The resistance values of the Wheatstone bridge shown are \(P=2000\ \Omega\), \(Q=500\ \Omega\), \(R=3000\ \Omega\). The battery voltage is \(E=50\ \text{V}\).
The battery has an internal resistance of \(1\ \Omega\) and the Galvanometer (G) has a resistance of \(50\ \Omega\).
The value of the resistance S for balanced condition is \(\Omega\) (Answer in integer)
Show Hint
Balance in a Wheatstone bridge needs the ratio of adjacent arms to match; the battery and galvanometer resistances play no part in that ratio.
Step 1: Identify the balance condition for a Wheatstone bridge.
A Wheatstone bridge reaches balance when no current flows through the galvanometer branch. In the arm arrangement shown, P and Q sit as the ratio arms at the top, and R and S sit as the arms that carry the standard resistor and the unknown resistor. At balance, the ratio of the two top arms equals the ratio of the two bottom arms:
\[
\frac{P}{Q}=\frac{R}{S}
\] Step 2: Write down the known values.
From the circuit,
\[
P=2000\ \Omega,\quad Q=500\ \Omega,\quad R=3000\ \Omega
\] Step 3: Solve for S.
Rearrange the balance condition to isolate S:
\[
S=\frac{R\times Q}{P}
\]
Put in the numbers:
\[
S=\frac{3000\times500}{2000}=\frac{1500000}{2000}=750
\] Step 4: Check why the battery and galvanometer resistances do not matter.
The balance condition comes purely from the null condition, that is, zero current in the galvanometer branch. Once that branch carries zero current, the galvanometer resistance drops out of the equation completely, since no current flows through it to create a voltage drop that could shift the balance. The internal resistance of the battery only limits how much total current the source can supply to the bridge, it does not change the ratio at which the bridge balances. So the \(1\ \Omega\) internal resistance and the \(50\ \Omega\) galvanometer resistance given in the question are extra information meant to test this understanding, and they do not change the value of S. Final Answer:
The resistance S needed for the balanced condition is
\[
\boxed{750\ \Omega}
\]
