Question

An object is placed \(20.0 \mathrm{cm}\) from a converging lens with focal length \(15.0 \mathrm{cm}\) (see the figure, not drawn to scale). A concave mirror with focal length \(10.0 \mathrm{cm}\) is located \(75.0 \mathrm{cm}\) to the right of the lens. (a) Describe the final image- -is it real or virtual? Upright or inverted? (b) What is the location of the final image? (c) What is the total transverse magnification?

Short answer

The final image is virtual, upright with respect to the mirror but inverted overall, located 6 cm in front of the mirror, and the total transverse magnification is -1.2.

Step-by-step solution

Identify the Lens Formula

To find the image distance for the converging lens, we use the lens formula: \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length, \( d_o \) is the object distance, and \( d_i \) is the image distance.

Calculate Image Distance for Lens

Plugging in the given values for the lens: \( f = 15 \text{ cm} \) and \( d_o = 20 \text{ cm} \), we get:\[ \frac{1}{15} = \frac{1}{20} + \frac{1}{d_{i1}} \]Solving this equation gives \( d_{i1} = 60 \text{ cm} \). This image is real and inverted.

Determine Object Distance for Mirror

The image formed by the lens becomes the object for the mirror. Since the mirror is \(75 \text{ cm}\) from the lens, the object distance for the mirror \( d_{o2} = 75 - 60 = 15 \text{ cm} \), as the image from the lens is on the same side as the mirror.

Identify the Mirror Formula

To find the image distance for the concave mirror, use the mirror formula: \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \). Here, \( f = -10 \text{ cm} \) (negative for concave mirror) and \( d_{o2} = 15 \text{ cm} \).

Calculate Image Distance for Mirror

Plug the values into the mirror formula:\[ \frac{1}{-10} = \frac{1}{15} + \frac{1}{d_{i2}} \]Solving this gives \( d_{i2} = -6 \text{ cm} \). The negative sign indicates that the image is virtual and located on the same side as the object for the mirror.

Determine Final Image Characteristics

Given that the final image formed by the mirror is virtual, it is upright relative to the mirror's object (which was inverted from the lens). Thus, in reference to the original object, it remains inverted.

Calculate Total Transverse Magnification

The magnification of the lens is given by \( m_1 = -\frac{d_{i1}}{d_o} = -\frac{60}{20} = -3 \), and for the mirror, \( m_2 = -\frac{d_{i2}}{d_{o2}} = \frac{-6}{15} = 0.4 \). The total magnification is the product: \( m = m_1 \times m_2 = -3 \times 0.4 = -1.2 \).

Key concepts

Converging Lens

A converging lens, also known as a convex lens, is designed to redirect parallel rays of light towards a common focal point. These lenses are thicker at the center than at the edges. Converging lenses are often used in devices like cameras, magnifying glasses, and eyeglasses, helping to focus incoming light to produce a clear image.
When an object is placed in front of a converging lens, light rays from the object pass through the lens and initially converge at a point. This phenomenon is known as image formation. Depending on the distance of the object from the lens compared to the lens's focal length, the image may appear in various forms:

• Real and inverted: When the object is beyond the focal point.
• Virtual and upright: When the object is within the focal length.

Understanding these principles is crucial for analyzing how images are formed, especially when combining lenses and mirrors in optical systems.

Concave Mirror

A concave mirror has a reflective surface that curves inward, resembling the inside of a sphere. This type of mirror is effective in concentrating light to a focal point and is commonly found in telescopes, headlights, and shaving mirrors.
Concave mirrors follow a set pattern of behavior when forming images. They can produce both real and virtual images based on the object's location relative to the mirror's focal point:

• Real and inverted: If the object is positioned beyond the focal length from the mirror.
• Virtual and upright: If the object is within the focal length.

In our exercise, a concave mirror is located a certain distance away from a converging lens, together forming a more complex optical system. Understanding each component's role aids in predicting the final image characteristics and locations.

Image Formation

Image formation refers to the process by which lenses and mirrors redirect light rays to create an image. This foundational concept is essential in optics and is used in a variety of applications, from eyeglasses to cameras.
With a converging lens, image formation depends on the relative position of the object:

• If the object is outside the focal length, the lens forms a real, inverted image on the opposite side.
• If the object is inside the focal length, a virtual, upright image is formed on the same side as the object.

Similarly, a concave mirror forms images based on the object's distance from the mirror:

• Real and inverted images if positioned beyond the focal length.
• Virtual and upright images if within the focal length.

In the given exercise, understanding these principles is key to determining the resulting image characteristics across different optical elements.

Magnification

Magnification is a measure of how much larger or smaller an image is compared to the object. It is determined by the ratio of image height to object height and is crucial in diagnosing optical performances. Magnification can also be calculated by the formula:\[ m = -\frac{d_i}{d_o} \]where \( d_i \) is the image distance and \( d_o \) is the object distance.
In the context of the exercise, the converging lens and concave mirror both affect the overall magnification. Each element contributes its own magnification factor, which can be calculated using the respective distances:

• For the lens: \( m_1 = -\frac{d_{i1}}{d_o} \)
• For the mirror: \( m_2 = -\frac{d_{i2}}{d_{o2}} \)

The total transverse magnification is the product of the individual magnifications, reflecting the cumulative effect on the final image's size.

Lens Formula

The lens formula is crucial for determining the image distance given a specific focal length and object distance. The formula is expressed as:\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]where \( f \) is the lens's focal length, \( d_o \) is the object's initial distance, and \( d_i \) is the image distance.
This formula provides a mathematical way to predict where an image will form when light passes through a lens. By rearranging and solving the equation for \( d_i \), we can ascertain the image's precise location on the opposite side of the lens.
Applying this in the exercise, we first use known values for the converging lens, allowing us to compute where the lens would direct light to form an image. This is the starting point for following calculations involving other optical components, like mirrors.

Mirror Formula

Similar to lenses, mirrors have a specific formula to calculate image distances. The mirror formula is written as:\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]where \( f \) represents the mirror's focal length (negative for concave mirrors), and \( d_o \) and \( d_i \) are the object and image distances respectively.
This formula helps determine how mirrors will reflect light and position an image. In analyzing optical systems with mirrors, knowing the focal length and object distance allows us to find out where the image will form.
In the outlined exercise, employing the mirror formula helps identify the characteristics and location of the virtual image formed by the concave mirror. The calculations included ensure proper understanding of how light reflects and images are projected.