(f) analysis of circuits with more than one source of e.m.f.

The idea

When a circuit contains more than one source of e.m.f., you cannot treat the current as driven by a single p.d. Instead, apply Kirchhoff's laws: at any junction, the sum of currents entering equals the sum leaving; around any closed loop, the algebraic sum of potential differences is zero. Assign a current direction to each branch, write loop equations using the e.m.f.s and the p.d.s across resistors, and solve the simultaneous equations. For example, two batteries of 6.0 V and 4.0 V in parallel with resistors require you to define loop directions and signs consistently. The method works for any number of sources and resistors, including opposing e.m.f.s.

What earns the marks

• State Kirchhoff's junction rule: the sum of currents entering a junction equals the sum leaving.
• State Kirchhoff's loop rule: around any closed loop, the algebraic sum of e.m.f.s equals the sum of p.d.s, or the total p.d. is zero.
• Assign a current direction to each branch and apply consistent sign conventions for e.m.f.s and p.d.s.
• Write simultaneous equations for each independent loop and junction, then solve for unknown currents or p.d.s.
• Check the solution: a negative current indicates the actual direction is opposite to the assumed direction.

Common mistakes and how to put them right

• Assuming that the larger e.m.f. always determines the current direction in every branch.

  Put it right: The actual direction depends on the full circuit, and a branch may carry current driven by the smaller source if the larger source is opposed by other p.d.s.

• Adding e.m.f.s arithmetically without considering their polarities.

  Put it right: e.m.f.s in opposing senses in a loop must be subtracted, not added.

• Ignoring internal resistance of the sources.

  Put it right: Each source has internal resistance, which must be included as a resistor in series with its e.m.f. when analysing the circuit.

• Applying the loop rule to a non-closed path.

  Put it right: Kirchhoff's loop rule applies only to closed loops.

Exam tips

• Draw the circuit clearly and label all currents with assumed directions before writing equations.
• Use a consistent sign convention: for example, traversing a loop in the chosen direction, treat e.m.f.s that drive current in that direction as positive and those opposing as negative.
• Solve the simultaneous equations carefully, then interpret any negative result as a current flowing opposite to your assumed direction.