With Biot–Savart we can find the field of a current loop, the building block of solenoids, electromagnets and, on a tiny scale, magnetic atoms. As with the charged ring, symmetry does most of the work.
Why it works
On the axis: each piece is $\sqrt{z^2 + R^2}$ away; the sideways parts cancel around the loop, and the parts along the axis carry a factor $R/\sqrt{z^2 + R^2}$.
Key idea
Worked examples
Worked example: A flat coil
- $B = \dfrac{\mu_0 NI}{2R} = \dfrac{(4\pi \times 10^{-7})(20)(2.0)}{2(0.050)} = 5.03 \times 10^{-4}\ \text{T}$, about $503\ \mu\text{T}$, ten times Earth's field.
Answer: $B \approx 503\ \mu\text{T}$
Worked example: A quarter circle
- $\phi = \pi/2$: $B = \dfrac{\mu_0 I\phi}{4\pi R} = \dfrac{(4\pi \times 10^{-7})(5.0)(\pi/2)}{4\pi(0.040)} = 1.96 \times 10^{-5}\ \text{T}$.
- That is one quarter of a full loop's $\mu_0 I/(2R)$, as it should be.
Answer: $B \approx 19.6\ \mu\text{T}$
Worked example: Moving out along the axis
- $\dfrac{B(z)}{B(0)} = \dfrac{R^3}{(z^2 + R^2)^{3/2}} = \dfrac{1}{(4 + 1)^{3/2}} = 0.0894$.
- Less than a tenth, two radii out. Far away, $B \approx \dfrac{\mu_0 IR^2}{2z^3}$.
Answer: about $0.0894$ of the center value
Far from a loop, its field has the same shape as a bar magnet's, and both are called magnetic dipoles. Earth's field far out in space looks much like this. Close to a pair of coils the field can be shaped instead: two identical coils, one radius apart and carrying the same current, make a field that is very nearly uniform in the space between them, a common way to make a known field in a lab.
Common mistake