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Resistors Connected in Series

If several resistors (e.g. three) are connected in a series connection with a power source, we obtain the following circuit diagram.

Resistor Circuits !

If measurements of current I are carried out in this circuit at different locations using an ammeter, it is discovered that the same value is always obtained regardless of the location of the measurement. In the above example this means that I1 = I2. In general the following holds true:

In a series connection of resistors the current I through each of the resistors is the same.

Now if you measure the voltage drops U1, U2 and U3 across the individual resistors and you compare this with the total voltage U (supplied by the voltage source), you arrive at the following law:

In a series connection the total voltage is equal to the sum of the partial voltages, i.e. it holds that Resistor Circuits !

The following circuit diagram illustrates this relationship.

Resistor Circuits !

This law is a simplified version of Kirchhoff's second law, which states that

The algebraic sum of all voltages in a closed circuit is equal to the sum of all voltage drops across the circuit's resistors.

This law is frequently referred to as Kirchhoff's voltage law.

The voltage drops across the individual resistors can be computed with the aid of Ohm's law from the resistor values and the current I, whereby

Resistor Circuits !

If the total resistance of a series connection is termed Rtot then, of course, Ohm's law also applies to the entire circuit, i.e. it holds true that

Resistor Circuits !

Rges = Rtot

If you now apply Kirchhoff's law and Ohm's law for the individual resistors, we obtain

Resistor Circuits !

For the total resistance of the series circuit we thus obtain the following expression

Resistor Circuits !

i.e.

In a series circuit the total resistance (also called equivalent resistance) is equal to the sum of the partial resistances.

Since the same current flows through all of the resistors, the partial voltages dropping across the resistors on account of Ohm's law and the resistors themselves respond as follows:

In a series circuit the ratio of the partial voltages is equal to the ratio of corresponding partial resistances.

Example: A voltage of U = 10 V is applied to a threeresistors R1 = 10 W, R2 = 40 W and R3 = 50 W connected in series. This results in a total resistance Rtot = 100 W and thus a current I = U/Rtot = 0.1 A. For the voltage drops across the partial resistors, the following values result U1 = I·R1 = 1 V, U2 = I·R2 = 4 V and U3 = I·R3 = 5 V.

The following interactive animation shows a circuit with two resistors connected in series. When you modify the resistor values and the voltage supply of the circuit, observe how the current and the voltage drops across the resistors change

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