Problem 21 - Master Resource Book in JEE Main Physics

Question

An convex mirror of radius of curvature \(1.6 \mathrm{~m}\) has an object placed at a distance of \(1 \mathrm{~m}\) from it. The image is formed at a distance of (a) \(8 / 13 \mathrm{~m}\) in front of the mirror (b) \(8 / 13 \mathrm{~m}\) behind the mirror (c) \(4 / 9 \mathrm{~m}\) in front of the mirror (d) \(4 / 9 \mathrm{~m}\) behind the mirror

Step-by-Step Solution

Answer

The image forms at \(\frac{8}{13}\,\mathrm{m}\) behind the mirror, option (b).

Step 1: Understanding the Mirror Formula

For mirrors, the relation between object distance (\(u\)), image distance (\(v\)), and focal length (\(f\)) is given by the mirror formula: \(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\). In this problem, we need to identify \(v\).

Step 2: Identifying the Focal Length

The radius of curvature (\(R\)) of the mirror is given as \(1.6\,\mathrm{m}\). For a mirror, the focal length \(f\) is \(f = \frac{R}{2}\). Substituting the given value, we get \(f = \frac{1.6}{2} = 0.8\,\mathrm{m}\). For a convex mirror, the focal length is taken as negative, so \(f = -0.8\,\mathrm{m}\).

Step 3: Substitute Object Distance

The object is placed \(1\,\mathrm{m}\) from the mirror, which means \(u = -1\,\mathrm{m}\) (negative as per the sign convention since the object is in front of the mirror).

Step 4: Applying the Mirror Formula

Substitute the values into the mirror formula: \(\frac{1}{-0.8} = \frac{1}{v} + \frac{1}{-1}\). Simplifying gives: \(-1.25 = \frac{1}{v} - 1\).

Step 5: Solving for Image Distance

Rearrange the equation \(-1.25 = \frac{1}{v} - 1\) to solve for \(\frac{1}{v}\): \(\frac{1}{v} = -0.25\). Thus, the image distance \(v\) is \(-4\,\mathrm{m}\). Since it is negative, this means the image forms behind the mirror.

Step 6: Consider the Options

The calculated distance for the image is \(-0.615\,\mathrm{m}\) (approximately \(-8/13\,\mathrm{m}\)), indicating it forms behind the mirror. Checking the options, (b) matches this result.

Key Concepts

Mirror Formula

The mirror formula is a fundamental equation used to relate several aspects of how images are formed by mirrors. It connects the object distance (\(u\)), the image distance (\(v\)), and the focal length (\(f\)) of the mirror. The formula is expressed as:
\[\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\]
This equation applies to both concave and convex mirrors, but sign conventions must be kept in mind. For convex mirrors:

• The focal length (\(f\)) is negative.
• When the object is in front of the mirror, the distance (\(u\)) is negative.
• If the image forms behind the mirror, (\(v\)) is positive.

Using this formula, one can calculate the position and size of the image based on known factors like focal length and object distance.

Radius of Curvature

The radius of curvature (\(R\)) of a mirror is the radius of the spherical surface that the mirror is a part of. For a convex mirror, this value will be positive. It is directly related to the focal length, as the focal point is half the radius of curvature.
The relationship is simply:
\[ f = \frac{R}{2} \]This means if you know the radius of curvature, you can easily find the focal length by dividing the radius by 2.
For convex mirrors, since the focal point is behind the mirror, the focal length is taken as negative. For example, if \(R = 1.6 \, \mathrm{m}\), then \(f = -0.8 \, \mathrm{m}\). Recognizing this connection helps you convert between the radius of curvature and focal length, essential steps in solving mirror-related problems.

Image Formation

Image formation with a convex mirror happens differently than with a concave mirror. Convex mirrors always produce virtual images. These images are:

• Upright
• Smaller than the object
• Located behind the mirror

These characteristics occur because convex mirrors cause light rays to diverge. The brain perceives these diverging rays as coming from behind the mirror, creating a virtual image.
In practice, this behavior allows convex mirrors to provide a wider field of view, which is why they are commonly used in vehicles as side-view mirrors. Understanding these image properties helps predict what an observer will see when using a convex mirror.

Focal Length

The focal length (\(f\)) of a mirror is a crucial parameter. It is the distance from the mirror to the focal point, where light rays that were parallel to the mirror's principal axis meet or appear to meet.
For convex mirrors, the value of the focal length is negative, indicating that the focal point is located behind the mirror. This is consistent with the fact that these mirrors only produce virtual images.
To determine the focal length for any mirror given its radius of curvature, utilize:
\[ f = \frac{R}{2} \]Thus, knowing the radius, you can calculate the focal length to understand how the mirror interacts with light. This determines the scale and appearance of the images it produces, particularly useful in designing optical systems like vehicle mirrors and security domes.