Image Formed by a Convex Mirror: Virtual and Upright Image Explained

Correct Answer: (D) Virtual and upright

Understanding the Given Convex Mirror Problem

This question asks us to determine the nature of the image formed by a convex mirror. The object is placed 15 cm in front of the mirror, while the radius of curvature of the mirror is given as 30 cm. To solve the question completely, we can use both the standard image-forming property of a convex mirror and the mathematical mirror formula.

A convex mirror is also known as a diverging mirror. When light rays from a real object strike a convex mirror, the reflected rays spread out or diverge. These reflected rays do not actually meet in front of the mirror. However, when their backward extensions are drawn, they appear to meet behind the mirror. Therefore, a convex mirror forms a virtual image of a real object.

For any position of a real object in front of a convex mirror, the image is always formed behind the mirror between the pole and the principal focus. The image is always virtual, upright and diminished. Therefore, even before performing any calculation, we can identify option (D) as the correct answer.

Relationship Between Radius of Curvature and Focal Length

The radius of curvature R and focal length f of a spherical mirror are related by:

f = R/2

The given radius of curvature is:

R = 30 cm

Therefore:

f = 30/2

Hence:

f = 15 cm

According to the Cartesian sign convention, the principal focus of a convex mirror lies behind the mirror. Distances measured behind the mirror are taken as positive. Therefore, the focal length of the convex mirror is:

f = +15 cm

Applying the Cartesian Sign Convention

Correct use of the sign convention is essential when solving numerical problems involving spherical mirrors. The pole of the mirror is taken as the origin, and all distances are measured from the pole along the principal axis.

The object is placed 15 cm in front of the convex mirror. Since the object lies on the side opposite to the positive direction of incident light, its object distance is negative:

u = −15 cm

The focal length of the convex mirror is positive because its principal focus lies behind the mirror:

f = +15 cm

We now use these values to determine the image distance.

Calculating the Image Distance Using the Mirror Formula

The mirror formula is:

1/f = 1/v + 1/u

Substituting the given values:

1/15 = 1/v + 1/(−15)

Therefore:

1/15 = 1/v − 1/15

Adding 1/15 to both sides:

1/v = 1/15 + 1/15

Therefore:

1/v = 2/15

Hence:

v = 15/2 cm

Therefore:

v = +7.5 cm

The positive value of the image distance shows that the image is formed behind the mirror. An image formed behind a mirror by the apparent intersection of reflected rays is a virtual image.

Position of the Image Formed by the Convex Mirror

The focal length of the convex mirror is 15 cm, while the calculated image distance is 7.5 cm. Therefore, the image is formed 7.5 cm behind the mirror.

The principal focus is located 15 cm behind the mirror. Since the image is formed at 7.5 cm behind the mirror, it lies between the pole and the principal focus. This is exactly the expected image position for a real object placed in front of a convex mirror.

Thus, the calculated result agrees with the general rule that a convex mirror always forms the image of a real object behind the mirror, between the pole and the principal focus.

Calculating the Magnification of the Image

The magnification produced by a spherical mirror is given by:

m = −v/u

Substituting the values:

m = −(+7.5)/(−15)

Therefore:

m = +0.5

The positive sign of magnification indicates that the image is upright or erect with respect to the object. The magnitude of magnification is less than 1, which shows that the image is smaller than the object.

Therefore, the image formed by the convex mirror is not only virtual and upright but also diminished to half the size of the object.

Why Does a Convex Mirror Form a Virtual Image?

When light rays from the object strike the reflecting surface of a convex mirror, they are reflected outward and diverge from one another. Since the reflected rays move away from each other, they cannot physically meet in front of the mirror.

However, an observer looking into the mirror sees the reflected rays as if they were coming from a point behind the mirror. When the reflected rays are extended backward, their extensions meet behind the mirror. The image formed by this apparent intersection is called a virtual image.

Because the actual light rays do not pass through the image position, the image cannot be obtained on a screen. This is one of the main characteristics of a virtual image.

Why Is the Image Upright?

The image formed by a convex mirror has the same orientation as the object. This is confirmed mathematically by the positive value of magnification:

m = +0.5

In mirror optics, positive magnification indicates an upright image, whereas negative magnification indicates an inverted image. Since the magnification is positive in this problem, the image is upright.

This result is also consistent with the general behavior of convex mirrors. For every real object placed in front of a convex mirror, the image is always virtual and upright.

Detailed Analysis of Each Option

Option (A): Real and Inverted

Option (A) is incorrect. A real and inverted image can be formed by a concave mirror when the object is placed beyond its principal focus. A convex mirror, however, causes reflected rays to diverge and cannot form a real image of a real object.

In the given problem, the calculated image distance is +7.5 cm, which means the image is formed behind the mirror. Therefore, the image is virtual rather than real. The positive magnification also shows that the image is upright rather than inverted.

Option (B): Real and Upright

Option (B) is incorrect because the image formed by the convex mirror is not real. The reflected rays from a convex mirror do not actually converge at a point. They only appear to originate from a point behind the mirror.

Although the image is upright, it is virtual. Therefore, describing the image as real and upright is incorrect.

Option (C): Virtual and Inverted

Option (C) is incorrect. The first part of the statement is correct because the image formed by a convex mirror is virtual. However, the image is not inverted.

The calculated magnification is +0.5. A positive magnification indicates an upright image. Therefore, the complete description “virtual and inverted” is not correct.

Option (D): Virtual and Upright

Option (D) is correct. The image distance is +7.5 cm, which shows that the image is formed behind the mirror and is therefore virtual. The magnification is +0.5, and the positive sign confirms that the image is upright.

Thus, the image formed by the convex mirror is virtual, upright and diminished.

General Image-Forming Property of a Convex Mirror

A convex mirror always forms a virtual, upright and diminished image of a real object, irrespective of the distance between the object and the mirror. The image is always located behind the mirror between the pole and the principal focus.

When the object is very far away, the image forms close to the principal focus and is highly diminished. As the object moves closer to the mirror, the image moves toward the pole and becomes relatively larger. However, it still remains smaller than the object, virtual and upright.

This ability to produce a wide field of view and upright images is why convex mirrors are commonly used as rear-view mirrors and side-view mirrors in vehicles.

Final Answer

The radius of curvature of the convex mirror is:

R = 30 cm

Therefore, its focal length is:

f = R/2 = 15 cm

Using the Cartesian sign convention:

f = +15 cm

u = −15 cm

Applying the mirror formula:

1/f = 1/v + 1/u

we obtain:

v = +7.5 cm

The positive image distance shows that the image is formed behind the mirror and is therefore virtual. The magnification is:

m = −v/u = +0.5

The positive magnification shows that the image is upright.

Therefore, the correct answer is (D) Virtual and upright.