

$$B_z=\frac{\mu I}{2\pi}\frac{1}{\sqrt{(R+r)^2+(z-A)^2}}\left (K(k^2)+\frac{R^2-r^2-(z-A)^2}{(R-r)^2+(z-A^2)} E(k^2) \right ),\\ B_r=\frac{\mu I}{2\pi r}\frac{z-A}{\sqrt{(R+r)^2+(z-A)^2}}\left (K(k^2)+\frac{R^2-r^2-(z-A)^2}{(R-r)^2+(z-A^2)} E(k^2) \right ),\\ B_\phi=0.$$ where $K(k^2)$ and $E(k^2)$ are complete elliptic integrals of the first and second kind, respectively. The argument $k^2$ is given by

$$k^2=\frac{4Rr}{(R+r)^2+(z-A)^2}$$

If your wire has a certain thickness, you can approximate the magnetic field generated by this ring as the sum of the field generated by a number of infinitisimal current rings that together form the loop of the desired dimensions.

• Thank you for the answer! Could you please clarify what "r", "z", and "A" are? Is "r" the diameter of the wire making the loop and "z" a point on the x-axis? – Jeremy Adams Nov 9 '16 at 0:02
• I should have been clearer indeed. $r$ is the radial position at which you want to know the field and $z$ is the longitudinal position (in cylindrical coordinates). For your problem you want to construct the field of a wire of finite thickness out off a number of thin wires. The center of the loops are along the $z$ axis but can have a shift $A$. By changing $A$ and $R$ for the different loops and summing you construct the field you want. – Paul Nov 9 '16 at 0:40