Show by a diagram how resistors $R_{1}, R_{2} a n d R_{3}$ are joined in parallel. Obtain an expression for the total resistance of the combination.
[5 MARKS]

Open in App

Solution

Circuit diagram: 2 Marks
Derivation: 3 Marks

In the parallel combination of resistors, one end of each resistor is connected at one point (say A) and the other end of each resistor is connected at the other point (Say B).
A cell is then connected in between these points A and B through a key as shown in Fig. 8.26.

Let R be the equivalent resistance, V be the potential difference applied between the ends A and B and I be the current that enters at A. Then the current I divides itself into three branches $I_{1}, I_{2} a n d I_{3}$ flowing through the resistors $R_{1}, R_{2} a n d R_{3}$ respectively.
Total current, $I = I_{1} + I_{2} + I_{3}$ ...(i)
Potential difference between A and B,
$V = I_{1} R_{1} = I_{2} R_{2} = I_{3} R_{3}$
Or $I_{1} = \frac{V}{R_{1}}$ , $I_{2} = \frac{V}{R_{2}}$ ,and $I_{3} = \frac{V}{R_{3}}$
As R is the equivalent resistance and I the total current,
$∴ I = \frac{V}{R}$
Substituting $I = \frac{V}{R}$ in Eq. (i), we get :
$\frac{V}{R} = \frac{V}{R_{1}} + \frac{V}{R_{2}} + \frac{V}{R_{3}}$
Or $\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}$
Thus, when three or more resistors are connected in parallel, the reciprocal of the combined resistance is equal to the sum of the reciprocal of the individual resistances.

Suggest Corrections

Similar questions

Q. Three resistors

R_{1}, R_{2}, R_{3}

are joined in series. Draw labelled diagram and obtain expression of its equivalent resistance.

Q. Resistors are sometimes joined together and they have several applications in electronics. Obtain the expression for the effective resistance in the combination of two resistors

R_{1}

and

R_{2}

in parallel.

Write the expressions for the equivalent resistance R of three resistors $R_{1}, R_{2}$ and $R_{3}$ joined in (a) parallel, and (b) series.

The circuit diagram given below shows the combination of three resistors $R_{1}, R_{2}, a n d R_{3}$
Find

(i) total resistance of the circuit.
(ii) total current flowing in the circuit.
(iii) the potential difference across $R_{1}$

Q. The equivalent resistance R of three resistors

R_{1}

R_{2}

and

R_{3}

joined in parallel is

Join BYJU'S Learning Program

Show by a diagram how resistors R1,R2 and R3 are joined in parallel. Obtain an expression for the total resistance of the combination. [5 MARKS]

Show by a diagram how resistors $R_{1}, R_{2} a n d R_{3}$ are joined in parallel. Obtain an expression for the total resistance of the combination.
[5 MARKS]