Two Experiments — Question 9

Question 9

A symmetric magnet passage has zero net charge but nonzero heating

A magnet traverses a pickup so its linkage starts at zero, rises linearly to Lambda0 in time T and falls linearly to zero in another time T. Pickup resistance is R. 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 current and total signed charge.

2. Find the heat dissipated over the full passage and its dependence on T.

Question 9 - Solution

A symmetric magnet passage has zero net charge but nonzero heating

(a) Method and calculation

$$i_{\mathrm{rise}}=-\frac{\Lambda_0}{RT},\quad i_{\mathrm{fall}}=+\frac{\Lambda_0}{RT}.$$$$\boxed{Q=i_{\mathrm{rise}}T+i_{\mathrm{fall}}T=0}.$$

(b) Interpretation and further calculation

$$H=\int_0^{2T}i^2Rdt=2T\frac{\Lambda_0^2}{RT^2}.$$$$\boxed{H=\frac{2\Lambda_0^2}{RT}\propto T^{-1}}.$$

Physical check

Signed charge can cancel while heat, proportional to current squared, accumulates in both halves.