Two Experiments — Question 6

Question 6

Co-moving a magnet and coil suppresses the relative-motion signal

A rigid magnet and pickup coil move on a common rail. Their separation is $d=x_c-x_m$, and the measured linkage is $\Lambda(d)=C/d^3$, with C positive and d positive. External uniform fields are constant. For this calibrated experiment, use signed flux linkage $\Lambda=N\Phi$, $\mathcal E=-d\Lambda/dt$ and $i=\mathcal E/R$; neglect pickup self-inductance.

1. Find the emf in terms of the two velocities.

2. For equal velocities, find the signal. If the coil speed exceeds magnet speed by u, find its sign and magnitude.

Question 6 - Solution

Co-moving a magnet and coil suppresses the relative-motion signal

(a) Method and calculation

$$\dot d=v_c-v_m,\quad\dot\Lambda=-3Cd^{-4}(v_c-v_m).$$$$\boxed{\mathcal E=3C(v_c-v_m)/d^4}.$$

(b) Interpretation and further calculation

$$v_c=v_m\Rightarrow\boxed{\mathcal E=0}.$$$$v_c-v_m=u>0\Rightarrow\boxed{\mathcal E=3Cu/d^4>0}.$$

Physical check

The prescribed quasistatic linkage model depends on relative separation; uniform translation does not change it.