Resistors are everywhere in electric circuits. Resistors are not only limited to a small device called resistor whose main work is to provide specific resistance in an electric circuit. Anything that provides resistance acts as a resistor. You may have used an electric circuit with a lot of light bulbs for decoration purposes and each light bulb acts as a resistor.
We often need to combine two or more resistors in an electric circuit. We can change the value of resistance or get the required resistance by combining resistors in various ways. We have two ways to combine resistors which are resistors in series and resistors in parallel combination. Understanding the relationship of current, voltage and resistance, that is \(V = IR\) obtained from Ohm's law is required.
Resistors in Series
In resistors in series combination we combine resistors one after the other, that is one resistor is directly connected to another as shown in Figure 1 below. In the figure below we have three resistors of resistances \(R_1\), \(R_2\) and \(R_3\) respectively and they are connected in a way that the right end of one resistor is connected to the left end another resistor and so on.
In the case of the resistors in series, the current is the same through each resistor, that is if \(I\) is the current through all resistors, the potential difference across resistor of resistance \(R_1\) is \(IR_1\) and across resistance \(R_2\) is \(IR_2\) and so on
So, the total voltage drop in the entire circuit is the sum of the voltage drops across individual resistances, that is if \(R\) is the equivalent resistance of the circuit, the voltage drop across \(R\) is \(IR\) and hence we have
\[\begin{align*} IR &= IR_1 + IR_2 + IR_3 \\ \text{or,} \quad R &= R_1 + R_2 + R_3 \tag{1} \label{1} \end{align*}\]
Hence the equivalent resistance of resistors in series combination is the sum of the resistances of all resistors connected in an electric circuit. If you compare this result with the combination of capacitors in series, you'll find the result of capacitors in series to be similar to the combination of resistors in parallel explained below.
If an electric circuit has multiple bulbs connected in series (which you may have used for decorations at special occasions), all bulbs stop to glow even if the connection to one bulb breaks. An electric bulb acts as a resistor and in the series combination all of them are connected end to end one after the other and if the connection to one bulb breaks, the connection to all bulbs breaks!
One practical benefit of light bulbs connected in series is the total voltage is divided into smaller voltages for each light bulb and each bulb consumes less power which means less heat and they are cooler, and therefore the bulbs can have longer lifetime. Note from electric power that if \(V\) is the voltage, then power \(P\) is \(V^2/R\).
Resistors in Parallel
Now we consider resistors in parallel combination. If you arrange resistors in an order as shown in Figure 2 below, that is all left ends are connected to one terminal and all right ends are connected to another terminal of the battery (source of emf). Such a combination is called the parallel combination of resistors.
In parallel combination of resistors, the potential difference or voltage across each resistor is the same (because both of their ends are directly connected to the terminals of the battery). If the above figure there are three resistors connected in parallel combination having resistances \(R_1\), \(R_2\) and \(R_3\).
In this case the current through each resistor is not the same. Since the voltage is the same, the current through higher resistances is lower and the current through lower resistances is higher, and therefore maintains the constant voltage. Let \(I_1\), \(I_2\), \(I_3\) be the currents through resistances \(R_1\), \(R_2\) and \(R_3\) respectively. If \(I\) is the total current \(R\) is the equivalent resistance of the entire circuit, we know that \(IR = I_1R_1\), \(IR = I_2R_2\) and \(IR = I_3R_3\) which implies \(I_1 = IR/R_1\), \(I_2 = IR/R_2\) and \(I_3 = IR/R_3\) respectively.
The total current \(I\) is the sum of all currents across all resistors, so
\[\begin{align*} I &= I_1 + I_2 + I_3 \\ \text{or,} \quad I &= \frac{IR} {R_1} + \frac{IR}{R_2} + \frac{IR}{R_3}\\ \text{or,} \quad \frac{1}{R} &= \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \tag{2} \label{2} \end{align*}\]
For the resistors in parallel combination the reciprocal of equivalent resistance is equal to the sum of the reciprocals of all resistances. The result of the equivalent capacitance for the capacitors in parallel combination is not similar to the result of resistors in parallel combination. Make sure that you do not confuse with combinations of capacitors and resistors. It is very easy to distinguish if you know what's going on.
If we consider only two resistors connected in series combination of reisistances \(R_1\) and \(R_2\) respectively. Then, the equivalent resistance \(R\) is
\[\begin{align*} \frac{1}{R} &= \frac{1}{R_1} + \frac{1}{R_2} = \frac{R_1 + R_2}{R_1R_2} \\ \text{or,} \quad R &= \frac{R_1R_2}{R_1+R_2} \tag{3} \end{align*}\]
The above expression for the equivalent resistance for two resistors connected in series shows that the equivalent resistance is always less than each resistance. The equivalent resistance is always less than the smallest of all resistances regardless of the number of resistors.
Since the voltage is the same across each resistor, \(I_1R_1 = I_2R_2\) for any two resistors, so
\[\frac{I_1}{I_2} = \frac{R_2}{R_1}\]
which means the current is inversely proportional to the resistance, that is current through higher resistance is lower and vice-versa.
There may arise a misconception regarding the resistors in parallel considering the current only takes the path of the smallest resistance. The common sense behind this is "if there is a lower resistance available for conduction, then why it is necessary for the current to take the path of any higher resistance ". But that is not true. Current takes the path of all resistances. It is true that the current is lower for higher resistances and higher for lower resistances. At least a little current exists through a resistor even if its resistance is very high. If the resistance is infinite, you can expect the current to be zero but this is only the ideal situation. No real resistance is infinite.
In the light bulb example as before if the light bulbs are connected in parallel, the rest of them continue to glow even if the connection to one or more of them is broken. The downside of parallel connection is that the voltage across each bulb is the same equal to the total applied voltage in the circuit and they are brighter and hotter than those connected in series. The light bulbs in parallel are more risky as they can even cause fire, and on the other hand if the circuit is broken in series connection, it is a lot tedious task to find the faulty one to fix it.
An workaround on this problem is to develop the bulbs in series with a jumper or shunt wire (anti-fuse) loop such that if one bulb is broken, the circuit is automatically connected and the rest of the bulbs still glow. The shunt wire loop is initially insulated and as one of the bulbs break, the entire voltage appears across the broken light bulb which breaks the insulation completing the circuit.
One problem still arises from this that if one bulb is broken, the equivalent resistance decreases which increases the current and decreases the lifetime of other bulbs. If that bulb is not repaired, more bulbs will break, so the bulbs must be repaired as soon as they fail. If the bulbs are not repaired in time, the current will be much higher. This situation can be the cause of short circuit and it is dangerous.
