Question

An object is located at a distance of \(100 . \mathrm{cm}\) from a concave mirror of focal length \(20.0 \mathrm{~cm}\). Another concave mirror of focal length \(5.00 \mathrm{~cm}\) is located \(20.0 \mathrm{~cm}\) in front of the first concave mirror. The reflecting sides of the two mirrors face each other. What is the location of the final image formed by the two mirrors and the total magnification by the combination?

Short answer

The final image formed by the combination of two concave mirrors is located at infinity, and the total magnification is 0.

Step-by-step solution

Find the image formed by the first mirror

To find out the image formed by the first mirror, we'll use the mirror formula: \( \frac{1}{f} = \frac{1}{d_o}+\frac{1}{d_i}\) Here, \(f = 20.0\;\text{cm}\) (focal length of the first mirror), \(d_o = 100\;\text{cm}\) (object distance from the first mirror), and \(d_i\) is the image distance we need to find. Plug in the values: \( \frac{1}{20} = \frac{1}{100}+\frac{1}{d_i} \) Solve for \(d_i\): \( d_i = \frac{100 \times 20}{100 - 20} = \frac{2000}{80} = 25\;\text{cm}\) So, the first image is formed at a distance of 25 cm in front of the first mirror.

Calculate magnification for the first mirror

Now, let's find the magnification of the first mirror using the formula: \( m_1 = -\frac{d_i}{d_o}\) Plug in the values we found in the previous step: \( m_1 = -\frac{25}{100} = -0.25 \) So the magnification of the first mirror is \(-0.25\).

Determine the new object distance for the second mirror

Since the second mirror is located \(20\;\text{cm}\) in front of the first mirror, the distance between the first image and the second mirror is: \( d_o' = 25 - 20 = 5\;\text{cm}\) Thus, the new object distance for the second mirror is 5 cm.

Find the final image formed by the second mirror

Now, we'll use the mirror formula again for the second mirror: \( \frac{1}{f'} = \frac{1}{d_o'}+\frac{1}{d_i'}\) Here, \(f' = 5.00\;\text{cm}\) (focal length of the second mirror), \(d_o' = 5\;\text{cm}\) (the new object distance), and \(d_i'\) is the final image distance we want to find. Plug in the values: \( \frac{1}{5} = \frac{1}{5}+\frac{1}{d_i'} \) Solve for \(d_i'\): \( d_i' = \infty\) This implies that the final image formed by the second mirror is at infinity.

Calculate magnification for the second mirror

Now, let's find the magnification of the second mirror using the formula: \( m_2 = -\frac{d_i'}{d_o'}\) Since \(d_i' = \infty\), the magnification of the second mirror is zero: \( m_2 = 0 \)

Calculate the total magnification

Finally, to find the total magnification by the combination of the two mirrors, we need to multiply the magnifications of both mirrors: \( m_{total} = m_1 \times m_2 \) \( m_{total} = -0.25 \times 0 \) \( m_{total}=0\) In conclusion, the final image formed by the combination of the two mirrors is located at infinity, and the total magnification is 0.

Key concepts

Concave Mirror

A concave mirror, also known as a converging mirror, has a reflective surface that curves inward, resembling the interior of a sphere. This design allows the mirror to focus parallel light rays towards a single point, known as the focal point. Such mirrors are often used in applications like telescopes, headlights, and for cosmetic mirrors to provide magnified images.
A key feature of concave mirrors is their ability to form real images when the object is placed beyond the focal point. These real images are inverted and can be projected onto a screen. However, if the object is placed between the focal point and the mirror, the image formed will be virtual, upright, and magnified.

Mirror Formula

The mirror formula is a mathematical relationship crucial for solving problems involving mirrors, particularly concave and convex mirrors. It is given by:

• \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \)

This formula connects three important quantities:

• \( f \): the focal length of the mirror, which is the distance between the mirror's surface and its focal point.
• \( d_o \): the object distance, which is how far the object is from the mirror.
• \( d_i \): the image distance, or the distance from the image to the mirror.

Using this formula, you can solve for any unknown parameter when the other two are known. It's essential for determining where an image will form relative to the mirror and its characteristics.

Magnification

Magnification is a measure of how much larger or smaller the image appears compared to the actual object. For mirrors, magnification can be defined using the formula:

• \( m = -\frac{d_i}{d_o} \)

Here:

• \( m \) is the magnification.
• \( d_i \) is the image distance.
• \( d_o \) is the object distance.

The negative sign is included to indicate whether the image is inverted (which happens when the magnification is negative) or upright (when the magnification is positive).
In the exercise, the first mirror's magnification was found to be \(-0.25\), showing the image is smaller than the object and inverted. The total magnification is the product of individual magnifications, showing overall size change of the image through multiple mirrors.

Image Distance

In optics, image distance (\(d_i\)) measures how far the image is from the mirror along the principal axis. It is a crucial aspect of image formation.

• A positive image distance indicates a real image formed on the same side as the object in the case of mirrors.
• A negative image distance suggests a virtual image formed on the opposite side.

In concave mirrors, real images are formed when light converges and passes through a point after reflection. In the problem given, the first image distance calculated was \(25\;\rm{cm}\), meaning it was a real image formed in front of the first mirror.

Focal Length

The focal length of a mirror is the distance from the mirror's surface to its focal point, where parallel light rays either converge (concave mirror) or appear to diverge from (convex mirror). It is a vital feature because it determines the converging or diverging power of the mirror.

• The focal length (\( f \)) is half of the radius of curvature of the mirror.
• A shorter focal length means greater converging power in concave mirrors, thus producing larger magnified images close by.

In the exercise described, two concave mirrors with focal lengths of \(20.0\;\rm{cm}\) and \(5.0\;\rm{cm}\) are used. Their variable focal lengths impact the formation of images and the magnification effects in the optical setup.