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Simple dc Circuits

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In the previous class we studied the simplest possible circuit, a single emf and a single resistor. Today we examine combinations of resistors that can be reduced to an equivalent circuit involving just one equivalent resistor.

Series Circuits

Three resistors in series
  Equivalent simple circuit
Current is the same through each resistor; in essence that is the definition of "series," that one end of each resistor is hooked together and nothing else at that point.

i = i1 = i2 = i3

The potential difference supplied by the battery is shared by the three resistors. How the voltage is shared is determined by the relative resistance of each resistor.

V = V1 + V2 + V3

   
Combining the two equations via Ohm's law:

V = i1R1 + i2R2 + i3R3
  The simple equivalent circuit is required to have the same current through the battery, given the same applied voltage:
V = i (R1 + R2 + R3)
 
V = i Req
 

Comparison of the equations above yields:

Req = R1 + R2 + R3
Generalizing:

Resistors in series

Concept Check 3

Parallel Circuits

Three resistors in parallel
  Equivalent simple circuit
The voltage difference is the same across each resistor; in essence that is the definition of "parallel."

V = V1 = V2 = V3

The current through the battery is shared by the three resistors. How the current splits among the three branches is determined by the relative resistance of each resistor.

i = i1 + i2 + i3

   
Combining the two equations via Ohm's law:

i = V1/R1 + V2/R2 + V3/R3

  The simple equivalent circuit is required to have the same current through the battery, given the same applied voltage:
i = V (1/R1 + 1/R2 + 1/R3)
 
i = V /Req
 

Comparison of the equations above yields:

1/Req = 1/R1 + 1/R2 + 1/R3
Generalizing:

Resistors in parallel

Concept Check 4

Series and Parallel Combinations

Two resistors in series
Resistors R1 and R2 are in series; one end of each is connected, with nothing else at junction. Same current flows through each resistor.
  R1 and R2 are not in series
R1 and R2 are NOT in series!
 
Two resistors in parallel
Resistors R2 and R3 are in parallel; ends of both are directly connected to each other, with nothing between. Same voltage difference across each resistor.
  R2 and R3 are not in parallel
R2 and R3 are NOT in parallel!
(or in series either.)

Detailed Series/Parallel Circuit Example

Quiz 1: Electric Fields  --  (Solution)

 

Relevant Online Resources for This Class

dc Circuits tutorial from the Department of Physics at the University of Guelph


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"http://www.physics.udel.edu/~watson/phys208/clas0220.html"
Last updated Feb. 21, 1998.
Copyright George Watson, Univ. of Delaware, 1997.