# Series and Parallel Circuits

## Series and Parallel Circuits Revision

**Series and Parallel Circuits**

A **series circuit** is a circuit that consists of one closed loop in which all the components are joined by **wires** end to end. A **parallel circuit** consists of two or more loops connected to the same **power** source.

**Kirchhoff’s Laws**

**Kirchhoff’s first law **states that “the sum of all currents entering a junction is always equal to the current leaving a junction”. This shows the ideas of **conservation of energy** and conservation of charge.

**Example: **In the circuit below, A_1 measures a current of 10 \text{ A}. Assuming both branches of the circuit are identical, what would the other ammeter readings show?

**[5 marks]**

Applying Kirchoff’s first law:

\begin{aligned} \bold{A_1} &= \bold{10} \textbf{ A} \\ \bold{A_2} &= \bold{5} \textbf{ A} \\ \bold{A_3} &= \bold{5} \textbf{ A} \\ \bold{A_4} &= \bold{5} \textbf{ A} \\ \bold{A_5} &= \bold{5} \textbf{ A} \\ \bold{A_6} &= \bold{10} \textbf{ A} \end{aligned}

**Kirchhoff’s second law** states that the total emf in a circuit is equal to the sum of the **potential differences** across each component.

**Example: **Calculate the reading on both voltmeters assuming that the resistors are identical.

**[2 marks]**

Kirchoff’s seconds law:

\bold{V_1 + V_2 = 12}\textbf{ V} \\ V_1=V_2= \bold{6}\textbf{ V}

**Voltage and Current**

It is important to understand the characteristics of **series** and **parallel** circuits. These characteristics can then be used to form a basis for our calculations.

**Series circuits**

In a series circuit, the **current** is the same at all points of the circuit and therefore, all components have the same **current**.

I_t= I_1 = I_2=I_3 ....

However, **potential difference** is shared among the components of a **series** circuit.

V_t=V_1+V_2+V_3

If more than one cell is connected in **series**, the total **voltage** produced is equal to the sum of all the cells.

**Parallel circuits **

In a **parallel** circuit, the **current** is shared among the branches of the circuit. Therefore:

I_t=I_1+I_2+I_3 ...

However, **potential difference** is equal in all branches of a **parallel** circuit.

V_t=V_1=V_2=V_3

If cells are connected in **parallel**, the total **terminal pd **is the same as one of the cells.

**Calculating Resistance**

The **total resistance **in a circuit is calculated differently depending upon if the circuit is **series** or **parallel**.

**Resistors** in a series circuit are added together to find the total **resistance**. This can be summarised in the equation:

R_t=R_1+R_2+R_3...

**Example:** what is the total resistance in this circuit?

**[1 mark]**

\begin{aligned} R_t &= R_1 + R_2 + R_3... \\ &= 20+12 \\ &= \bold{32 \, \Omega} \end{aligned}

**Resistors** in **parallel** are more difficult to combine. The total **resistance** when **resistors** are added in **parallel** can be calculated using the equation:

\dfrac{1}{R_t} = \dfrac{1}{R_1}+\dfrac{1}{R_2}+\dfrac{1}{R_3}...

**Example:** what is the total resistance in this circuit?

**[3 marks]**

This example is made of resistors in series with one in parallel. Firstly, we must add the two resistors in parallel.

\begin{aligned}\dfrac{1}{R_t} &= \dfrac{1}{R_1}+\dfrac{1}{R_2}+\dfrac{1}{R_3}... \\ &= \bold{\dfrac{1}{200}+\dfrac{1}{750}} \\ &= 6.33 \times 10^{-3} \\ R_t &= \dfrac{1}{6.33 \times 10^{-3}} \\ &= \bold{158 \, \Omega} \end{aligned}

Now add the other resistor in series:

\begin{aligned} R_t&=R_1+R_2+... \\ &= 158 + 500 \\ &= \bold{658 \Omega} \end{aligned}

Using this equation, the **total resistance** of resistors combined in **parallel** is less than the **resistance** of the smallest resistor. Therefore, combining resistors in **parallel** can be used to reduce the **resistance** in a circuit.

**Electrical Energy and Power**

**Power** is defined as **“the rate of doing ****work****”** or how much **energy** is used by a circuit component per unit of time. It is measured in Watts \text{(W)} or \text{(Js}^{-1}\text{)} as an alternative unit.

There are several different ways of calculating **electrical power**, depending on what information you have been given:

P=\dfrac{E}{t}

- P=
**power**in watts \text{(W)} - E=
**energy**in joules \text{(J)} - t=
**time**in seconds \text{(s)}

P=VI

- P=
**power**in watts \text{(W)} - V=
**potential difference**in volts \text{(V)} - I=
**current**in amps \text{(A)}

P=\dfrac{V^2}{R}

- P=
**power**in watts \text{(W)} - V=
**potential difference**in volts \text{(V)} - R=
**resistance**in ohms (\Omega)

P=I^2R

- P=
**power**in watts \text{(W)} - I=
**current**in amps \text{(A)} - R=
**resistance**in ohms (\Omega)

## Series and Parallel Circuits Example Questions

**Question 1:** Three 10 \Omega resistors are connected in parallel. What is the total resistance of the parallel circuit?

**[2 marks]**

**Question 2:** Two resistors are connected in series with a 6 \text{ V} power supply. R_1 receives 3.5 \text{ V} how much potential difference does R_2 receive and explain why.

**[2 marks]**

V_t = V_1+V_2 + V_3... for a series circuit.

Therefore, V_2 receives 6 - 3.5 =\bold{2.5} \textbf{ V}

This is due to **Kirchhoff’s second law **which stated that “the total emf in a circuit is equal to the sum of the potential differences across each component.”

**Question 3:** In 20 minutes, 150 \text{ kJ} is transferred to a heater. Calculate the power of the heater.

**[2 marks]**