Derive the equivalent resistance of three parallel resistors R₁,R₂,R₃ from Kirchhoff's laws and V=IR. Explain why voltage is common and current is summed, and use the result to show that R_eq is smaller than every positive branch resistance.

Question

Accepted Answer

Shared nodes give common V; junction law gives I=ΣI_i. Thus V/R_eq=V/R₁+V/R₂+V/R₃ and 1/R_eq=Σ1/R_i. Since the reciprocal sum exceeds each individual reciprocal, R_eq is below each branch resistance. Explain common p.d. from same node pair/loop law. Use junction current conservation. Substitute each I_i=V/R_i and I=V/R_eq. Cancel V to obtain reciprocal sum. Use positivity to justify equivalent resistance below each branch.

A symbolic parallel derivation

Derive the equivalent resistance of three parallel resistors R₁,R₂,R₃ from Kirchhoff's laws and V=IR. Explain why voltage is common and current is summed, and use the result to show that R_eq is smaller than every positive branch resistance.

What a complete response should do

Your self-assessment

Derive the equivalent resistance of three parallel resistors R₁,R₂,R₃ from Kirchhoff's laws and V=IR. Explain why voltage is common and current is summed, and use the result to show that Req is smaller than every positive branch resistance.

What a complete response should do

Your self-assessment

Derive the equivalent resistance of three parallel resistors R₁,R₂,R₃ from Kirchhoff's laws and V=IR. Explain why voltage is common and current is summed, and use the result to show that R_eq is smaller than every positive branch resistance.