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The resistance of two conductors in parallel is \( 12 \Omega \) and in series is \( 50 \Omega \). Find the resistance of each conductor.
Give:
Resistance of two conductor-
In parallel = $ 12 \Omega $
In series = $ 50 \Omega $
To find: The resistance of each conductor.
Solution:
Let $R_1$ and $R_2$ are the resistance of the two conductors.
Now, according to the question,
When the resistance of two conductors are connected in series, then the total resistance is given as-
$R_T=R_1+R_2$
$50=R_1+R_2$ ----------------(i)
When the resistance of two conductors are connected in parallel, then the total resistance is given as-
$\frac {1}{R_T}=\frac {1}{R_1}+\frac {1}{R_2}$
$\frac {1}{12}=\frac {1}{R_1}+\frac {1}{R_2}$
$\frac {1}{12}=\frac {R_2+R_1}{R_1\times R_2}$
$12=\frac {R_1\times R_2}{R_2+R_1}$ ---------------(ii)
Putting the value of $R_1+R_2$ from (i) in the above equation we get-
$\frac {R_1\times R_2}{50}=12$
$R_1\times {R_2}=600$ ---------------(iii)
Substituting the value of $R_2$ from (i) in the equation (iii) we got a quadratic equation:
$R_1\times ({50-R_1})=600$ $[from\ (i)\ we\ can\ obtain\ (R_2=50-R_1)]$
$50R_1-R_1^2=600$
$R_1^2-50R_1+600=0$
$R_1^2-20R_1-30R_1+600=0$
$R_1(R_1-20)-30(R_1-20)=0$
$(R_1-30)(R_1-20)=0$
$(R_1-30)=0\Rightarrow R_1=30$
$(R_1-20)=0\Rightarrow R_1=20$
We get 2 values of $R 1$, one is 30, and the other is 20.
Now, putting the value of $R_1=30$ in (i) we get-
$50=30+R_2$
$R_2=50-30$
$R_2=20$
If we put the value of $R_1=20$ in (i) we get-
$R_2=30$
Thus, we can see that both $R_1$ and $R_2$ can have the resistance $ 20 \Omega $ or $ 30 \Omega $ interchangeably.
So, the resistance of one conductor is $ 20 \Omega $ and the resistance of the other conductor is $ 30 \Omega $.