Question

derive an expression for equivalent resistance in parallel combination for 3 resistors​

Answer 1

Answer:

THE POINTS TO BE NOTED FOR PARALLEL COMBINATION :

• The potential difference across each resistor is same (= Va = Vb = V, say) which is equal to the potential difference across the terminals of the battery(or source).
• The current in a resistor is inversely proportional to its resistance (by the relation V= IR). The sum of currents I1 + I2 + I3 + . . . in the separate branches of the parallel circuit is equal to the total current I drawn from from the source.

i.e.,

$I = I_{1} + I_{2} + I_{3}$ + ...

Ohm's Law,

V = IR

here,

1. I = current flowing in the conductor
2. R = resistance of the conductor
3. V = potential difference across its ends

                                                                                                                           

EXPRESSION FOR EQUIVALENT RESISTANCE :

Let, $I_{1} , I_{2} ,and I_{3 }$ be the currents through the resistors $R_{1}, R_{2} , and R_{3}$ respectively, then total Current drawn form the circuit

$I = I_{1} + I_{2} + I_{3}$ +  .  .  .  .  .  . (i)

If potential difference across the ends is V, then by Ohm's Law :-

Current in $R_{1}$ = $\frac{V}{R_{1}}$

Current in $R_{2} = \frac{V}{R_{2}}$

Current in  $R_{3} = \frac{V}{R_{3}}$

On adding these,

$I_{1} + I_{2} + I_{3}$ =  $\frac{V}{R_{1}}$ + $\frac{V}{R_{2}}$ + $\frac{V}{R_{3}}$  .  .  .  . (ii)

If the equivalent resistance of the combination is $R_{p}$, then total current drawn from the source is

$I = \frac{V}{R_{p} }$  .   .   .   .   .   . (iii)

Substituting the values of  $I and I_{1} + I_{2} + I_{3}$  from eqns (iii) and (ii) in eqn (i), we get

=> $\frac{V}{R_{p} } = V (\frac{1}{R_{1} } + \frac{1}{R_{2} } + \frac{1}{R_{3} } )$

=> $\boxed{\sf{ \dfrac{1}{R_{eq}} = \dfrac{1}{R_1} + \dfrac{1}{R_2} } + \dfrac{1}{R_3} } }$

Thus, in the parallel combination of three resistors, the reciprocal of the equivalent resistance is equal to the sum of the reciprocals of the individual resistances.