Series and parallel combinations of resistors

Resistors are primarily used in two types of combinations. These two types are two different ways of arranging the resistances. Firstly, the resistors can be connected end to end. In this type of combination, the beginning of one resistor is connected to the end of another resistor. The remaining ends of the resistors are connected to the circuit. This combination of resistors is known as a series combination. Another combination is when we combine both resistors ends, and their joint ends are connected to the circuit. This is known as parallel connection. In a parallel connection, the equivalent resistor is smaller than the value of the smallest resistor used.The net effective value of the resistor is called the equivalent resistance.

Body

A continuous flow of charged particles is referred to as electric current or current electricity. Current electricity flows from higher potential to lower potential. A closed-loop of conducting material allows the current to flow. This closed-loop made of wire is called an electric circuit. A circuit necessarily comprises a cell or battery. The cell has two terminals which provide potential differences allowing current to flow. One terminal is positive, and the other is negative. The circuit also requires a resistor or a combination of resistors which limit the amount of current which can flow in the circuit. Otherwise, an infinite amount of current will start flowing and burn the circuit. A switch is used to turn on or turn off the circuit. On closing the key, the circuit is complete, and a stream of free electrons moves through the conductor, constituting the electric current. Conventionally, the direction of current is taken to be opposite to the flow of electrons.

Resistance

Georg Simon, a well-known physicist, observed the current I flowing in the metallic wire when a potential difference is applied across the two ends. He stated that the electric current in the metallic wire is proportional to the potential difference in volts applied across the terminals of the closed circuit, provided that the temperature remains constant.

This observation, after being verified, came to be known as OHM’s law. In other words, it can be written as:

The potential difference between the terminals ∝ Current

V ∝ I

Or V = I R, here R is a constant of proportionality called resistance.

Resistance is the property of a circuit that opposes the flow of current. It can also be stated that resistance controls the magnitude of current flowing through the circuit.

Mathematically, resistance can be defined as

R=V/I

Combination of resistances

Resistors are used in combination to attain any particular value of resistance with the available number of resistors of any value. Hence we can also say that resistors are combined to get the desired value of electric current in the circuit with any number and value of resistances available. There are two kinds of combinations in which the resistances are used. They are:

1. Series combination of resistances.

2. Parallel combination of resistances

Equivalent resistance

If a single resistance can replace a combination of resistances without changing the value of current in that circuit, then that value of resistance is called equivalent resistance for the combination.

Series combination of resistances

Resistances are said to be connected in series when they are connected end to end such that the same current flows through every resistor. The beginning of one resistor is connected to the end of another resistor. In this case, the net resistance or the equivalent resistance is the algebraic sum of the value of all the resistances connected in a series combination. Mathematically, if we connect some resistances in series, for example (R₁, R₂, R₃), then the equivalent resistance can be given as:

R_eq = R₁ + R₂ + R₃

Here the ammeter measures the current flowing in the circuit.the current through each resistor is the same i.e., I = I₁ = I₂ = I₃.

On replacing the three resistors with an equivalent resistance Req, such that the current in the circuit as measured by the ammeter I remain the same, then by applying Ohm’s law,

V_eq = IR_eq

Here, we can also note that the net sum of the voltage difference across each resistor is equal to the total potential difference across the battery’s terminals.

V_eq = V₁ + V₂ + V₃

Individually by applying Ohm’s law to each resistor,we get

V₁ = IR₁

V₂ = IR₂

V₃= IR₃

Hence, IR= IR₁+ IR₂ + IR₃

or

R_eq = R₁ + R₂ + R₃

Properties of the series combination of resistance

Some important points which are to be noted in the series combination of resistance are as follows:

1. The current through the entire circuit and each resistance in the circuit will remain the same.

2. The equivalent potential difference across the series combination is equal to the sum total of individual potential differences across each resistor.

3.  Because of the point stated above, the value of equivalent resistance is equal to the algebraic sum of the total resistance.

4. Thus, the value of equivalent resistance is greater than the value of the largest resistance used in the combination.

Uses of a series combination of resistances

The applications of series combinations of the circuit are as given below:

1. Since the value of net resistance is greater than the value of the largest resistance used, the series combination increases the value of resistance in the circuit.

2. Since the potential difference across the individual resistance is proportional to the current flowing through it, this combination divides high possible differences across many resistance.

3. Series combination is used in laboratory resistance boxes, decorative lights, etc.

4. Series combination has a significant disadvantage: if any resistors break or are disrupted or any kind of failure occurs in even one of the resistors, the whole circuit is switched off. The entire circuit becomes non-functional.

Parallel combination of resistances

Resistances are connected in parallel when all the resistors are connected between two fixed points. The resistors can be said to have common ends. Therefore, they have common potential differences across them. The ends of all the resistors intersect at a common point. The net resistance is given by

1/R_eq = 1/R₁ + 1/R₂ + 1/R₃ + 1/R₄ + 1/R₅ + …

Thus, the value of the equivalent resistance of the combination is smaller than the smallest resistance used in the circuit.

Properties of a parallel combination of resistances

1. Since all the resistances have common ends, they must have common potential differences across them.

2. Since the potential difference across them is the same, the direction of current in all of them must also be the same.

3. But, since the magnitude of resistances might be different, the magnitude of current in all of them can be different.

4.  Current diverges from one end into different paths and then converge to the same intersecting point at the other end.

5. One significant advantage of a parallel combination of resistors is that failure of any resistors does not change the quantity of current flowing in any other resistor in the combination. But, the magnitude of current in the entire circuit changes on the failure of any resistance in the circuit.

Uses of the parallel combination of resistances

The applications of the parallel combination of resistances are as given.

1. Every household uses a parallel combination of resistances for power points.

2. The automobile industry also uses parallel circuits for DC power supply.

3. It protects every kind of appliance when connecting fuses and circuit breakers.

Conclusion

Both series and parallel combinations of resistors are useful and used in different places as per usage convenience. The combinations are categorised based on which their ends are connected. Series connection refers to the end to end connection, whereas parallel connection refers to both ends being connected together.

Series connection increases the overall resistance, and parallel connection reduces the equivalent resistance.