Calculating the Current in a Single-Loop Circuit — Question 8

Question 8

A small series-resistance change has a controlled nonlinear effect on current

A loop has fixed emf E and initial total resistance S. One element changes so total resistance becomes $S+\delta R>0$.

1. Find the exact fractional current change.

2. Expand to second order for $|\delta R|\ll S$, and compare the exact current changes for $\delta R=+.10S$ and $-.10S$.

Question 8 - Solution

A small series-resistance change has a controlled nonlinear effect on current

(a) Method and calculation

$$I_0=E/S,\quad I_1=E/(S+\delta R).$$$$\frac{I_1-I_0}{I_0}=-\frac{\delta R}{S+\delta R}=-\frac x{1+x},\quad x=\delta R/S.$$

(b) Interpretation and further calculation

$$\Delta I/I_0=-x+x^2+O(x^3).$$$$x=.10:\ \Delta I/I_0=-.0909091;\qquad x=-.10:\ \Delta I/I_0=+.111111.$$

Physical check

Equal positive and negative resistance changes do not produce exactly symmetric current changes.