Magnets — Question 3

Question 3

A compass calibration separates two perpendicular field components

A horizontal compass sees an unknown field $B_E\hat y$ and a coil field $kI\hat x$. Its equilibrium deflection from +y is $30^\circ$ at current I0. Ignore other fields.

1. Find BE in terms of k and I0.

2. At current 2I0 find the deflection and the ratio of total field magnitudes at the two currents.

Question 3 - Solution

A compass calibration separates two perpendicular field components

(a) Method and calculation

$$\tan\theta=\frac{kI}{B_E}\Rightarrow\boxed{B_E=\sqrt3kI_0}.$$$$B_1=\sqrt{B_E^2+(kI_0)^2}=2kI_0.$$

(b) Interpretation and further calculation

$$\theta_2=\tan^{-1}(2/\sqrt3)=49.1066^\circ.$$$$\boxed{B_2/B_1=\sqrt7/2=1.32288}.$$

Physical check

The angle does not double when current doubles because the compass follows the vector sum.