DC Circuits

RESISTORS IN PARALLEL

http://www.physics.uoguelph.ca/tutorials/ohm/blueball.gifResistors can be connected such that they branch out from a single point (known as a node), and join up again somewhere else in the ciruit. This is known as a parallel connection. Each of the three resistors in Figure 1 is another path for current to travel between points A and B.

http://www.physics.uoguelph.ca/tutorials/ohm/ohmE10.gif
Figure 1 Example of a circuit containing three resistors connected in parallel

http://www.physics.uoguelph.ca/tutorials/ohm/ohmE10b.gif
Figure2 Circuit containing resistors in parallel, equivalent to Figure 1

Note that the node does not have to physically be a single point; as long as the current has several alternate paths to follow, then that part of the circuit is considered to be parallel. Figures 1 and 2 are identical circuits, but with different appearances.

http://www.physics.uoguelph.ca/tutorials/ohm/blueball.gifAt A the potential must be the same for each resistor. Similarly, at B the potential must also be the same for each resistor. So, between points A and B, the potential difference is the same. That is, each of the three resistors in the parallel circuit must have the same voltage.

[1]http://www.physics.uoguelph.ca/tutorials/ohm/ohmE11.gif

Also, the current splits as it travels from A to B. So, the sum of the currents through the three branches is the same as the current at A and at B (where the currents from the branch reunite).

[2]http://www.physics.uoguelph.ca/tutorials/ohm/ohmE12.gif

By Ohm's Law, equation [2] is equivalent to:

[3]http://www.physics.uoguelph.ca/tutorials/ohm/ohmE13.gif

By equation [1], we see that all the voltages are equal. So the V's cancel out, and we are left with

[4]http://www.physics.uoguelph.ca/tutorials/ohm/ohmE14.gif

This result can be generalized to any number of resistors connected in parallel.

[5]http://www.physics.uoguelph.ca/tutorials/ohm/ohmE15.gif

Since resistance is the reciprocal of conductance, equation [5] can be expressed in terms of conductances.

[6]http://www.physics.uoguelph.ca/tutorials/ohm/ohmE16.gif


http://www.physics.uoguelph.ca/tutorials/ohm/blueball.gifExample Problem on Resistors in Parallel
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