Derive the combined resistance of three series resistors R₁,R₂,R₃ using both Kirchhoff's laws and V=IR. Explain why the same I may be used for all three. Then generalise to any number of resistors.

Question

Accepted Answer

Charge conservation gives common I. Loop law gives V=V₁+V₂+V₃=I(R₁+R₂+R₃)=IR_eq, so R_eq=R₁+R₂+R₃ and generally ΣR_i. With no branch junctions, charge conservation gives the same steady current I through every series resistor. Kirchhoff's loop law gives total p.d. V = V₁ + V₂ + …, taking each resistive drop in the current direction. Substitute each Vᵢ = IRᵢ and the equivalent relation V = IR_eq. Factor and cancel the common nonzero I: IR_eq = I(R₁ + R₂ + …), so R_eq = R₁ + R₂ + … . The result applies to any number of series resistors at the specified operating conditions, not only two equal resistors.

A symbolic series derivation

Derive the combined resistance of three series resistors R₁,R₂,R₃ using both Kirchhoff's laws and V=IR. Explain why the same I may be used for all three. Then generalise to any number of resistors.

What a complete response should do

Your self-assessment