16.1 Potential difference  (Page 2/2)

Potential difference and series resistors

When resistors are in series, one after the other, there is a potential difference across each resistor. The total potential difference across a set of resistors in series is the sum of the potential differences across each of the resistors in the set. This is the same as falling a large distance under gravity or falling that same distance (difference) in many smaller steps. The total distance (difference) is the same.

Look at the circuits below. If we measured the potential difference between the black dots in all of these circuits it would be the same just like we saw above. So we now know the total potential difference is the same across one, two or three resistors. We also know that some work is required to make charge flow through each one, each is a step down in potential energy. These steps add up to the total drop which we know is the difference between the two dots.

Let us look at this in a bit more detail. In the picture below you can see what the different measurements for 3 identical resistors in series could look like. The total voltage across all three resistors is the sum of the voltages across the individual resistors.

Ohm's law

The voltage is the change in potential energy or work done when charge moves between two points in the circuit. The greater the resistance to charge moving the more work that needs to be done. The work done or voltage thus depends on the resistance. The potential difference is proportional to the resistance.

Ohm's Law

Voltage across a circuit component is proportional to the resistance of the component.

Use the fact that voltage is proportional to resistance to calculate what proportion of the total voltage of a circuit will be found across each circuit element.

We know that the total voltage is equal to ${\mathrm{V}}_1$ in the first circuit, to ${\mathrm{V}}_1$ + ${\mathrm{V}}_2$ in the second circuit and ${\mathrm{V}}_1$ + ${\mathrm{V}}_2$ + ${\mathrm{V}}_3$ in the third circuit.

We know that the potential energy lost across a resistor is proportional to the resistance of the component. The total potential difference is shared evenly across the total resistance of the circuit. This means that the potential difference per unit of resistance is

Then the voltage across a resistor is just the resistance times the potential difference per unit of resistance

Emf

When you measure the potential difference across (or between) the terminals of a battery you are measuring the “electromotive force” (emf) of the battery. This is how much potential energy the battery has to make charges move through the circuit. This driving potential energy is equal to the total potential energy drops in the circuit. This means that the voltage across the battery is equal to the sum of the voltages in the circuit.

We can use this information to solve problems in which the voltages across elements in a circuit add up to the emf.

What is the voltage across the resistor in the circuit shown?

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.

   $$V_{battery}=2\mathrm{V}$$
2. Applicable principles

   We know that the voltage across the battery must be equal to the total voltage across all other circuit components.

   $$V_{battery}=V_{total}$$

   There is only one other circuit component, the resistor.

   $$V_{total}=V_1$$

   This means that the voltage across the battery is the same as the voltage across the resistor.

   $$V_{battery}=V_{total}=V_1$$
   $$V_{battery}=V_{total}=V_1$$
   $$V_1=2\mathrm{V}$$

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What is the voltage across the unknown resistor in the circuit shown?

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.

   $$V_{battery}=2\mathrm{V}$$
   $$V_{\mathrm{A}}=1\mathrm{V}$$
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.

   $$V_{battery}=V_{total}$$

   The total voltage in the circuit is the sum of the voltages across the individual resistors

   $$V_{total}=V_{\mathrm{A}}+V_{\mathrm{B}}$$

   Using the relationship between the voltage across the battery and total voltage across the resistors

   $$V_{battery}=V_{total}$$
   $$\begin{array}{ccc}\hfill V_{battery}& =& V_1+V_{resistor}\hfill \\ \hfill 2\mathrm{V}& =& V_1+1\mathrm{V}\hfill \\ \hfill V_1& =& 1\mathrm{V}\hfill \end{array}$$

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What is the voltage across the unknown resistor in the circuit shown?

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.

   $$V_{battery}=7\mathrm{V}$$
   $$\begin{array}{ccc}\hfill V_{known}& =& V_{\mathrm{A}}+V_{\mathrm{C}}\hfill \\ & =& 1\mathrm{V}+4\mathrm{V}\hfill \end{array}$$
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.

   $$V_{battery}=V_{total}$$

   The total voltage in the circuit is the sum of the voltages across the individual resistors

   $$V_{total}=V_{\mathrm{B}}+V_{known}$$

   Using the relationship between the voltage across the battery and total voltage across the resistors

   $$V_{battery}=V_{total}$$
   $$\begin{array}{ccc}\hfill V_{battery}& =& V_{\mathrm{B}}+V_{known}\hfill \\ \hfill 7\mathrm{V}& =& V_{\mathrm{B}}+5\mathrm{V}\hfill \\ \hfill V_{\mathrm{B}}& =& 2\mathrm{V}\hfill \end{array}$$

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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.

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 $V_{parallel}=2\mathrm{V}$ .

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, $V_{parallel}$ .

   $$V_{battery}=7\mathrm{V}$$
   $$V_{known}=1\mathrm{V}+4\mathrm{V}$$
3. Applicable principles

   We know that the voltage across the battery must be equal to the total voltage across all other circuit components.

   $$V_{battery}=V_{total}$$

   Voltages only add 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.

   $$V_{total}=V_{parallel}+V_{known}$$

   Using the relationship between the voltage across the battery and total voltage across the resistors

   $$V_{battery}=V_{total}$$
   $$\begin{array}{ccc}\hfill V_{battery}& =& V_{parallel}+V_{known}\hfill \\ \hfill 7\mathrm{V}& =& V_1+5\mathrm{V}\hfill \\ \hfill V_{parallel}& =& 2\mathrm{V}\hfill \end{array}$$

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