Question

derive an expression for the effective resistance R when R1 R2 R 3 are connected in parllel

Answer 1

Answer:

let i1 I2 and i3 the current flow through R1 R2 and R3,then total current drawn from source be:

I=i1+i2+i3.....(i)

if the potential difference between the two ends of a and b is B then:

according to Ohm's law :

Current in R1 is i1= V/R1

" " R2 is i2= V/R2

". ". R3 is i3= V/R3

on aadding these:

i1+i2+i3=V/R1+V/R2+ V/R3......(ii)

if equivalent resistance between point A and B is Rp then total current drawn from source will be:

I=V/Rp..........(iii)

substituting the value of I and i1+ i2+ i3 from eq. (iii) And (ii) in eq. (i).

V/ Rp= V(1/ R1+ 1/ R2+ 1/ R3).

Hence:

1/ Rp= 1/ R1+ 1/R2+ 1/R3.

proved an expression of equivalent resistance

Answer 2

Answer

Consider two resistors connected in parallel, as shown in figure. It is clear, from the figure, that the potential drop 'V' across the two resistors is the same. In general, however, the currents I1 and I2 which flow through resistors R1 and R2, respectively, are different. According to Ohm's law, the equivalent resistance Req between B and A is the ratio of the potential drop V across these points and the current I which flows between them. This current must equal the sum of the currents I1 and I2 flowing through the two resistors, otherwise charge would build up at one or both of the junctions in the circuit.

Thus,

I=I1+I2

It follows that

1/Req= I/V= (I1+I2)/V= I1/V +I2/V

giving,

1/Req=1/R1 + 1/R2

Here, we have made use of the fact that the potential drop V is common to all three resistors. Clearly, the rule is

'The reciprocal of the equivalent resistance of two resistances connected in parallel is the sum of the reciprocals of the individual resistances. '