Question

Five persons \(P_1, P_2, P_3, P_4\) and \(P_5\) recorded object distance (\(u\)) and image distance (\(v\)) using same convex lens having power \(+5\) D as (25,96), (30,62), (35,37), (45,35) and (50,32) respectively. Identify correct statement.

• A. Readings recorded by \(P_4\) and \(P_5\) persons are incorrect
• B. Readings recorded by \(P_3\) and \(P_2\) persons are incorrect
• C. Readings recorded by all persons are correct
• D. Readings recorded by \(P_3\) persons are incorrect

Hint:

Always verify experimental readings using the lens formula to check consistency.

Answer 1

The Correct Option is D

To solve this problem, we must verify the recorded values of object distance \(u\) and image distance \(v\) using the lens formula for a convex lens. The power of the lens (\(P\)) is given as \(+5\) D. The lens formula is given by:

\(\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\)

Where \(f\) is the focal length of the lens. The focal length \(f\) can be calculated using the formula:

\(f = \frac{1}{P} = \frac{1}{5} = 0.2 \, \text{m} = 20 \, \text{cm}\)

Let us now verify each person's readings:

1. For \(P_1\): \(u = 25\) cm, \(v = 96\) cm

Applying the lens formula:

\(\frac{1}{f} = \frac{1}{96} - \frac{1}{25}\)

Calculate:

\(\frac{1}{f} = \frac{25 - 96}{2400} = \frac{-71}{2400}\)

We should have \(\frac{1}{f} = \frac{1}{20}\), which is approximately \(0.05\). The calculation shows a significant error from the expected result. This indicates a possibility of incorrect reading here.

1. For \(P_2\): \(u = 30\) cm, \(v = 62\) cm

Applying the lens formula:

\(\frac{1}{f} = \frac{1}{62} - \frac{1}{30}\)

Calculate:

\(\frac{1}{f} = \frac{30 - 62}{1860} = \frac{-32}{1860}\)

This value is incorrect as well.

1. For \(P_3\): \(u = 35\) cm, \(v = 37\) cm

Applying the lens formula:

\(\frac{1}{f} = \frac{1}{37} - \frac{1}{35}\)

Calculate:

\(\frac{1}{f} = \frac{35 - 37}{1295} = \frac{-2}{1295}\)

This calculation is significantly off from the expected value.

1. For \(P_4\): \(u = 45\) cm, \(v = 35\) cm

Applying the lens formula:

\(\frac{1}{f} = \frac{1}{35} - \frac{1}{45}\)

Calculate:

\(\frac{1}{f} = \frac{45 - 35}{1575} = \frac{10}{1575}\)

This result needs further validation, as minor measurement inaccuracies might affect it, however, it is closer to expectations.

1. For \(P_5\): \(u = 50\) cm, \(v = 32\) cm

Applying the lens formula:

\(\frac{1}{f} = \frac{1}{32} - \frac{1}{50}\)

Calculate:

\(\frac{1}{f} = \frac{50 - 32}{1600} = \frac{18}{1600} \approx 0.011\)

This result is incorrect.

After analysis, the readings recorded by \(P_3\) with \(u = 35\) cm and \(v = 37\) cm are more evidently incorrect as they deviate significantly, not aligning at all with the expected calculations.

Thus, the correct option is: Readings recorded by \(P_3\) persons are incorrect.