Problem 56 - Master Resource Book in JEE Main Physics

Question

The dimensions of the power of lens are (a) \(\left[\mathrm{LT}^{-2}\right]\) (b) \(\left[\mathrm{M}^{0} \mathrm{~L}^{-1} \mathrm{~T}^{0}\right]\) (c) \(\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\) (d) None of these

Step-by-Step Solution

Answer

The correct answer is (b) \([\mathrm{M}^0 \mathrm{~L}^{-1} \mathrm{~T}^0]\).

Step 1: Understand the Concept of Power of a Lens

The power of a lens, often denoted as \(P\), is defined as the reciprocal of the focal length \(f\) of the lens in meters. Mathematically, it is expressed as \(P = \frac{1}{f}\) and is measured in diopters.

Step 2: Determine the Dimensions of Focal Length

The focal length \(f\) is a measure of distance, so it has the dimensional formula \([\mathrm{L}^1]\), where \(\mathrm{L}\) stands for length.

Step 3: Calculate the Dimensional Formula of the Power

Since power \(P\) is the reciprocal of the focal length, its dimensions are \(\frac{1}{[\mathrm{L}^1]} = [\mathrm{L}^{-1}]\).

Step 4: Match with the Given Options

The dimensions of power of a lens are \([\mathrm{L}^{-1}]\), which corresponds to option (b) \([\mathrm{M}^0 \mathrm{L}^{-1} \mathrm{T}^0]\). It indicates that power is only dependent on length and has no mass or time dimensions.

Key Concepts

Power of a Lens

The power of a lens is a measure of its ability to bend light. When we talk about the power of a lens, we refer to its optical strength. This is denoted by the letter "P" and is typically expressed in diopters. Power is inversely related to the focal length, with the formula given by the equation: \( P = \frac{1}{f} \). This means as the focal length decreases, the power of the lens increases.

• Diopters are the units used to describe lens power.
• The greater the diopter value, the stronger the lens's ability to converge or diverge light.
• A positive power indicates a converging lens (like convex lenses), while negative indicates a diverging lens (like concave lenses).

The power of a lens is crucial for applications in optics, including eyeglasses, microscopes, and camera lenses. Understanding it helps us correct vision or design optical instruments for various purposes.

Dimensional Analysis

Dimensional analysis is a mathematical tool used to understand relationships between different physical quantities by analyzing their dimensions. This technique is fundamental when figuring out how to express a quantity's physical dimensions in terms of specific physical units like length (L), mass (M), and time (T).
For the power of a lens, we start by analyzing its formula: \( P = \frac{1}{f} \). Since focal length \( f \) has the dimension of length, its dimension is represented as \([L^1]\). The power then, which is the reciprocal of focal length, has dimensions of \([L^{-1}]\).

• In dimensional analysis, quantities not involving mass or time are indicated by zeros in those dimensions, hence \([M^0 T^0]\).
• This analysis confirms that power mainly relates to length, having no direct relation to mass or time.
• Dimensional analysis helps ensure equations are dimensionally consistent and aids in converting units from one system to another.

Using dimensional analysis allows us to simplify complex physical laws and helps predict the behavior of systems in different but similar conditions.

Focal Length

Focal length is a key concept in understanding lenses and their behavior. It is the distance between the lens and its focus, where light rays converge or diverge after passing through the lens. This distance is crucial as it determines the lens's power.

• A shorter focal length means a greater bending of the light rays, thus, a stronger lens.
• The focal length is measured in meters, leading to its dimensional formula being \([L^1]\).
• In optical systems like cameras, controlling the focal length adjusts the field of view and zoom capability.

Understanding focal length helps in designing lenses for various practical applications, such as eyeglasses for correcting vision, or in devices like telescopes that need precise distance measurements. It's essential for anyone studying optics to fully grasp the concept of focal length and how it relates to lens power.