Equivalent resistance in parallel

The idea

For resistors in parallel, the reciprocal of the equivalent resistance equals the sum of the reciprocals of the individual resistances: 1/Rp = 1/R1 + 1/R2 + 1/R3.

Joining three paths

Three resistors of 5 Ω, 10 Ω and 30 Ω are connected in parallel to a battery. They can be replaced by one equivalent resistance Rp.

Which of these gives the equivalent resistance Rp of resistors in parallel?

• Add the reciprocals (1 divided by each) of the resistances, and that sum is 1 divided by Rp
• Add the resistances themselves
• Multiply the resistances

Reciprocals

Choose a set of resistors. The line at the bottom shows the reciprocals adding up.

A different kind of sum

In each case 1 divided by Rp came out as the sum of 1 divided by each resistance. For 5, 10 and 30 Ω that sum is 1/3, so Rp is 3 Ω.

For resistors in parallel, the reciprocal of the equivalent resistance equals the sum of the reciprocals of the individual resistances: 1/Rp = 1/R₁ + 1/R₂ + 1/R₃.

Notes

For parallel resistors, the reciprocal of the equivalent resistance is the sum of the reciprocals: 1/Rp = 1/R₁ + 1/R₂ + 1/R₃.

Check yourself

Resistors of 6 Ω and 3 Ω are connected in parallel. What is the equivalent resistance, in Ω?

Answer: 2 Ω

1/Rp = 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2, so Rp = 2 Ω.

Resistors of 10 Ω and 40 Ω are connected in parallel. What is the equivalent resistance, in Ω?

Answer: 8 Ω

1/Rp = 1/10 + 1/40 = 4/40 + 1/40 = 5/40 = 1/8, so Rp = 8 Ω.

The sum of the reciprocals of the resistances in a parallel combination is 1/4. What is Rp?

Which equation is correct for three resistors in parallel?

• 1/4 Ω. The sum gives 1/Rp, the reciprocal. Rp is its reciprocal, 4 Ω.
• 12 Ω. Rp is found by taking the reciprocal of 1/4, which is 4 Ω.
• 4 Ω — correct. Yes!
• 1/Rp = 1/R₁ + 1/R₂ + 1/R₃ — correct. Yes!
• Rp = R₁ + R₂ + R₃. That is the equation for resistors in series.
• Rp = 1/R₁ + 1/R₂ + 1/R₃. The sum of reciprocals gives 1/Rp, not Rp.