Capacitors in Parallel and in Series — Question 4

Question 4

Series and parallel measurements determine an unknown pair up to interchange

Two positive capacitances have parallel equivalent 10.0 nF and series equivalent 2.40 nF.

1. Find the individual capacitances.

2. Derive the condition on general measured values Cp and Cs for two real positive capacitances to exist.

Question 4 - Solution

Series and parallel measurements determine an unknown pair up to interchange

(a) Method and calculation

$$C_1+C_2=10.0,\quad C_1C_2=C_sC_p=24.0\quad\text{in nF units}.$$$$C^2-10C+24=0\Rightarrow\{C_1,C_2\}=\{4.00,6.00\}\,\mathrm{nF}.$$

(b) Interpretation and further calculation

$$C_{1,2}=\frac{C_p\pm\sqrt{C_p^2-4C_pC_s}}2.$$$$C_p>0,\quad0<C_s\le C_p/4.$$

Physical check

Equality corresponds to identical capacitors; the two measurements cannot identify which physical device has which value.