Current Electricity — Advanced Circuit Practice (Hard)

Quick Recap — Advanced Circuits (Multi-step)

• EMF from two loads: $\varepsilon=I_1(R_1+r)=I_2(R_2+r)$ — two conditions fix $\varepsilon$ and $r$.
• Grouping cells (total $N=mn$, $m$ rows of $n$ in series, load $R$): current is maximised when $R=\dfrac{nr}{m}$ (i.e. external $=$ internal).
• Balanced Wheatstone / symmetry: in a balanced bridge the galvanometer branch carries no current, so it can be removed or shorted when finding equivalent resistance.
• Capacitor in a DC network: in the steady state no current flows through a capacitor branch; use that branch only to find the voltage across (hence charge on) the capacitor.
• Power split: for equal resistors, series draws $\tfrac14$ the power of parallel from the same source.

Worked mini-example. A potentiometer with external 5 $\Omega$ shifts its balance from 100 cm to 80 cm on closing the cell's circuit: $r=R\dfrac{l_1-l_2}{l_2}=5\times\dfrac{20}{80}=1.25\ \Omega$.