Resistors in Parallel

Resistors connected in parallel have the same voltage across them, however, the current splits at the branch node, and each resistor has a different current flowing through it. The sum of the current through each branch is equal to the dc current provided by the power source. Although many examples show two resistor circuits, you can have any number of resistors connected in parallel. The calculator provided here might be useful for basic GCSE style questions.

Calculator

Derivation

The voltage across resistor R1 is V, therefore the above expression using Ohm’s Law describes the current through it.

The voltage across resistor R2 is also V because both resistors, R1 and R2 are in parallel with the same voltage source across them. Therefore, the above expression describes the current through resistor R2.

This expression describes the total current in terms of voltage and total effective resistance.

The total current is the sum of individual currents flowing through each resistor. They combine at the node and equate to the current from the source. The current therefore obeys Kirchhoff’s first Law, that the currents entering the node equal the current leaving the node.

If we substitute the expressions for current in the previous stages in this formula, we get the expression shown below.

As you can see, the voltage V is the same throughout the expression, and therefore we divide throughout by V to cancel it out thus leaving the familiar formula for resistors in parallel.

This is the formula for resistors in parallel.

Reciprocal Formula

This is the reciprocal formula for resistors in parallel and provides the total resistance.

Shortcut Formula

This is a simplified shortcut formula, which is easier to work with.

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