Combining resistors and Kirchhoff's Laws

In a nutshell

Kirchhoff's second law states that in any circuit, the sum of the electromotive forces is equal to the sum of all the potential differences in a closed loop. The total resistance of resistors connected in series is the sum of all the resistances. The reciprocal of the total resistance of resistors connected in parallel is the sum of all the reciprocal resistances.

Equations

Variable definitions

Kirchhoff's laws

Kirchhoff's first law derives from the conservation of charge and it states that at any point in an electrical circuit, the sum of the currents going into that point is equal to the sum of the currents going out of it. It can be represented with the equation:

​$$\Sigma I_{in}=\Sigma I_{out}$$​​

Kirchhoff's second law derives from the conservation of energy. It states that for any circuit, the sum of the electromotive forces is equal to the sum of all the potential differences in a closed loop. Mathematically it can be represented with the equation:

​$$\Sigma\varepsilon=\Sigma V$$​​

Circuits in series

Current, voltage and resistance behave differently depending on whether components are connected in series or in parallel.

When connected in series, according to Kirchhoff's first law the current must be the same before and after going through the component, hence it is the same everywhere. For voltage Kirchhoff's second law applies meaning that the sum of the voltages in the loop is equal to the sum of the electromotive forces. 

​$$\varepsilon_1+\varepsilon_2=V_1+V_2+V_3$$ $$I$$ is the same everywhere​​

Circuits in parallel

When components are connected in parallel, due to Kirchhoff's first law, the current is split between them. For voltage instead Kirchhoff's second law states that the sum of voltages is equal to the electromotive forces in a loop, which means that the voltages will be equal to the electromotive forces for each loop.

​$$\varepsilon_1+\varepsilon_2=V_1=V_2$$ $$I=I_1+I_2$$​​​

Resistance in series and parallel

Resistance is related to current and voltage through the equation:

​$$V=IR$$​​

This means that it can be used together with the relations in series and parallel above to find how resistance behaves in those situations.

In a series circuit, the total resistance of the circuit is equal to the sum of the resistance of all the components:

​$$R=R_1+R_2+...$$​​

When resistors are placed in parallel, the reciprocal of the total resistance is the sum of all the reciprocals of the resistance of all the components:

​$$\dfrac{1}{R}=\dfrac{1}{R_1}+\dfrac{1}{R_2}+...$$​​

Example 

In the circuit below $$R_1$$ has a resistance of $$10\ \Omega$$ and both $$R_2$$​ and $$R_3$$ have a resistance of $$5\ \Omega$$. What is the total resistance of the circuit?

Firstly, write down all the known values:

$$\begin{aligned}R_1&=10\ \Omega \\R_2&=5\ \Omega \\R_3&=5\ \Omega\end{aligned}$$​​

Next, since $$R_2$$ and $$R_3$$ are connected in series, you can apply the equation for resistances in series:

$$R=R_1+R_2+...$$​​

In this case it becomes:

$$R_{23}=R_2+R_3$$​​

Substitute the values into the equation and calculate their combined resistance:

​$$R_{23}=5+5=10\ \Omega$$​​

This combined resistance can be treated as one resistor connected in parallel to $$R_1$$. To find the total resistance then you need to use the equation for resistances in parallel:

​$$\dfrac{1}{R}=\dfrac{1}{R_1}+\dfrac{1}{R_2}+...$$​​

For this case it becomes:

​$$\dfrac{1}{R}=\dfrac{1}{R_1}+\dfrac{1}{R_{23}}$$​​

Substitute the values and calculate the reciprocal of the total resistance:

$$\dfrac{1}{R}=\dfrac{1}{10}+\dfrac{1}{10}=\dfrac{1}{5}\$$​​

Finally to find the total resistance calculate the reciprocal of this value:

$$\left(\dfrac{1}{5}\right)^{-1}=5\ \Omega$$​​

The total resistance of this circuit is $$\underline{5\ \Omega}$$.