CK-12-Physics - Intermediate

http://www.ck12.org Chapter 17. Circuits

V 1 +V 2 →IR 1 +IR 2 =I(R 1 +R 2 )→

Vtotal=I(Requivalent)→R 1 +R 2 = 10. 0 Ω+ 6. 0 Ω= 16. 0 Ω

In general, the total or equivalent resistance of a series circuit is the sum of the individual resistors.

Rseries equivalent=R 1 +R 2 +...

The circuit inFigure17.12 is equivalent to a circuit with one 16Ωresistor.

The total current in the circuit is

I=RequivalentV = 1612. 0 VΩ= 0. 75 A

http://demonstrations.wolfram.com/ResistorsInSeries/

Resistors in Parallel

In aparallel circuitthe current has more than one pathway available. The current encounters one or more junctions.

A junction is a point at which two or more wires connect. Resistors will not necessarily have the same current

passing through them. The voltage drop across each of the resistors in parallel, however, must be the same.

Electrical devices connected in parallel (say, two light bulbs) permit one bulb to burn out and the other to remain lit.

This is how homes are wired. If you’ve ever experienced the consequences of a burned-out fuse or tripped a circuit

breaker (we’ll discuss these in greater detail in the next section) then you’re familiar with some electrical appliances

remaining on, while others are off. The appliances (and lights) that won’t turn on must be in series with a failed part

of the circuitry.

Figure17.13 shows two resistors connected in parallel. The currentI 1 at junctionA, flows through the branch of

the circuit containing theR 1 = 10. 0 Ωresistor. The remaining currentI 2 continues on into the branch containing

theR 2 = 6. 0 Ωresistor. The total current must be equal to the sum of currentsI 1 andI 2 , as shown inFigure17.13.

Otherwise, the charge would accumulate or disappear at a junction and this would violate the law of conservation of

charge. What goes into a junctionIinmust come out of a junctionIout.

Itotal=I 1 +I 2 orIin=Iout

Notice that the voltage drop across each resistor (from red to purple) is 12 volts.

Computing the Equivalent Resistance of a Parallel Circuit

Keeping in mind that the voltage drops are the same, we useV=IR.

V=I 1 R 1 →I 1 =RV 1

V=I 2 R 2 →I 2 =RV 2

ButItotal=I 1 +I 2 →RV 1 +RV 2 =V

(

1

R 1 +

1

R 2

)

And,Itotal=RequivalentV →V

(

1

R 1 +

1

R 2

)

→Requivalent^1 =R^11 +R^12.

In general, the reciprocal of the equivalent resistance for a parallel circuit is the sum of the reciprocals of the

individual resistances.

A little algebra will show that for two resistors the total, or equivalent, resistance is

Rparallel equivalent=RR 11 +RR^22.

The total resistance inFigure17.13 is