Understanding Optical Formulas: Mirror and Lens

In the study of optics, specifically geometric optics, two fundamental equations are used to relate the object distance, image distance, and focal length for spherical mirrors and thin lenses: the mirror formula and the lens formula. It is crucial to remember the correct sign conventions and the mathematical form of these formulas to accurately solve problems related to reflection and refraction.

Mirror Formula Analysis

Let's examine the first statement regarding the mirror formula:

• Statement I: Mirror formula : \((1/v) - (1/u) = (1/f)\)

The standard and correct mirror formula, which describes the relationship between the object distance \((u)\), image distance \((v)\), and focal length \((f)\) for spherical mirrors (both concave and convex), is given by:

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

Here, \(u\) is the distance of the object from the mirror, \(v\) is the distance of the image from the mirror, and \(f\) is the focal length of the mirror. It's important to apply proper sign conventions (e.g., New Cartesian Sign Convention) when using this formula.

Comparing Statement I with the correct mirror formula, we observe a difference in the sign between the terms \((1/v)\) and \((1/u)\). Statement I uses a minus sign, whereas the correct formula uses a plus sign. Therefore, Statement I is incorrect.

Lens Formula Analysis

Now, let's analyze the second statement concerning the lens formula:

• Statement II: Lens formula : \((1/v) + (1/u) = (1/f)\)

The standard and correct lens formula (also known as the thin lens equation), which relates the object distance \((u)\), image distance \((v)\), and focal length \((f)\) for thin spherical lenses (both concave and convex), is given by:

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

Similar to mirrors, \(u\) is the distance of the object from the optical center of the lens, \(v\) is the distance of the image from the optical center, and \(f\) is the focal length of the lens. Sign conventions must be consistently applied.

Upon comparing Statement II with the correct lens formula, we see that Statement II uses a plus sign between the terms \((1/v)\) and \((1/u)\), while the correct formula uses a minus sign. Thus, Statement II is also incorrect.

Summary of Correct Optical Formulas

To summarize, here are the universally accepted and correct formulas for spherical mirrors and thin lenses:

Based on our detailed analysis, both Statement I (for the mirror formula) and Statement II (for the lens formula) are incorrectly stated. The given question presents inversions of the correct signs for each respective formula.

Conclusion

Since both the provided statements for the mirror formula and the lens formula are incorrect, the pair presented in the question is not correct. Therefore, the option indicating that neither I nor II is correct aligns with our findings.