Question

The resistance of two conductors in parallel is 12 ohm and in series is 50 Ohm then find value of resistance of each conductor.​

Answer 1

Let the resistance of each conductor be ${R_{1}\:and\:R_{2}}$.

In parallel connection,

$R_{p} = \frac{R_{1}R_{2}}{R_{1} + R_{2}}$ - - - - (1)

$R_{s} = R_{1}+ R_{2}$ - - - - (2)

Multiplying equation (1) and (2),

We get,

$R_{p}R_{s} = R_{1}R_{2}$

$R_{1}R_{2} = 600$

$R_{1} = \frac{600}{R_{2}}$

Substituting in equation (2),

We get,

$R_{2}^{2} - 50R_{2}+ 600 = 0$

$R_{2} = 20 \: or \: 30$

The value of each resistance is ${30\:and\:20}$ ohms.

Answer 2

Solution:- It is given that the resistance of two conductors in parallel is 12 ohm and in series is 50 Ohm.

Let r1 and r2 be the unknown resistance.

(i).

When the resistances are in series combination.

r1 + r2 = 50 Ohm.

(ii).

When the resistances are in parallel combination.

=>1/r1 + 1/r2

=> r1r2/r1 + r2 = 1/R(say)

where R is given as 12 Ohm.

Therefore, r1 + r2 = 50 Ohm

and r1r2/r1 + r2 = 12 Ohm.

=> r1r2 = 12 × 50

=> r1r2 = 600

Also,

=> ( r1 - r2 )^2 = ( r1 + r2 )^2 - 4r1r2

=> ( r1 - r2 )^2 = 2500 - 4 × 600

=> ( r1 - r2 )^2 = 100

=> ( r1 - r2 )^2 = (10)^2

=> r1 - r2 = 10

From (i) and (iii), 2r1 = 60

Therefore, r1 = 30 Ohm and r2 = 20 Ohm.