Three resistances A , B and C are connected in such a way that their combined equivalent resistance is equal so that of B. IF A and B are `10 Omega` and `30 Omega` respectively, find the three possible values of C and draw the corresponding circuits.

Explanation

In this case there are only three possible arrangements which are shown in the fig.1.39 If the three resistances are arranged in any other alternative way the equivalent resistance will be greater than B or less than B.

In each arrangement, the equivalent resistance is equal to B (given)
So, in the arrangement (a),
`A+(BC)/(B+C)=B or,10+(30C)/(30+C)=30`
or,`30C=600+20Cor,10C=600or,C=60Omega`
In the arrangement (b)
`(AB)/(A+B)+C=B or,(10times30)/(10+30)+C=30`
or,`C=30-30/4=30-7.5=22.5 Omega`
In the arrangement ( c)
`((A+B)C)/((A+B)+C)=B or,((10+30)timesC)/((10+30)+C)=30`
or,`(40C)/(40+C)=30or,40C=1200+30C`
or,`10C=1200or,C=120 Omega`