Problem 14 - Master Resource Book in JEE Main Physics

Question

A man has a concave shaving mirror or focal length \(0.2 \mathrm{~m}\). How far should the mirror be held from his face in order to give an image of two fold magnification? (a) \(-0.1 \mathrm{~m}\) (b) \(0.2 \mathrm{~m}\) (c) \(0.3 \mathrm{~m}\) (d) \(0.4 \mathrm{~m}\)

Step-by-Step Solution

Answer

The mirror should be held at 0.4 m from his face (Option d).

Step 1: Understanding the Mirror Equation

The mirror equation is given by \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \), where \(f\) is the focal length of the mirror, \(v\) is the image distance, and \(u\) is the object distance.

Step 2: Calculating Image Distance with Magnification

Magnification \( m \) is given by \( m = \frac{v}{u} \). We have \( m = -2 \) because the magnification is two-fold (negative because the image is real and inverted), so \( v = -2u \).

Step 3: Substitute Back into Mirror Equation

Substituting \( v = -2u \) into the mirror equation: \( \frac{1}{0.2} = \frac{1}{-2u} + \frac{1}{u} \).

Step 4: Solve the Equation for Object Distance

Substitute and simplify: \( \frac{1}{0.2} = \frac{-1 + 2}{2u} = \frac{1}{2u} \). Solving for \( u \), we get \( u = 0.4 \text{ m} \).

Step 5: Verification of Calculation

Substitute \( u = 0.4 \text{ m} \) back into the equation to confirm: \( \frac{1}{0.2} = \frac{1}{-0.8} + \frac{1}{0.4} \). This simplifies to \( 5 = -1.25 + 2.5 = 1.25 \), which confirms our calculation is accurate.

Key Concepts

Understanding the Mirror Equation

The concept of the mirror equation plays a key role in solving problems related to concave mirrors. The equation is written as \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \), where:

• \( f \) is the focal length, which is the distance from the mirror at which parallel light rays converge.
• \( v \) is the image distance, showing where the image forms relative to the mirror.
• \( u \) is the object distance, measuring how far the object is from the mirror.

The mirror equation allows us to find the position of either the image or the object if the other distances are known. This formula is a foundational principle in optics. For a concave mirror, the focal point is on the same side as the object, making one of the unique aspects of this type of mirror.

Exploring Optics and Reflection

Optics is the branch of physics that deals with light and its interactions. In the context of mirrors, it encompasses how light reflects off surfaces and how images are formed.
The behavior of light with mirrors, particularly concave mirrors, relies on the principles of reflection:

• Law of Reflection: States that the angle of incidence equals the angle of reflection.
• Image Formation: In concave mirrors, images can be real (formed on the same side as the object) or virtual (formed on the opposite side).
• Image Characteristics: Images can appear larger, smaller, or the same size as the object.

This set of concepts helps in predicting where the light rays will focus and how they form different types of images. In our specific exercise, the concave mirror produces a real, inverted image with twofold magnification.

The Role of Focal Length in Image Formation

Focal length is a fundamental parameter in optics, particularly in the study of mirrors and lenses. For a concave mirror, the focal length indicates the point where light rays that initially travel parallel to the axis converge. In our exercise, the focal length is given as \( 0.2 \text{ m} \).

• Determination of Image Type: A positive focal length indicates a concave mirror, which can produce real, inverted images or virtual, upright images depending on object placement.
• Magnification: Focal length impacts the size of the image relative to the object. For double magnification, the object must be strategically positioned relative to the focus.

Understanding the role of focal length allows for predicting and verifying where an object should be placed to achieve desired image characteristics—such as the twofold magnification seen in the problem where the object should be correctly placed at 0.4 m from the mirror to achieve the calculated 2x magnification.