Problem 100 - University Physics with Modern Physics

Question

A convex spherical mirror with a focal length of magnitude 24.0 \(\mathrm{cm}\) is placed 20.0 \(\mathrm{cm}\) to the left of a plane mirror. An object 0.250 \(\mathrm{cm}\) tall is placed midway between the surface of the plane mirror and the vertex of the spherical mirror. The spherical mirror forms multiple images of the object. Where are the two images of the object formed by the spherical mirror that are closest to the spherical mirror, and how tall is each image?

Step-by-Step Solution

Answer

First image: -7.06 cm from the mirror, height 0.177 cm; second image: 120 cm from the mirror, height 1.5 cm.

Step 1: Determine the Object Distance for the Spherical Mirror

The object is placed midway between the plane mirror and the spherical mirror, so it is 10 cm from the spherical mirror.Given:\[ d_o = 10 \text{ cm} \]

Step 2: Apply Mirror Formula to Find the Image Distance

The mirror formula is given by:\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]For a convex mirror, the focal length (\(f\)) is negative, so \( f = -24 \text{ cm} \).Substitute the values:\[ \frac{1}{d_i} = \frac{1}{-24} - \frac{1}{10} = -0.0417 - 0.1 = -0.1417 \]Solving for \(d_i\):\[ d_i = \frac{1}{-0.1417} \approx -7.06 \text{ cm} \]This is the image distance for the first image formed by the spherical mirror.

Step 3: Determine the Height of the First Image

Using the magnification formula:\[ m = -\frac{d_i}{d_o} \]\[ m = -\frac{-7.06}{10} = 0.706 \]Height of the image (\(h_i\)):\[ h_i = m \times h_o = 0.706 \times 0.250 \approx 0.1767 \text{ cm} \]The height of the first image is approximately 0.177 cm.

Step 4: Consider Reflected Image from Plane Mirror

The light first reflects off the plane mirror, creating a virtual object for the spherical mirror at a distance of 10 cm to the right of the spherical mirror. The virtual object is at a distance:\[ d'_o = -20 \text{ cm} \](because the object is behind the mirror with respect to incoming light)

Step 5: Calculate Image Distance for Virtual Object using Mirror Formula

Substitute \(d'_o = -20\) cm into the mirror formula (\(f = -24\) cm):\[ \frac{1}{d'_i} = \frac{1}{-24} - \frac{1}{-20} = -0.0417 + 0.05 = 0.0083 \]Solving for \(d'_i\):\[ d'_i = \frac{1}{0.0083} \approx 120 \text{ cm} \]This distance is the image of the virtual object created by the plane mirror.

Step 6: Calculate Height of the Second Image

The magnification is calculated similarly:\[ m' = -\frac{d'_i}{d'_o} = -\frac{120}{-20} = 6 \]Height of the second image:\[ h'_i = m' \times h_o = 6 \times 0.250 = 1.5 \text{ cm} \]The height for the second image is 1.5 cm.

Key Concepts

Convex Mirrors

Convex mirrors are a type of spherical mirror that curves outward, resembling the outside of a sphere. Unlike concave mirrors, convex mirrors diverge light rays that strike their surface. This means that when parallel rays of light hit a convex mirror, the reflected rays appear to diverge from a single point behind the mirror. That point is called the focal point. For convex mirrors, the focal point, and therefore the focal length, is considered to be negative in optical calculations. This characteristic makes convex mirrors useful for a wide array of applications, such as rear-view mirrors in vehicles, providing a wider field of view.

Mirror Formula

The mirror formula is an essential equation used to establish the relationship between the object distance (\(d_o\)), the image distance (\(d_i\)), and the focal length (\(f\)) of spherical mirrors. The formula can be stated as:

\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]
This equation is fundamental in optics for calculating where an image will form when light reflects off a mirror.

• For convex mirrors, the focal length (\(f\)) is always taken as a negative value.
• The object distance (\(d_o\)) is usually measured from the object to the mirror, and it is positive if the object is on the same side as the incoming light.
• The image distance (\(d_i\)) can be negative when the image is virtual, typical in convex mirrors, indicating that the image appears to be on the opposite side of the mirror.

Image Distance Calculation

Calculating image distance is done using the mirror formula, by rearranging it as:

\[\frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o}\]
We substitute known values of the focal length and object distance into this revised formula. Let's use an example:

• In problem scenarios involving convex mirrors, always make sure the focal length (\(f\)) is negative. For instance, if it is given as 24 cm, it should be used as -24 cm in calculations.
• Place the object distance (\(d_o\)) under the appropriate condition, whether it needs a negative or positive designation.
• For a real-life scenario like our given problem, after solving the equation, you might find \(d_i\) could be a negative value, revealing that the image is virtual and on the opposite side of the mirror.

In essence, understanding what each sign means is crucial for correct interpretation of optical results.

Magnification Formula

The magnification formula is a useful tool in determining how the size of an image compares to the size of the object itself. It can be expressed as:

\[m = -\frac{d_i}{d_o}\]
and the height of the image (\(h_i\)) is measured using:

\[h_i = m \times h_o\]
Here, \(m\) is the magnification.

• A magnification (\(m\)) greater than 1 means the image is larger than the object, while a magnification less than 1 means the image is smaller.
• The negative sign in the formula indicates that if the image distance (\(d_i\)) is negative (which means the image is virtual), the image will be upright.
• In the context of convex mirrors, since they always form virtual, erect, and reduced images, it's common to find that the magnitude of \(m\) is less than 1.

Understanding these relationships aids in predicting how the convex mirror will alter the perception of the original object.