What Produces a Magnetic Field? — Question 2

Question 2

A transverse coil field can be calibrated by the direction of a compass

A horizontal background field is $B_0\hat x$. A coil produces $\kappa I\hat y$, with $\kappa=20.0$ microtesla per ampere. At I=1.50 A, a small compass aligns at 30.0 degrees counterclockwise from +x. The compass does not appreciably alter the field.

1. Find B0 and the magnitude of the total field.

2. Predict the direction when the current is reversed and explain why the two deflections have opposite signs.

Question 2 - Solution

A transverse coil field can be calibrated by the direction of a compass

(a) Method and calculation

$$\tan\theta=\frac{\kappa I}{B_0}\Rightarrow B_0=\frac{30.0}{\tan30^\circ}=30\sqrt3\,\mathrm{\mu T}.$$$$B=\sqrt{B_0^2+(\kappa I)^2}=60.0\,\mathrm{\mu T}.$$

(b) Interpretation and further calculation

$$\vec B(-I)=B_0\hat x-\kappa I\hat y.$$$$\theta(-I)=\tan^{-1}(-\kappa I/B_0)=-30.0^\circ.$$

Physical check

Current reversal reverses the coil contribution, while the fixed background remains unchanged.