In this explainer, we will learn how to calculate the potential difference, current, and resistance at different points within simple series circuits.
The diagram below shows a circuit consisting of a cell and a resistor. The potential difference provided by the cell is , the current in the circuit is , and the resistance of the resistor is .
A circuit does not always consist of a single component as in the diagram above. When multiple components are found together in a closed loop, one after another in a line, they are said to be in series.
When components are in series, their resistances can be treated as if they were a single component, as seen in the diagram below.
This is done by taking the sum of all the resistances of all components in the circuit.
Rule: Total Resistance in Series
The total resistance in a circuit, , is given by where is the resistance of the first component, is the resistance of the second component, and so on.
This rule holds true no matter how many components are in the circuit, so long as they are in series.
The placement of components within a circuit will not affect the total resistance as long as the components are in series with each other. The circuit diagrams below show a cell connected with two resistors, placed in different ways.
These circuits are all equivalent. They can all be described as a cell, followed by , then .
Example 1: Identifying Equivalent Circuit Diagrams
The diagram shows two circuits. Are the two circuits equivalent to each other? If not, then why?
- No, because the current goes a different way around the circuit in the second circuit.
- No, because the resistors are in different positions.
- No, because the cell is the opposite way around in the second circuit.
- No, because the first circuit shows which side of the cell is positive and which is negative but the second circuit does not.
- Yes, the two circuits are equivalent.
Components can be connected either way around in the circuit and the total resistance will be the same. Resistors behave the same no matter how they are connected in a circuit, so the direction of the current does not matter. It is not A or C, since reversing the cell is the same as switching the current direction.
Resistors being in different positions does not affect their equivalency, so long as they are on the same path in the circuit, in series. It is not B.
The first circuit labels its current, , and the positive and negative terminals of its cell. The second circuit does not label the current or cell terminals, but we still understand that it has a cell and current. Labeling or not labeling a circuit does not affect its equivalency. It is not D.
There is a cell with two resistors of 5 Ω after it. This phrase describes both circuits. The correct answer is E: these circuits are equivalent.
The total resistance in a series circuit is found by summing the resistances of the components in series with each other.
In contrast to this, measuring current in a series circuit requires no summations at all. This is because it is the same at every point in the circuit. Looking at the diagram below, we see several selected points.
The current is the same at all of these points. The number of electrons moving past these points is the same, as they all only have one path to take. Cells and components with resistance do not add or subtract the amount of electrons present at specific points in the circuit.
Rule: Current in Series
For a series circuit, the current at point 1, , is the same as the current at point 2, , and so on for any point measured:
Let’s look at an example.
Example 2: Finding the Current through Components in Series
The circuit in the diagram consists of two resistors in series, with ammeters, , , and , placed at different points in the circuit. reads 4 A.
- What is the current given by the second ammeter, ?
- What is the current given by the third ammeter, ?
An ammeter is a device that can measure current at specific points in the circuit.
We are given the values of the resistances of these resistors, but we do not actually need these, since we are directly given the value of . This is a series circuit, so the current is the same at all points in the circuit. This means that
So the current given by the second ammeter, , is the same as , which is 4 A.
In the same way as part 1, the third ammeter, , is equivalent to and . It is also 4 A.
So far we have looked at resistance and current in series circuits, but how can we work out the potential difference across a component in a series circuit?
In a series circuit, the sum of the potential differences across each component is equal to the potential difference provided by the cell or battery. In the circuit shown in the diagram below, there is just one component, a resistor. The potential difference provided by the cell is 10 V; therefore, the potential difference across the resistor—the only component in the circuit—is also 10 V.
We can measure the potential difference between any two points in the circuit, but note that if we were to measure the potential difference between two points that have no component between them, such as A and B, or C and D, we would find it to have a value of 0 V.
When multiple components are in the same series circuit, potential difference is spread between them. Let’s see this demonstrated with the following diagram.
In the diagram above, there is a total of 10 V from the cell. Measuring across points A and E, there is the full 10 V.
The total voltage can be written as where and denote the potential differences across resistors and respectively.
This relationship holds true no matter how many components there are in a circuit, so long as they are in series, as seen in the equation below.
Rule: Total Potential Difference in Series
The total potential difference in a circuit, , is where is the potential difference across component 1, is that across component 2, and so on.
If we say that resistors and have the same resistance, then each one would account for half of the potential difference:
This means that the potential difference across the first resistor, , will be equal to the potential difference across the second one, . Thus, the potential difference across the two, from points B to C and C to D, will be the same:
This would make the total potential difference in the circuit look like
For the circuit above, is 10 V. If we substitute this value in, we can find the potential difference across and by dividing both sides by 2:
This causes the 2 on the right side to cancel, leaving behind
Not every circuit will have resistors with the same values of resistance.
Let’s look at an example question.
Example 3: Finding the Potential Difference across a Component in Series
A cell supplying 12 V is connected in series with two resistors. The potential difference across the first resistor is 4 V. What is the potential difference across the second resistor?
Let’s start by drawing a diagram of the circuit, showing the potential difference across each resistor.
The total potential difference in the circuit, , is supplied by the 12 V cell. The equation relating the total potential difference to the potential difference across each resistor is
For two resistors, the equation is just
These resistors have different values of resistance, since they do not have the same potential difference across them. The first resistor has a potential difference across it, , of 4 V.
We want to find , so let’s isolate it. We can do this by subtracting from both sides of the equation, and canceling it on the right side to give
We know the total potential difference supplied by the cell is 12 V, and is 4 V. Substituting these values into the equation gives
The potential difference across the second resistor is, therefore, 8 V.
Example 4: Finding the Total Resistance of Components in Series
A student sets up the circuit shown in the diagram. If the value of is 3 Ω, what is the current through the circuit?
We need to find the current in this circuit. We have been given the potential difference of the cell and the values of the resistances. In order to find the total current of the circuit, we will have to use Ohm’s law.
Recall the Ohm’s law equation: where is the total potential difference, is the total current, and is the total resistance of the circuit.
We wish to find the total current, so let’s isolate . We can divide both sides by to get rid of the on the right side:
We have the value we need for the potential difference, 10 V, but we need the total resistance of the circuit, not just the individual resistor values. Since the resistors are in series, we can use the rule
To simply add the resistor’s values together. is 3 Ω and the given resistor is 7 Ω, meaning the total resistance is just
Now we just put in the values of potential difference, 10 V, and total resistance, 10 Ω, to find the current:
The units of volts over ohms are amperes, so the answer is
The current in the circuit is 1 ampere.
Example 5: Finding the Resistance of Components in Series
The circuit shown in the diagram has a current of 0.5 A through it. What is the value of ?
The current is given, but one of the values of resistance is not. Let’s look at the Ohm’s law equation, understanding that we need the total resistance, :
This series circuit contains two resistors, which we will call (the unknown) and (the known 7 Ω). The total resistance in this case is given by
We can substitute this expression for the total resistance in Ohm’s law to obtain
The unknown value is , so let’s isolate it. We should start by dividing both sides by : which cancels the on the right side to give
Now we just need to subtract from both sides: which cancels the on the right side:
The potential difference is 10 V, the current is 0.5 A, and is 7 Ω. Substituting these values in gives
The units of volts over amperes give units of ohms, so the equation becomes which means the unknown resistance, , is 13 Ω.
Let’s summarize what we have learned in this explainer.
- Components in a circuit are in series if they are placed one after another on the same path.
- For components in series, the total resistance can be found using the following rule: where is the total resistance of the circuit.
- For components in series, the total potential difference can be found using the following rule: where is the total potential difference of the circuit.
- For components in series, the rule for current applies: