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## High school physics

### Course: High school physics>Unit 12

Lesson 3: Series and parallel resistors

# Resistors in series and parallel review

Review how to find the equivalent resistance for resistors in parallel and series configurations. Recall the current and voltage properties of series and parallel configurations of resistors.

## Key terms

Term (symbol)Meaning
Equivalent resistance (R, start subscript, start text, e, q, end text, end subscript)The total resistance of a configuration of resistors.

## Equations

EquationSymbol breakdownMeaning in words
\begin{aligned} R_s &= \displaystyle\sum_{i} R_i \\ &\text {or} \\ R_s &=R_1+R_2+... \end{aligned}R, start subscript, s, end subscript is equivalent series resistance and sum, start subscript, i, end subscript, R, start subscript, i, end subscript is the sum of all individual resistances R, start subscript, i, end subscript.Equivalent series resistance is the sum of all the individual resistances.
\begin{aligned} \dfrac{1}{R_p} &= \displaystyle\sum_{i} \dfrac{1}{R_i} \\ &\text {or} \\ \dfrac{1}{R_p} &=\dfrac{1}{R_1}+\dfrac{1}{R_2}+...\end{aligned}R, start subscript, p, end subscript is equivalent parallel resistance and sum, start subscript, i, end subscript, start fraction, 1, divided by, R, start subscript, i, end subscript, end fraction is the sum of all individual resistances R, start subscript, i, end subscript reciprocals.The reciprocal of the equivalent parallel resistance is the sum of all the individual resistance reciprocals.

## Resistors in series and parallel

### Series resistor properties

Any time we have more than one resistor in a row, the configuration is described as having the resistors in series or series resistors (Figure 1).
Resistors in series have some special characteristics worth remembering. Any configuration of resistors in a series will have the following properties.
• The same current flows through each resistor: I, start subscript, 1, end subscript, equals, I, start subscript, 2, end subscript, equals, …, equals, I, start subscript, n, end subscript
• Potential difference is distributed among series resistors: delta, V, start subscript, s, end subscript, equals, delta, V, start subscript, 1, end subscript, plus, delta, V, start subscript, 2, end subscript, plus, …, plus, delta, V, start subscript, n, end subscript
• The resistor with the biggest resistance has the greatest voltage.
• The equivalent resistance R, start subscript, s, end subscript is always more than any resistor in the series configuration.

### Parallel resistor properties

Another possible way to arrange resistors in a circuit is to have multiple resistors branch off from a single junction in the circuit (Figure 2).
Resistors in parallel also have some special characteristics:
• The current is distributed across resistors: I, equals, I, start subscript, 1, end subscript, plus, I, start subscript, 2, end subscript, plus, …, plus, I, start subscript, n, end subscript
• Potential difference is the same across all resistors in parallel: delta, V, start subscript, 1, end subscript, equals, delta, V, start subscript, 2, end subscript, equals, …, equals, delta, V, start subscript, n, end subscript
• The smallest resistance gets the most current.
• The equivalent resistance R, start subscript, p, end subscript is always less than any resistor in the parallel configuration.
Keep in mind that not all circuits are strictly series or parallel. Sometimes they can be a combination of both. We will learn how to analyze more complicated circuits in the next few lessons.

## How to calculate equivalent resistance

Resistors in series or parallel can be replaced by a single resistor of equivalent resistance. This strategy is helpful for solving complex circuit problems because it let’s us simplify the circuit.

### Equivalent series resistance

We can redraw the circuit with the resistors in series replaced by a single equivalent resistor (Figure 3).
We can calculate R, start subscript, s, end subscript from the resistances of the individual resistors in series. If R, start subscript, 1, end subscript, equals, 4, \Omega and R, start subscript, 2, end subscript, equals, 8, \Omega, then the equivalent resistance is the sum of R, start subscript, 1, end subscript and R, start subscript, 2, end subscript:
\begin{aligned}R_s &= R_1 + R_2 \\\\ &= 4 \, \Omega + 8\, \Omega \\\\ &= 12\, \Omega \end{aligned}

