Using Magnetic Fields

Section 27.5 Using Magnetic Fields

A circular loop of wire carries a constant clockwise current \(I\text{.}\)

What direction is the magnetic field at the center of the loop?

Hint.

Try adding up the magnetic fields from the current at a few different points around the loop.

What strategy do you think you could use to find the magnitude of the magnetic field?

Solution.

Chop: \(vdq = \frac{ds}{dt}dq = ds\frac{dq}{dt} = Ids\)

Multiply: \(d\vec{B} = \frac{\mu_0}{4\pi}\frac{dq v}{R^2}(-\hat{z}) = \frac{\mu_0}{4\pi}\frac{IRd\theta}{R^2}(-\hat{z})\)

Add: \(\vec{B} = \int d\vec{B} = \frac{\mu_0}{2}\frac{I}{R}(-\hat{z})\)

The magnitude of the magnetic field created by a circular loop of radius \(R\) carrying current \(I\) at its center is

\begin{equation*} \vec{B} = \frac{\mu_0}{2}\frac{NI}{R} \end{equation*}

Here \(N\) is the number of coils in the loop: \(N = 1\) for a single loop.