6 DC Circuits

Introduction

Kirchhoﬀ’s Rules

In Lab 5,you explored the behavior of circuits with resistors connected in series and in

parallel.You were able to analyze the behavior of these circuits using equations for equivalent

resistance.This approach does not generally work for circuits with more than one power

source.While equivalent resistance still has some application,in a multi-loop,multi-source

circuit like the one you will be investigating,you will need a more general approach to circuit

analysis —Kirchhoﬀ’s rules.

The ﬁrst rule,the loop rule,states that the total change in electric potential around a

closed loop must be zero.This means that the sum of the potential diﬀerences across all of

the devices in a closed loop must be zero,

loop

ΔV

i

= 0 (1)

The second rule,known as the junction rule,applies to currents.This rule states that no

net current enters or leaves a junction,

junction

i

i

= 0 (2)

These rules became evident in your measured potential diﬀerences and currents in Lab 5.

Here,you will use Kirchhoﬀ’s rules to predict the behavior of a more complex circuit.

Real Batteries:Internal Resistance

You will be investigating a multi-loop circuit powered by two D-cell batteries.In order to

make sense of the behavior of the batteries,you will need to account for the fact that real

batteries have internal resistance.This internal resistance reduces the potential diﬀerence

between the battery terminals whenever it is delivering current.

ℰ

r

i

ΔV

Figure 1:Determining the internal resistance r of a battery.

If you measure the potential diﬀerence ΔV across the terminals of a battery as shown in

Fig.1 while it is delivering a current i,you will not measure its emf (E).Instead you will

measure

ΔV = E −ir (3)

where r is the internal resistance of the battery.When the battery is disconnected from the

circuit,so that i = 0,then a measurement of the potential diﬀerence across its terminals gives

E.It is therefore possible to calculate the internal resistance of a battery frommeasurements

of ΔV,i and E using

r =

E −ΔV

i

(4)

Experiment:a Multi-loop Circuit with Two Power Sources

ℰ

2

R

2

R

3

i

3

i

1

R

1

i

2

ℰ

1

r

1

r

2

B

CA

D

Figure 2:A multi-loop DC circuit with two power sources and three resistors.

You have been supplied with two D-cell batteries in holders,a set of three resistors mounted

on a Plexiglas stand,a digital multimeter (DMM),and several leads for connecting these

components.

1.Turn the knob on the DMMto measure resistance on the hundreds of Ω scale.Measure

and record the resistance of each resistor with the DMM by connecting one end of to

the V/Ω terminal and the other end to the COM terminal.Check your results with the

color codes.

2.Measure and record the emfs (E values) of your batteries.It is important not to connect

them to anything other than the DMM when you make these measurements.(Why?)

3.Construct the circuit shown in Figure 2.Make sure that the batteries’ orientations are

correct and that you record which E value goes with which battery.

4.Turn the knob on the DMM to measure potential diﬀerence on a scale of a few Volts.

Measure the potential diﬀerence ΔV

BA

= V

B

−V

A

between the points labeled A and B

in Fig.2 by connecting the V/Ω terminal of the DMMto point B and the COM terminal

of the DMM to point A.Similarly measure ΔV

CB

,ΔV

CA

,ΔV

DC

,and ΔV

DA

.

5.Use the DMM to measure magnitudes and directions the three unique currents in the

circuit i,i

1

,and i

2

.This will require you to connect the DMM in series with each

separate branch of the circuit.Remember never to connect an ammeter in parallel

with anything!Use the red current terminal on your DMMbest suited to measure mA

currents.Be careful not to change the way the circuit is arranged as you make these

measurements.If you’re not sure how,ask for help!

In order to determine the direction of the current,remember that a positive current

ﬂows into the red A (or mA) terminal and out of the black COM terminal.The directions

you measure may not agree with the arrows in Figure 2,and that’s OK!Record what

you actually observe.

6.Check your measurements by using your measured currents and voltages ΔV

CB

,ΔV

DC

,

and ΔV

CA

along with Ohm’s Law to calculate the resistances R

1

,R

2

,and R

3

.Com-

pare them with the actual values measured with the DMM in step 1.Resolve any

inconsistencies before you move on!

Analysis

1.Use Eq.4 and your measurements of ΔV

BA

and i

1

to ﬁnd the internal resistance r

1

of

battery 1.Similarly,use ΔV

DA

and i

2

to ﬁnd r

2

.

ℰ

2

R

2

R

3

i

3

i

1

R

1

i

2

ℰ

1

Figure 3:Simpliﬁed circuit diagram.

2.Use Kirchhoﬀ’s rules and your measured resistances and emfs to predict the magnitudes

and directions of each of the unique currents i

1

,i

2

,and i

3

.It simpliﬁes the mathematics

somewhat to combine the pairs of resistors connected in series R

′

1

= R

1

+ r

1

and

R

′

2

= R

2

+r

2

as illustrated in Fig.3.

Questions

1.Why was it important to measure the emfs (E values) of your batteries while they were

not connected to anything other than the DMM?

2.What are the internal resistances of your batteries?Show your calculations.

3.Are your measured potential diﬀerences compatible with the loop rule?

4.Show the calculations you made to predict the currents in the circuit shown in Figure 2.

Compare your predictions to your measurements,and discuss any signiﬁcant (> 5%)

discrepancies.

## Comments 0

Log in to post a comment