### Equivalent parallel resistance

We can redraw a circuit with all resistors in parallel replaced by a single equivalent resistor (Figure 4).
We can calculate R, start subscript, p, end subscript from the resistances of the individual resistors in parallel. If R, start subscript, 1, end subscript, equals, 4, \Omega and R, start subscript, 2, end subscript, equals, 8, \Omega, then the equivalent resistance R, start subscript, p, end subscript is:
\begin{aligned} \dfrac{1}{R_p} &= \dfrac{1}{R_1} + \dfrac{1}{R_2} \\\\ &= \dfrac{1}{4\,\Omega} + \dfrac{1}{8\,\Omega} \\\\ &= \dfrac{2}{8\,\Omega} + \dfrac{1}{8\,\Omega} \\\\ \dfrac{1}{R_p} &= \dfrac{3}{8\,\Omega}\end{aligned}
Now, let’s be careful here. Lots of people make a mistake here: start fraction, 3, divided by, 8, end fraction is not the equivalent parallel resistance R, start subscript, p, end subscript yet, it is the reciprocal. To solve for R, start subscript, p, end subscript, we need to take the reciprocal of both sides:
\begin{aligned} \left (\dfrac{1}{R_p} \right )^{-1} &= \left ( \dfrac{3}{8\,\Omega} \right )^{-1} \\\\ R_p &= \dfrac{8}{3} \,\Omega \\\\ &\approx 2.7\,\Omega \end{aligned}

For deeper explanations, see our video on series resistors and video on parallel resistors.
To check your understanding and work toward mastering these concepts, check out the exercise on calculating equivalent resistance for series and parallel resistors.

## Want to join the conversation?

• Why in a circuit with several branches the voltage drop across each of the branches is the same?
Why doesn't it also split up as the current?
• Hi,

Here's an analogy that might help you understand:

Imagine there is a bakery, and it produces 5 slices of bread/min and there's 60g of carbo per slice of bread.
If you own the production for yourself, you get 5 bread/min, and 60g carbo/bread.
If you are sharing with 4 other friends and you are dividing the bread evenly, you are getting only 1 bread/min, but still 60g carbo/bread.

In this analogy,
the bakery=battery
You and your friends=//resistors
Bread production=flow of electrons
Bread=electrons' charge, measured in C.
Carbo=energy, measured in J.
Time is, well, time.
Therefore carbo/bread=J/C, which is the voltage;
bread/min=C/s, which is the current.

Here we can see that the voltage doesn't drop with //resistors, but the current does.

(This is not a proof, but only a way to understand)
• It is given that the resistor with the biggest resistance has the greatest voltage. We know that electric current is directly proportional to the potential difference across a circuit. So, this statement now means that the biggest resistance allows the largest amount of current to flow through it. Isn't this fact contradictory to what we have learnt? (Current is inversely proportional to resistance)
• V=IR can be rearranged to I=V/R therefore while the resistance increases the current decreases. In series, the current is the same throughout the circuit. So, a larger resistor will have a larger voltage drop.

If I have a larger resistor in parallel with a smaller resistor and they both have the same voltage applied since they are in parallel, which one do you think has more current flowing through?
• what if there are 2 resistors in series that are parallel to 1 other resistor?
• In these types of cases it is best to simplify parts of the circuit and then find out the net resultant. In your example let each of the resistors individually have a value of resistance R. Then:
The equivalent resistance of the resistors in series
Req1= R + R = 2R
Now as you mentioned, these two in series, are in parallel to one other resistor R.
Finding Req1 basically gives us that a single resistor of 2R would perform the same function(in this case) as the 2 resistors R in series.

So we can instead, now consider, a resistor with resistance Req1 to be in parallel combination to a resistor with resistance R. And we know how to do this:
1/ReqFinal= 1/Req1 + 1/R
Now, as Req1=2R, we get:
1/ReqFinal= 1/2R + 1/R = 3/2R
i.e: ReqFinal=2R/3
Which is the equivalent resistance of the whole combination!
You can extend this to more complex circuits as well, simplify parts of it and keep going!
Cheers!