Question

A light bulb is 3.00 m from a wall. You are to use a concave mirror to project an image of the bulb on the wall, with the image 3.50 times the size of the object. How far should the mirror be from the wall? What should its radius of curvature be?

Short answer

Place the mirror 4.50 m from the wall. Its radius of curvature is 8.40 m.

Step-by-step solution

Understanding the Given Information

We are given that the magnification factor is 3.50, meaning the image is 3.50 times the size of the object. The object (the light bulb) is placed 3.00 m from the wall. We need to determine both the distance of the mirror from the wall and the radius of curvature of the mirror.

Applying the Magnification Formula

The formula for magnification is given by \( M = \frac{-d_i}{d_o} \), where \( d_i \) is the image distance and \( d_o \) is the object distance. Here, \( M = 3.50 \) and \( d_o = 3.00 \) m. Therefore, the image distance, \( d_i = -3.50 \times 3.00 \) m.

Solving for Image Distance

Using the magnification formula, we find \( d_i = -10.50 \) m. The negative sign indicates that the image is formed on the same side as the object.

Relating Distances to the Mirror's Position

Let \( D \) be the distance from the wall to the mirror. Since \( d_i = D - d_o \), substituting the known, \( -10.50 = D - 3.00 \). Solving gives \( D = -10.50 + 3.00 = -7.50 \). Thus, the mirror is positioned 7.50 m from the light source, or equivalently, 3.00 m - 7.50 m = -4.50 m from the wall.

Calculating the Radius of Curvature

The mirror formula is \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length of the mirror. Using the previously calculated \( d_i \) and \( d_o \), solve for \( f \). After finding \( f \), use \( R = 2f \) to find the radius of curvature \( R \). Substituting, \( \frac{1}{f} = \frac{1}{3.00} + \frac{1}{-10.50} \), solve for \( f \). \( f \approx 4.20 \) m.

Determining the Radius of Curvature

Once the focal length is calculated, use the formula \( R = 2f \) to find the radius of curvature of the mirror. Thus, \( R = 2 \times 4.20 = 8.40 \) m.

Key concepts

Concave Mirror

A concave mirror is a spherical mirror where the reflecting surface curves inward, resembling a portion of the inside of a sphere or a bowl.
It is known for converging light rays that strike it parallel to its principal axis.
This convergence forms images and enables magnification, a key feature in optics.

Properties and Image Formation

Concave mirrors can produce both real and virtual images.
When an object is placed far from the mirror, beyond its focal point, it generates a real, inverted, and possibly magnified image.
Conversely, objects placed between the mirror and its focal point form a virtual, upright, and magnified image on the same side as the object.

• Real images can be projected on a screen, making them practical in many optical devices.
• The distance between the object and the mirror, as well as the mirror's curvature, plays a critical role in determining what type of image will form.

Magnification

Magnification is a measure of how much larger or smaller an image is compared to the object itself.
It is denoted by the letter 'M' and is calculated using the formula \[ M = \frac{-d_i}{d_o} \], where \( d_i \) is the image distance and \( d_o \) is the object distance.

Understanding the Concept

In our exercise, the magnification is 3.50.
This tells us that the image is 3.50 times the size of the light bulb.
The negative sign in the magnification formula indicates the nature of the image:

• - Negative magnification means the image is inverted compared to the object.
• By knowing \( M \), we can determine the relationship between object and image distances, crucial for further calculations.

Radius of Curvature

The radius of curvature of a concave mirror is the total diameter of the sphere of which the mirror is a part of.
This radius is represented by the letter 'R' and is twice the focal length, mathematically given by the formula \( R = 2f \).

Importance in Optics

Having a clear understanding of the radius of curvature helps us understand the mirror's power to focus light:

• The larger the radius of curvature, the flatter the mirror, impacting the clarity and size of the actual image produced.
• It directly influences focal length, which, in turn, dictates how the light behaves as it interacts with the mirror.

Application to the Exercise

For the given exercise, once we calculate the focal length (\( f \approx 4.20 \) m),
figuring out the radius of curvature becomes straightforward with \( R = 2 \times 4.20 = 8.40 \) m.
This helps determine how far and how we should position the mirror to achieve the desired image size.

Focal Length

Focal length is the distance between the mirror's surface and its focal point, the spot where parallel rays of light meet after reflection.
It's denoted as 'f' and plays a vital role in determining the mirror's behavior in focusing light.

Relation to Curvature and Image Formation

A shorter focal length implies a more curved mirror, while a longer focal length suggests a flatter surface.
The concave mirror formula \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \) relates it with object distance \( d_o \) and image distance \( d_i \).

Application in the Exercise

In the exercise, we used this formula to find the focal length needed to project the desired image size.
Solving the equation revealed \( f \approx 4.20 \) m, enabling us to also determine the radius of curvature by doubling this focal length value.
Such calculations are key for setting up real-world optical devices and applications.