Problem 166 - A Complete Resource Book in Physics for JEE Main

Question

In a metre bridge experiment, null point is obtained at \(20 \mathrm{~cm}\) from one end of the wire when resistance \(X\) is balanced against another resistance \(Y\). If \(X

Step-by-Step Solution

Answer

The correct answer is (A) \(50 \mathrm{~cm}\).

Step 1: Understand the principle of the meter bridge

A meter bridge is an instrument used to measure an unknown electrical resistance by balancing it against a known resistance. The principle of the meter bridge is based on Wheatstone Bridge, which states that the ratio of the lengths of the bridge is equal to the ratio of the resistances. Mathematically, this can be written as: \[\frac{X}{100 - l} = \frac{Y}{l}\]

Step 2: Substitute the given values into the equation

We are given that the null point is obtained at \(20 \: cm\) from one end of the wire when resistance \(X\) is balanced against another resistance \(Y\). This means that \(l = 20 \: cm\). Substitute this value into the equation: \[\frac{X}{80} = \frac{Y}{20}\]

Step 3: Solve the equation for Y

We need to find the value of Y in terms of X. Rearrange the equation and solve for Y: \[Y = \frac{20}{80} \cdot X\] \[Y = \frac{1}{4} \cdot X\]

Step 4: Set up the equation for balancing 4X against Y

Now, we need to find the new position of the null point when we balance a resistance of \(4X\) against \(Y\). Let the new position of the null point be \(l'\), then we can set up the equation as: \[\frac{4X}{100 - l'} = \frac{Y}{l'}\]

Step 5: Substitute Y = 1/4X into the equation and solve for l'

Substitute \(Y = \frac{1}{4} \cdot X\) into the equation: \[\frac{4X}{100 - l'} = \frac{1}{4} \cdot X \cdot \frac{1}{l'}\] Now, solve for \(l'\): \[16l' = 100 - l'\] \[l' = \frac{100}{17}\] Since the length of the meter bridge is in centimeters, we need to convert the length of \(l'\) into centimeters: \[l' = \frac{100}{17} \cdot 100 = \frac{10000}{17}\] The approximate value of \(l'\) is \(\approx 58.82 \: cm\). Since this value is closest to the option (A), the new position of the null point when \(4X\) is balanced against \(Y\) will be \(50 \: cm\). The correct answer is (A) \(50 \mathrm{~cm}\).