Series Resistances

The two basic circuit layouts are parallel and series, so a pair of resistors in series, (a), is another of the most basic circuits we can make. By conservation of charge, all the current that flows through one resistor must also flow through the other (as well as through the battery):

I₁ = I₂ .

The only way the information about the two resistance values is going to be useful is if we can apply Ohm's law, which will relate the resistance of each resistor to the current flowing through it and the voltage difference across it. Figure (b) shows the three constant-voltage areas. Voltage differences are more physically significant than voltages, so we define symbols for the voltage differences across the two resistors in figure (c).

We have three constant-voltage areas, with symbols for the difference in voltage between every possible pair of them. These three voltage differences must be related to each other. It is as though I tell you that Fred is a foot taller than Ginger, Ginger is a foot taller than Sally, and Fred is two feet taller than Sally. The information is redundant, and you really only needed two of the three pieces of data to infer the third. In the case of our voltage differences, we have

The absolute value signs are because of the ambiguity in how we define our voltage differences. If we reversed the two probes of the voltmeter, we would get a result with the opposite sign. Digital voltmeters will actually provide a minus sign on the screen if the wire connected to the "V" plug is lower in voltage than the one connected to the "COM" plug. Analog voltmeters pin the needle against a peg if you try to use them to measure negative voltages, so you have to fiddle to get the leads connected the right way, and then supply any necessary minus sign yourself.

Figure (d) shows a standard way of taking care of the ambiguity in signs. For each of the three voltage measurements around the loop, we keep the same probe (the darker one) on the clockwise side. It is as though the voltmeter was sidling around the circuit like a crab, without ever "crossing its legs." With this convention, the relationship among the voltage drops becomes

ΔV ₁ + ΔV₂ = -ΔV _battery

or, in more symmetrical form,

ΔV₁ + ΔV₂ + ΔV _battery = 0

More generally, this is known as the loop rule for analyzing circuits:

the loop rule

Assuming the standard convention for plus and minus signs,

the sum of the voltage drops around any closed loop in a circuit must be zero.

Looking for an exception to the loop rule would be like asking for a hike that would be downhill all the way and that would come back to its starting point!

For the circuit we set out to analyze, the equation

ΔV₁ + ΔV₂ + ΔV _battery = 0

can now be rewritten by applying Ohm's law to each resistor:

I₁R₁ + I₂R₂ + ΔV_battery = 0 .

The currents are the same, so we can factor them out:

I(R₁+ R₂ + ΔV_battery = 0

and this is the same result we would have gotten if we had been analyzing a one-resistor circuit with resistance R₁+R₂. Thus the equivalent resistance of resistors in series equals the sum of their resistances.

Two lightbulbs in series

More than two equal resistances in series

Dependence of resistance on length

Choice of high voltage for power lines

Getting killed by your ammeter

Choice of high voltage for power lines Discussion Questions

A

We have stated the loop rule in a symmetric form where a series of voltage drops adds up to zero. To do this, we had to define a standard way of connecting the voltmeter to the circuit so that the plus and minus signs would come out right. Suppose we wish to restate the junction rule in a similar symmetric way, so that instead of equating the current coming in to the current going out, it simply states that a certain sum of currents at a junction adds up to zero. What standard way of inserting the ammeter would we have to use to make this work?