Problem 20 - Master Resource Book in JEE Main Physics

Question

An object \(5 \mathrm{~cm}\) tall is placed \(1 \mathrm{~m}\) from a concave spherical mirror which has a radius of curvature of \(20 \mathrm{~cm}\). The size of the image is (a) \(0.11 \mathrm{~cm}\) (b) \(-0.55 \mathrm{~cm}\) (c) \(0.55 \mathrm{~cm}\) (d) \(0.60 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The size of the image is (b) \(-0.55 \text{ cm}\).

Step 1: Convert Units

First, convert the distance of the object from meters to centimeters since the mirror's focal length and the image height are given in centimeters. We know that there are 100 cm in a meter, so the object distance (denoted as \( d_o \)) becomes \( 100 \text{ cm} \).

Step 2: Calculate Focal Length

Use the radius of curvature \( R = 20 \text{ cm} \) of the mirror to find the focal length \( f \). The focal length \( f \) of a spherical mirror is given by \( f = \frac{R}{2} \). Thus, \( f = \frac{20}{2} = 10 \text{ cm} \).

Step 3: Apply the Mirror Formula

Use the mirror formula, \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), to solve for the image distance \( d_i \). Substitute \( f = 10 \text{ cm} \) and \( d_o = 100 \text{ cm} \) into the equation: \( \frac{1}{10} = \frac{1}{100} + \frac{1}{d_i} \).

Step 4: Solve for Image Distance

Rearrange the equation to solve for \( \frac{1}{d_i} \): \( \frac{1}{d_i} = \frac{1}{10} - \frac{1}{100} = \frac{10 - 1}{100} = \frac{9}{100} \). Thus, \( d_i = \frac{100}{9} \approx 11.11 \text{ cm} \).

Step 5: Calculate Magnification

Calculate the magnification \( m \) using the formula \( m = -\frac{d_i}{d_o} \). Substituting \( d_i \approx 11.11 \text{ cm} \) and \( d_o = 100 \text{ cm} \), we get \( m = -\frac{11.11}{100} = -0.1111 \).

Step 6: Determine Image Height

Using the magnification, calculate the image height \( h_i \) using \( h_i = m \times h_o \), where \( h_o = 5 \text{ cm} \) is the object height. Thus, \( h_i = -0.1111 \times 5 = -0.5555 \text{ cm} \), which rounds to \(-0.55 \text{ cm} \).

Key Concepts

Concave Mirror

A concave mirror is a type of spherical mirror that curves inward, resembling a portion of the inside of a sphere. These mirrors are also known as converging mirrors because they can converge light to a focal point. When light rays parallel to the principal axis strike a concave mirror, they are reflected inward and meet at a point called the "focal point."

Concave mirrors are often used in:

• Telescopes, to focus light from distant stars and galaxies.
• Shaving or makeup mirrors, to provide a magnified reflection of the face.
• Headlights of vehicles, to focus the light in a beam.

They are versatile tools in both practical applications and scientific instruments. The behavior of a concave mirror is governed by the mirror formula and related principles that dictate how light behaves when it strikes the mirror surface.

Focal Length

The focal length of a concave mirror is the distance between its surface and its focal point. This distance is crucial because it determines how the mirror will reflect light and form an image.

For a spherical mirror, the focal length (\( f \)) is half of the radius of curvature (\( R \)). Given by the formula:\[f = \frac{R}{2}\]For example, if a concave mirror has a radius of curvature of 20 cm, its focal length will be 10 cm. The focal length tells us how strongly the mirror converges or diverges light. In practical terms, a shorter focal length indicates a stronger curvature and thus a more powerful converging effect, while a longer focal length reflects a gentler convergence.

Magnification

Magnification describes how much larger or smaller an image appears compared to the actual size of the object. For mirrors, it is defined by the formula:\[ m = -\frac{d_i}{d_o} \]where

• \( m \) is the magnification factor
• \( d_i \) is the image distance (distance from the mirror to the image)
• \( d_o \) is the object distance (distance from the mirror to the object)

A negative magnification indicates an inverted image, whereas a positive magnification implies an upright one.

An example from our exercise shows that when an object is located 100 cm from a concave mirror, and the resulting image is found 11.11 cm in front of the mirror, the magnification calculated as \(-0.1111\) signifies that the image is inverted and about 11.11% of the object's size.

Image Distance

Image distance refers to how far the image formed by a mirror is from the mirror itself. This is denoted as \( d_i \) in the mirror formula. Calculating the image distance is fundamental for understanding image characteristics such as size, orientation, and type.

The mirror formula for determining image distance is:\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]where

• \( f \) is the focal length of the mirror
• \( d_o \) is the object distance
• \( d_i \) is the image distance

By inserting known values for the focal length and object distance into this formula, one can solve for \( d_i \). For example, in the previous exercise, substituting \( f = 10 \text{ cm} \) and \( d_o = 100 \text{ cm} \) into the mirror formula helps us find that \( d_i \approx 11.11 \text{ cm} \). This indicates that the image appears very close to the mirror, illustrating how a concave mirror transforms reflections.