Resistors in Parallel

Definition

A Parallel circuit is a circuit that has more than one path for the electric current to flow, as shown in figure. The current branches so that electrons flow through each of the paths. If one path is broken, electrons continue to flow through the other paths. Adding or removing additional devices in one branch does not break the current path in the other branches, so the devices on those branches continue to work normally.

In a parallel circuit, the resistance in each branch can be different, depending on the devices in the branch. The lower the resistance is in a branch, the more current flows in the branch. So the current in each branch of a parallel circuit can be different.

Parallel circuits have several advantages. When one branch of the circuit is opened, such as when you turn a light off, the current continues to flow through the other branches. Houses, automobiles, and most electrical systems use parallel wiring so individual parts can be turned off without affecting the entire circuit.

Let us consider three resistors having resistances R₁, R₂, R₃ respectively. Let the voltage across the combination is ‘V’ and currents through R₁, R₂, R₃ are I₁, I₂, I₃. Voltage across all the resistors is the same as all of them have the same terminal points (A and B). (see figure)  

A parallel combination of resistors

V₁ = V₂ = V₃ = V (let) ... (1)

The total current ‘I’ entering through ‘A’ is divided among the three resistors (I₁, I₂ and I₃). Thus, total current ‘I’ is sum of individual currents through R₁, R₂ and R₃.

I= I₁ + I₂ + I₃ ... (2) 

Let the equivalent resistance of whole combination be R_p. 

$I=\frac{V}{R_p}$ ... (3)

From (2) and (3), we get,

$\frac{V}{R_p}=I_1+I_2+I_3$

or $\frac{V}{R_p}=\frac{V}{R_1}+\frac{V}{R_2}+\frac{V}{R_3}$ $\left[I=\frac{V}{R}\right]$

or $\frac{V}{R_p}=V\left(\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\right)$ _Or $\frac{1}{R_p}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}$

General formula for n resistors in parallel :

$\frac{1}{R_p}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+...+\frac{1}{R_n}$

Important Points Related to Parallel Combination 

1. The sum of the reciprocals of the separate resistances is equal to the reciprocal of equivalent resistance.

2. The voltage across each resistor of a parallel combination is the same, and is equal to the voltage across the whole group.

(a) V₁ = V₂ = V₃ = ……. = V_n = constant

(b) IR = V ⇒ IR = constant or $I\propto \frac{1}{R}$

3. In parallel combination,$I_1:I_2:I_3=\frac{1}{R_1}:\frac{1}{R_2}:\frac{1}{R_3}$

4. The total current is the sum of currents flowing in different branches. That is, I= I₁ + I₂ + I₃+ ……. I_n

5. For two resistors R₁ and R₂ in parallel, their equivalent resistance is given by, $R_e=\frac{R_1{R}_2}{R_1+R_2}$

6. In the case of resistors in parallel, if one resistance becomes ‘open’, all others will work as usual.

7. If ‘n’ equal resistors (R) are connected in parallel, then, their equivalent resistance is (R/n). If I is the total current passing through the combination, then current through each resistor is (I/n).

To get maximum resistance, resistors must be connected in series and to get minimum resistance, resistors must be connected in parallel.

Examples

1. In the circuits in diagram below, calculate the potential difference measurement on voltmeters P, Q and R.

Solution

In a series circuit, the potential difference across the combination is the sum of voltages across the various components. 

In figure (a), total voltage, V = 18 V, Voltage across component 1, V₁ = 12V, voltage on voltmeter P, V_p = ?

V = V₁ + V_p

18 V = 12 V + V_p

V_p = 6V

In a parallel circuit, the potential difference across each branch of the circuit is the same. When there is more than one component in a branch of a circuit, the potential difference across all the components add up to give the total potential difference supplied by the cell or battery.

In figure (b), voltage in each branch , 

V = 9 V 

In branch 1, Total voltage V = 6V + Voltage measured by voltmeter Q, V_Q

V_Q = 9 V – 6V = 3 V

In branch 3, Voltage measured by voltmeter R, 

V_R = V = 9 V

2. Figure shows a combination of four identical bulbs joined with a battery. Compare the brightness of the bulbs shown. What happens if bulb A fails, so that it cannot conduct current? What happens if bulb C fails? What happens if bulb D fails?

Explanation

Bulbs A and B are connected in series across the voltage of the battery, whereas bulb C is connected by itself across the battery. This means the voltage drop across C has the same magnitude as the battery voltage, whereas this same voltage is split between bulbs A and B. As a result, bulb C will glow more brightly than either of bulbs A and B, which will glow equally brightly. Bulb D has a wire connected across it i.e., a short circuit, so the potential difference across bulb D is zero and it doesn’t glow. If bulb A fails, B goes out, but bulb C will glow. If bulb C fails, there is no effect on the other bulbs. If bulb D fails, we cannot detect this event, because bulb D was not glowing initially; also, there is no effect on the other bulbs.

If positive and negative terminals of a battery are joined by a connecting wire i.e., it is short-circuited, then large current will flow through it as resistance of the circuit becomes negligible.

If both the terminals of a device (like a bulb) are joined by a connecting wire i.e., they are joined directly then, the device is said to be short circuited and potential difference across it becomes zero.

1.0Heating Effect of Electric Current

We know that to maintain the current, the source (cell or battery) has to keep expending its energy. A part of the source energy is consumed into useful work like in rotating the blades of an electric fan, producing light and sound in television, etc. Rest of the source energy is converted to heat to raise the temperature of the device. 

If an electric circuit is purely resistive, that is, it consists of resistors only connected to a battery, the source energy gets dissipated entirely in the form of heat. This effect is utilised in devices such as electric geysers, electric heater, electric iron, etc.

The conversion of a part (or whole) of the electric energy into heat energy when electric current flows through a device is called heating effect of current.