Question

In a Wheatstone bridge (see fig.), resistances $P$ and $Q$ are approximately equal. When $R = 400\,\Omega $, the bride is equal. When $R = 400\,\Omega $, the bridge is balanced. On interchanging $P$ and $Q$, the value of $R$, is balanced, is $405\,\Omega $. The value of $X$ is close to:
(A) $403\,\Omega $
(B) $404.5\,\Omega $
(C) $401.5\,\Omega $
(D) $402.5\,\Omega $

Answer 1

Hint: The value of the resistance $X$ can be determined by using the resistance formula of the wheatstone bridge. The resistance equation of the wheatstone bridge is written before and after the inter changing, then the two-resistance equation is formed, by multiplying the two equations, the solution can be determined.

Formula Used: The resistance formula of the wheat stone bridge is given by,
$\dfrac{P}{Q} = \dfrac{{{R_1}}}{X}$
Where $P$, $Q$ , ${R_1}$ and $X$ are the four resistance of the wheatstone bridge.

Complete step by step answer:
Given that,
The resistance of the resistor before interchanging is, ${R_1} = 400\,\Omega $ ,
The resistance of the resistor after interchanging is, ${R_2} = 405\,\Omega $ .
Now, the resistance before interchanging is given by,
The resistance formula of the wheat stone bridge is given by,
$\dfrac{P}{Q} = \dfrac{{{R_1}}}{X}\,...................\left( 1 \right)$
Now, the resistance after interchanging is given by,
The resistance formula of the wheat stone bridge is given by,
$\dfrac{Q}{P} = \dfrac{{{R_2}}}{X}\,...................\left( 2 \right)$
By multiplying the equation (1) and the equation (2), then
$\dfrac{P}{Q} \times \dfrac{Q}{P} = \dfrac{{{R_1}}}{X} \times \dfrac{{{R_2}}}{X}$
By cancelling the same terms in the numerator and in the denominator in the above equation, then
$1 = \dfrac{{{R_1}}}{X} \times \dfrac{{{R_2}}}{X}$
By multiplying the terms in the above equation, then the above equation is written as,
$1 = \dfrac{{{R_1}{R_2}}}{{{X^2}}}$
By rearranging the terms in the above equation, then the above equation is written as,
${X^2} = {R_1}{R_2}$
By substituting the resistance value before and after interchanging in the above equation, then
${X^2} = 400 \times 405$
By multiplying the terms in the above equation, then the above equation is written as,
${X^2} = 162000$
By taking the square root on both sides, then
$X = 402.5\,\Omega $
Hence, the option (D) is the correct option.

Note:The resistance of the wheatstone bridge is given before inter changing and the after interchanging, then the resistance is taken as ${R_1}$ and ${R_2}$. The resistance $P$ and $Q$ are cancelled because both are equal which is given in the question.