Question: An unknown resistance \[{{R}_{1}}\] is connected in series with a resistance of 10Ω. This combinatio...

Question

An unknown resistance ${{R}_{1}}$ is connected in series with a resistance of 10Ω. This combination is connected to one gap of a meter bridge while a resistance ${{R}_{2}}$ is connected in the other gap. The balance point is at 50 cm. Now when the 10Ω resistance is removed, the balance point shifts to 40 cm. The value of ${{R}_{1}}$ is (in ohms):
A. 20
B. 10
C. 60
D. 40

Answer 1

Solution

A meter bridge also known as slide wire bridge is an electrical instrument which works on the principle of a Wheatstone bridge. A meter bridge helps in finding the unknown resistance of a conductor as that of in a Wheatstone bridge. The Wheatstone Bridge helps in measuring the values of unknown resistances.

Complete step by step solution:
Given values: ${{R}_{1}}$ is connected in series with a resistance of 10Ω. And this is connected in meter bridge and the balancing point is 50 cm. thus, for first case:
$\dfrac{{{R}_{1}}+10}{{{R}_{2}}}=\dfrac{{{l}_{1}}}{100-{{l}_{1}}} \\\ \Rightarrow \dfrac{{{R}_{1}}+10}{{{R}_{2}}}=\dfrac{50}{100-50} \\\ \Rightarrow \dfrac{{{R}_{1}}+10}{{{R}_{2}}}=1 \\\$
$\therefore {{R}_{2}}={{R}_{1}}+10$ --(1)
Now for second case, doing in similar fashion:
$\dfrac{{{R}_{1}}}{{{R}_{2}}}=\dfrac{{{l}_{1}}}{100-{{l}_{1}}} \\\ \Rightarrow \dfrac{{{R}_{1}}}{{{R}_{2}}}=\dfrac{40}{100-40} \\\ \Rightarrow \dfrac{{{R}_{1}}}{{{R}_{2}}}=\dfrac{40}{60} \\\ \therefore {{R}_{2}}=\dfrac{3}{2}{{R}_{1}}$
From equation (1) we get,
${{R}_{1}}=20\Omega$
So, the correct option is A.

Additional Information:
We know in parallel combination the voltage across the two resistors remains same and the equivalent resistance is given by the formula, $R=\dfrac{{{R}_{1}}{{R}_{2}}}{{{R}_{1}}+{{R}_{2}}}$

Note: In the meter bridge, one of the lateral kinds of resistances is replaced by a wire having a length of the uniform cross section of about 1m. The other two resistance are known and an unknown one. The one part of the galvanometer is connected in between both resistances, while the other part, it is connected and the wire is used in finding the null point where the galvanometer is not showing any deflection. At this point, the bridge is said to be balanced.