Resistor Networks Revised

Series And Parallel Resistor Networks Revised

In previous lessons, you learnt about electric circuits and we introduced three quantities which are fundamental to dealing with electric circuits. These quantities are closely related and are current, voltage (potential difference) and resistance. To recap:

1. Electrical current, \(I\), is defined as the rate of flow of charge through a circuit.

2. Potential difference or voltage, \(V\), is related to the energy gained or lost per unit charge moving between two points in a circuit. Charge moving through a battery gains energy which is then lost moving through the circuit.

3. Resistance, \(R\), is an internal property of a circuit element that opposes the flow of charge. Work must be done for a charge to move through a resistor.

These quantities can be related, in circuit elements whose resistance remains constant, by Ohm's law.

We have focused on the properties of a single component. Now we need to look at a collection of components in a circuit.

Circuits don't consist of a single element and we've learnt about how voltage, current and resistance are affected in circuits with multiple resistors. There are two basic layouts we consider for a network of resistors, series and parallel. Resistors in series and resistors in parallel have different features when talking about current, voltage and equivalent resistance.

Example: Ohm's Law (Exam Question)

Question

Learners conduct an investigation to verify Ohm's law. They measure the current through a conducting wire for different potential differences across its ends. The results obtained are shown in the graph below.

1. Which ONE of the measured quantities is the dependent variable?

   (1 mark)

2. The graph deviates from Ohm's law at some point.

   1. Write down the coordinates of the plotted point on the graph beyond which Ohm's law is not obeyed.

      (2 marks)

   2. Give a possible reason for the deviation from Ohm's law as shown in the graph. Assume that all measurements are correct.

      (2 marks)

3. Calculate the gradient of the graph for the section where Ohm's law is obeyed.

   Use this to calculate the resistance of the conducting wire.

   (4 marks)

Question 1

Current OR I

(1 mark)

Question 2

The graph deviates from Ohm's law at some point.

1. (\(\text{4.0}\) ; \(\text{0.64}\))

   (2 marks)

2. Temperature was not kept constant.

   (2 marks)

Question 3

\begin{align*} \text{Gradient:} \quad m & = \frac{\Delta y}{\Delta x} \\ & = \frac{\text{0.64}-\text{0}}{\text{4}-\text{0}} \\ & = \text{0.16} \end{align*}\begin{align*} R & = \frac{1}{\text{0.16}} \\ & = \text{6.25}\ \Omega \end{align*}

(4 marks)

[TOTAL: 9 marks]

Note: Resistors connected in series

Resistors are in series if they are consecutive elements in the sequence of the circuit and there are no branches between them. For \(n\) resistors in series the equivalent resistance is:

\[R_{s} = R_{1} + R_{2} + R_{3} + \ldots + R_{n}\]

For \(n\) resistors in series the potential difference is split across the resistors:

\[V_{\text{Total}} = V_{1} + V_{2} + V_{3} + \ldots + V_{n}\]

The current is constant through the resistors.

\[I_{\text{Total}} = I_{1} = I_{2} = I_{3} = \ldots = I_{n}\]

It makes sense to remind ourselves of why these are consistent with other topics wehave covered previously:

• Conservation of charge: we have learnt that charges arenot created or destroyed. This is consistent with the current being the same throughout aresistor network that is in series. Charges aren't being added or lost or bunching up and, therefore, the rate at which charge moves past each point should be the same.
• Conservation of energy: we have learnt that energy isn't created or destroyed but transferred through work. The voltage across a resistor is the energy per unit charge (work) required to move through the resistor. The total work done to move through a network of resistors in series should be the sum of the work done to move through each individual resistor.