Superconductors — Question 3

Question 3

Zero resistance does not eliminate the voltage needed to change loop current

A superconducting coil remains below all critical limits. Its supplied model is $V=L\,dI/dt$ and stored energy $U=LI^2/2$. Take L=.400 H and ramp current uniformly from zero to 30.0 A in 6.00 s.

1. Find the required voltage and final stored energy.

2. Integrate terminal power during the ramp and compare with holding the final current steady.

Question 3 - Solution

Zero resistance does not eliminate the voltage needed to change loop current

(a) Method and calculation

$$dI/dt=30.0/6.00=5.00\,\mathrm{A/s},\quad V=.400(5.00)=2.00\,\mathrm V.$$$$U_f=\tfrac12(.400)(30.0)^2=180\,\mathrm J.$$

(b) Interpretation and further calculation

$$W=\int_0^{6}VI\,dt=\int_0^6(2.00)(5.00t)dt=180\,\mathrm J.$$$$dI/dt=0\Rightarrow V=0,\quad P=VI=0\quad\text{during an ideal steady hold}.$$

Physical check

Ramp energy is stored rather than dissipated; refrigerator power is separate from ideal coil terminal power.