What is the value of x when the Wheatstone bridge network shown in the figure is balanced? \( P=500\,\Omega \), \( Q=800\,\Omega \), \( R=(x+400)\,\Omega \) and \( S=1000\,\Omega \).
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( 225\,\Omega \)
A Wheatstone bridge is balanced when the galvanometer shows no deflection, that is, when its two ends are at the same potential. The ratio of the two resistances on one side of the galvanometer then equals the ratio on the other side. From the figure, P and Q meet at the upper end of G, and R and S meet at its lower end. So the balance condition is:
\( \displaystyle \frac{P}{Q}=\frac{R}{S} \)
The resistance of the arm R is:
\( \begin{aligned} R &= \frac{P\,S}{Q} \\ &= \frac{500\,\Omega\times1000\,\Omega}{800\,\Omega} \\ &= 625\,\Omega \end{aligned} \)
From the figure, R is made of \( x \) in series with \( 400\,\Omega \), so \( R=x+400\,\Omega \):
\( \begin{aligned} x &= R-400\,\Omega \\ &= 625\,\Omega-400\,\Omega \\ &= 225\,\Omega \end{aligned} \)