Page 959 - College Physics For AP Courses

Chapter 21 | Circuits, Bioelectricity, and DC Instruments 947
carefully and consistently determine the signs of the potential changes for each element using the four bulleted points discussed above in conjunction with Figure 21.26.
4. Solve the simultaneous equations for the unknowns. This may involve many algebraic steps, requiring careful checking and rechecking.
5. Check to see whether the answers are reasonable and consistent. The numbers should be of the correct order of magnitude, neither exceedingly large nor vanishingly small. The signs should be reasonable—for example, no resistance should be negative. Check to see that the values obtained satisfy the various equations obtained from applying the rules. The currents should satisfy the junction rule, for example.
The material in this section is correct in theory. We should be able to verify it by making measurements of current and voltage. In fact, some of the devices used to make such measurements are straightforward applications of the principles covered so far and are explored in the next modules. As we shall see, a very basic, even profound, fact results—making a measurement alters the quantity being measured.
Check Your Understanding
Can Kirchhoff’s rules be applied to simple series and parallel circuits or are they restricted for use in more complicated circuits that are not combinations of series and parallel?
Solution
Kirchhoff's rules can be applied to any circuit since they are applications to circuits of two conservation laws. Conservation laws are the most broadly applicable principles in physics. It is usually mathematically simpler to use the rules for series and parallel in simpler circuits so we emphasize Kirchhoff’s rules for use in more complicated situations. But the rules for series and parallel can be derived from Kirchhoff’s rules. Moreover, Kirchhoff’s rules can be expanded to devices other than resistors and emfs, such as capacitors, and are one of the basic analysis devices in circuit analysis.
Making Connections: Parallel Resistors
A simple circuit shown below – with two parallel resistors and a voltage source – is implemented in a laboratory experiment with ɛ = 6.00 ± 0.02 V and R1 = 4.8 ± 0.1 Ω and R2 = 9.6 ± 0.1 Ω. The values include an allowance for experimental uncertainties as they cannot be measured with perfect certainty. For example if you measure the value for a resistor a few times, you may get slightly different results. Hence values are expressed with some level of uncertainty.
Figure 21.28
In the laboratory experiment the currents measured in the two resistors are I1 = 1.27 A and I2 = 0.62 A respectively. Let us examine these values using Kirchhoff’s laws.
For the two loops,
E - I1R1 = 0 or I1 = E/R1
E - I2R2 = 0 or I2 = E/R2
Converting the given uncertainties for voltage and resistances into percentages, we get
E = 6.00 V ± 0.33%
R1 = 4.8 Ω ± 2.08%
R2 = 9.6 Ω ± 1.04%
We now find the currents for the two loops. While the voltage is divided by the resistance to find the current, uncertainties in voltage and resistance are directly added to find the uncertainty in the current value.
I1 = (6.00/4.8) ± (0.33%+2.08%) = 1.25 ± 2.4%
= 1.25 ± 0.03 A
I2 = (6.00/9.6) ± (0.33%+1.04%) = 0.63 ± 1.4%
= 0.63 ± 0.01 A
Finally you can check that the two measured values in this case are within the uncertainty ranges found for the currents. However there can also be additional experimental uncertainty in the measurements of currents.