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To balance the bridge in the circuit, the values of $R$ is:

(A) $8\,\Omega $
(B) $4\,\Omega $
(C) $20\,\Omega $
(D) $12\,\Omega $

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Last updated date: 11th Jun 2024
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Answer
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Hint The value of the $R$ can be determined by using the resistance balance equation of the wheatstone bridge. The resistance ${R_2}$ is taken as the sum of the $R$ and the other resistance which is given in the diagram, then the resistance $R$ can be determined.

Useful formula
The resistance equation of the wheat stone bridge is given by,
$\dfrac{{{R_1}}}{{{R_2}}} = \dfrac{{{R_3}}}{{{R_4}}}$
Where, ${R_1}$, ${R_2}$, ${R_3}$ and ${R_4}$ are the resistance of the individual resistors.

Complete step by step solution
From the diagram it is given that,
The resistance of the first resistor is given as, ${R_1} = 5\,\Omega $,
The resistance of the second resistor is given as, ${R_2} = \left( {R + 12} \right)\,\Omega $,
The resistance of the third resistor is given as, ${R_3} = 15\,\Omega $,
The resistance of the fourth resistor is given as, ${R_4} = 60\,\Omega $.
Now,
The resistance equation of the wheat stone bridge is given by,
$\dfrac{{{R_1}}}{{{R_2}}} = \dfrac{{{R_3}}}{{{R_4}}}\,................\left( 1 \right)$
By substituting the resistance values of the first resistor, second resistor, third resistor and fourth resistor in the above equation (1), then the above equation (1) is written as,
$\dfrac{5}{{\left( {R + 12} \right)}} = \dfrac{{15}}{{60}}$
By rearranging the terms in the above equation or cross multiplying the terms in the above equation, then the above equation is written as,
$5 \times 60 = 15 \times \left( {R + 12} \right)$
By multiplying the terms in the LHS in the above equation, then the above equation is written as,
$300 = 15 \times \left( {R + 12} \right)$
By rearranging the terms in the above equation, then the above equation is written as,
$\dfrac{{300}}{{15}} = \left( {R + 12} \right)$
By dividing the terms in the above equation, then the above equation is written as,
$20 = R + 12$
By keeping the term $R$ in one side and the other terms in the other side, then the above equation is written as,
$20 - 12 = R$
By subtracting the terms in the above equation, then the above equation is written as,
$R = 8\,\Omega $

Hence, the option (A) is the correct answer.

Note The balancing equation of the wheat stone bridge is completely depending on the resistance of the wheatstone bridge. The name for the resistance is given in the direction of the clockwise direction, during this calculation, the naming of the resistance is the most important.