Parallel Resistors
Parallel Resistors
When we add resistors in parallel to a circuit:
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There are more paths for current to flow which ensures that the current splits across the different paths.
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The voltage is the same across the resistors. The voltage across the battery in the circuit is equal to the voltage across each of the parallel resistors:
\[{V}_{\text{battery}} = {V}_{1} = {V}_{2} = {V}_{3} \ldots\] -
The resistance to the flow of current decreases. The total resistance, \({R}_{P}\), is given by:
\[\frac{1}{{R}_{P}} = \frac{1}{{R}_{1}} + \frac{1}{{R}_{2}} + \ldots\]
When resistors are connected in parallel the start and end points for all the resistors are the same. These points have the same potential energy and so the potential difference between them is the same no matter what is put in between them. You can have one, two or many resistors between the two points, the potential difference will not change. You can ignore whatever components are between two points in a circuit when calculating the difference between the two points.
Look at the following circuit diagrams. The battery is the same in all cases, all that changes is more resistors are added between the points marked by the black dots. If we were to measure the potential difference between the two dots in these circuits we would get the same answer for all three cases.
Lets look at two resistors in parallel more closely. When you construct a circuit you use wires and you might think that measuring the voltage in different places on the wires will make a difference. This is not true. The potential difference or voltage measurement will only be different if you measure a different set of components. All points on the wires that have no circuit components between them will give you the same measurements.
All three of the measurements shown in the picture below (i.e. A–B, C–D and E–F) will give you the same voltage. The different measurement points on the left have no components between them so there is no change in potential energy. Exactly the same applies to the different points on the right. When you measure the potential difference between the points on the left and right you will get the same answer.
Example: Voltages I
Question
Consider this circuit diagram:
What is the voltage across the resistor in the circuit shown?
Step 1: Check what you have and the units
We have a circuit with a battery and one resistor. We know the voltage across the battery. We want to find that voltage across the resistor.
\[{V}_{\text{battery}} = \text{2}\text{ V}\]Step 2: Applicable principles
We know that the voltage across the battery must be equal to the total voltage across all other circuit components.
\[{V}_{\text{battery}}={V}_{\text{total}}\]There is only one other circuit component, the resistor.
\[{V}_{\text{total}}={V}_{1}\]This means that the voltage across the battery is the same as the voltage across the resistor.
\begin{align*} {V}_{\text{battery}} & = {V}_{\text{total}} = {V}_{1} \\ {V}_{1} & = \text{2}\text{ V} \end{align*}Example: Voltages II
Question
Consider this circuit:
What is the voltage across the unknown resistor in the circuit shown?
Step 1: Check what you have and the units
We have a circuit with a battery and two resistors. We know the voltage across the battery and one of the resistors. We want to find that voltage across the resistor.
\begin{align*} {V}_{\text{battery}} & = \text{2}\text{ V} \\ {V}_{B} & = \text{1}\text{ V} \end{align*}Step 2: Applicable principles
We know that the voltage across the battery must be equal to the total voltage across all other circuit components that are in series.
\[{V}_{\text{battery}} = {V}_{\text{total}}\]The total voltage in the circuit is the sum of the voltages across the individual resistors
\[{V}_{\text{total}} = {V}_{A} + {V}_{B}\]Using the relationship between the voltage across the battery and total voltage across the resistors
\begin{align*} {V}_{\text{battery}} & = {V}_{\text{total}} \\\\ {V}_{\text{battery}} & = {V}_{1} + {V}_{\text{resistor}} \\ \text{2}\text{ V} & = {V}_{1} + \text{1}\text{ V} \\ {V}_{1} & = \text{1}\text{ V} \end{align*}Example: Voltages III
Question
Consider the circuit diagram:
What is the voltage across the unknown resistor in the circuit shown?
Step 1: Check what you have and the units
We have a circuit with a battery and three resistors. We know the voltage across the battery and two of the resistors. We want to find that voltage across the unknown resistor.
\begin{align*} {V}_{\text{battery}} & = \text{7}\text{ V} \\ {V}_{\text{known}} & = {V}_{A} + {V}_{C} \\ & = \text{1}\text{ V} + \text{4}\text{ V} \end{align*}Step 2: Applicable principles
We know that the voltage across the battery must be equal to the total voltage across all other circuit components that are in series.
\[{V}_{\text{battery}}={V}_{\text{total}}\]The total voltage in the circuit is the sum of the voltages across the individual resistors
\[{V}_{\text{total}} = {V}_{B} + {V}_{\text{known}}\]Using the relationship between the voltage across the battery and total voltage across the resistors
\begin{align*} {V}_{\text{battery}} & = {V}_{\text{total}} \\ {V}_{\text{battery}} & = {V}_{B} + {V}_{\text{known}} \\ \text{7}\text{ V} & = {V}_{B} + \text{5}\text{ V} \\ {V}_{B} & = \text{2}\text{ V} \end{align*}Example: Voltages IV
Question
Consider the circuit diagram:
What is the voltage across the parallel resistor combination in the circuit shown? Hint: the rest of the circuit is the same as the previous problem.
Step 1: Quick Answer
The circuit is the same as the previous example and we know that the voltage difference between two points in a circuit does not depend on what is between them so the answer is the same as above \({V}_{\text{parallel}} = \text{2}\text{ V}\).
Step 2: Check what you have and the units - long answer
We have a circuit with a battery and five resistors (two in series and three in parallel). We know the voltage across the battery and two of the resistors. We want to find that voltage across the parallel resistors, \({V}_{\text{parallel}}\).
\begin{align*} {V}_{\text{battery}} = \text{7}\text{ V} \\ {V}_{\text{known}} = \text{1}\text{ V} + \text{4}\text{ V} \end{align*}Step 3: Applicable principles
We know that the voltage across the battery must be equal to the total voltage across all other circuit components.
\[{V}_{\text{battery}}={V}_{\text{total}}\]Voltages only add algebraically for components in series. The resistors in parallel can be thought of as a single component which is in series with the other components and then the voltages can be added.
\[{V}_{\text{total}}={V}_{\text{parallel}}+{V}_{\text{known}}\]Using the relationship between the voltage across the battery and total voltage across the resistors
\begin{align*} {V}_{\text{battery}} & = {V}_{\text{total}} \\ {V}_{\text{battery}} & = {V}_{\text{parallel}} + {V}_{\text{known}} \\ \text{7}\text{ V} & = {V}_{\text{parallel}} + \text{5}\text{ V} \\ {V}_{\text{parallel}} & = \text{2}\text{ V} \end{align*}This lesson is part of:
Electric Circuits