Induction and Energy Transfers — Question 10

Question 10

A constant induced emf minimizes heat for a prescribed linkage change and duration

A resistive circuit of resistance R undergoes a prescribed total external linkage change DeltaLambda over duration T. Self-inductance is negligible, but the time profile can be chosen. Use the inequality $(\int_0^T fdt)^2\le T\int_0^T f^2dt$.

1. Derive a lower bound on heat.

2. Identify the linkage history that attains it and compare a triangular emf pulse with the same voltage-time integral.

Question 10 - Solution

A constant induced emf minimizes heat for a prescribed linkage change and duration

(a) Method and calculation

$$H=\frac1R\int_0^T\mathcal E^2dt,\quad\int_0^T\mathcal E dt=-\Delta\Lambda.$$$$\boxed{H\ge(\Delta\Lambda)^2/(RT)}.$$

(b) Interpretation and further calculation

$$\mathcal E=-\Delta\Lambda/T\text{ constant}\Rightarrow\Lambda(t)=\Lambda(0)+\Delta\Lambda\,t/T.$$$$\text{triangular pulse: }E_{\mathrm{peak}}=2|\Delta\Lambda|/T,\quad H=E_{\mathrm{peak}}^2T/(3R)=\boxed{\tfrac43H_{\min}}.$$

Physical check

The bound is on the prescribed resistive model; allowing self-inductance introduces stored magnetic energy and changes the current response.