Problem 23 - Physics

Question

When viewed in a spherical mirror, the image of a setting sun is a virtual image. The image lies \(12.0 \mathrm{~cm}\) behind the mirror. (a) Is the mirror concave or convex? Why? (b) What is the radius of curvature of the mirror?

Step-by-Step Solution

Answer

(a) The mirror is convex; it always forms virtual images behind. (b) The radius of curvature is \(-24.0 \mathrm{~cm}\).

Step 1: Identify Type of Mirror

Since the image is a virtual image located behind the mirror, we are dealing with a convex mirror. A concave mirror would create a real or virtual image in front of it (for object distances greater than its focal length), while a convex mirror always forms virtual images behind the mirror.

Step 2: Use the Mirror Formula

The mirror formula is \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length, \( d_o \) is the object distance, and \( d_i \) is the image distance. For a convex mirror, virtual images have negative image distances, so \( d_i = -12.0 \) cm.

Step 3: Determine Focal Length

In a typical setup for a distant object like the sun, the object distance \( d_o \) is approximately infinite, making \( \frac{1}{d_o} \approx 0 \). Therefore, \( \frac{1}{f} = \frac{1}{d_i} = -\frac{1}{12.0} \). Solve for \( f \): \( f = -12.0 \) cm.

Step 4: Compute Radius of Curvature

The radius of curvature \( R \) of a spherical mirror is \( R = 2f \). Substituting \( f = -12.0 \) cm, we find \( R = 2(-12.0) = -24.0 \) cm. The negative sign indicates the mirror is convex.

Key Concepts

Convex Mirrors

Convex mirrors are spherical mirrors where the reflective surface bulges outward. Unlike concave mirrors, they do not converge light rays to a focal point; instead, they diverge them.
This characteristic results in the formation of virtual images behind the mirror. Convex mirrors have several practical uses:

• Vehicle Side Mirrors: These mirrors provide a wider field of view, helping drivers to see more of the road and surrounding areas.
• Security and Surveillance: Often used in shops or public places to monitor larger areas.

Convex mirrors always produce images that appear reduced in size compared to the actual object. These images are upright and located behind the mirror, giving an additional confirmation that the mirror is convex.

Virtual Images

Virtual images are images that appear to be located behind the mirror's surface. Unlike real images, which can be projected onto a screen, virtual images cannot be projected. They are formed:

• When the reflected rays diverge.
• By extending the reflected rays backward, making them appear to converge behind the mirror.

In the context of convex mirrors, the nature of these images provides insights into the position and size of objects. For example, a virtual image of the setting sun, as described in our problem, indicates the reflective surface's unique properties that inherently redirect light rays in such a way that they seem to originate from a point behind the mirror.

Mirror Formula

The mirror formula is a crucial equation in understanding how spherical mirrors form images. Expressed as \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]where:

• \( f \): focal length of the mirror
• \( d_o \): object distance (distance from the object to the mirror)
• \( d_i \): image distance (distance from the image to the mirror)

For convex mirrors, the image distance (\( d_i \)) is negative since virtual images form behind the mirror. The focal length (\( f \)) also has a negative sign, confirming the convex nature of the mirror. This formula allows calculations involving either real or virtual images, making it versatile for solving numerous optical problems.

Radius of Curvature

The radius of curvature (\( R \)) of a spherical mirror is the radius of the sphere of which the mirror is a part. It is related to the mirror's focal length by the relation \( R = 2f \).

For convex mirrors, the negative focal length leads to a negative radius of curvature. This simply reinforces the concept that the mirror's surface curves outward:

• A convex mirror's negative radius of curvature signifies it diverges light, opposite to how concave mirrors converge light.

In the given problem, the computed radius of curvature of \(-24.0\) cm clearly indicates the presence of a convex mirror, supporting the characteristics of virtual image formation and extended field of view.