VOLTAGES ACROSS SERIES RESISTANCES

In a series circuit, the voltage is divided up among the components. The sum total of the potential differences across each resistance is equal to the dc power-supply or battery voltage. This is always true, no matter how large or how small the resistances and whether or not they’re all the same value. If you think about this for a moment, it’s easy to see why this is true.
Look at the schematic diagram of Fig. 1. Each resistor carries the same current. Each resistor R_n has a potential difference E_n across it equal to the product of the current and the resistance of that particular resistor. These E_n values are in series, like cells in a battery, so they add together. What if the E_n values across all the resistors added up to something more or less than the supply voltage E? Then there would be a “phantom emf” someplace, adding or taking away voltage. However, there can be no such thing. An emf cannot come out of nowhere.
Look at this another way. The voltmeter V in Fig. 1 shows the voltage E of the battery because the meter is hooked up across the battery. The meter V also shows the sum of the E_n values across the set of resistors simply because the meter is connected across the set of resistors. The meter

Fig. 1. Analysis of voltage in a series dc circuit. See text for discussion.

says the same thing whether you think of it as measuring the battery voltage E or as measuring the sum of the E_n values across the series combination of resistors. Therefore, E is equal to the sum of the E_n values.
This is a fundamental rule in series dc circuits. It also holds for common utility ac circuits almost all the time.
How do you find the voltage across any particular resistor Rn in a circuit like the one in Fig. 1? Remember Ohm’s law for finding voltage: E = IR. The voltage is equal to the product of the current and the resistance. Remember, too, that you must use volts, ohms, and amperes when making calculations. In order to find the current in the circuit I, you need to know the total resistance and the supply voltage. Then I = E/R. First find the current in the whole circuit; then find the voltage across any particular resistor.