Problem 8 - University Physics with Modern Physics

Question

An object is 18.0 cm from the center of a spherical silvered-glass Christmas tree ornament 6.00 cm in diameter. What are the position and magnification of its image?

Step-by-Step Solution

Answer

The image position is -1.38 cm, and the magnification is 0.077.

Step 1: Recognize the Problem Type

The problem involves determining the image position and magnification for an object and a spherical mirror (silvered-glass ornament). Knowing the properties of a spherical mirror, we identify that the given ornament behaves as a convex mirror.

Step 2: Determine the Mirror's Focal Length and Radius of Curvature

Since the ornament is a sphere, the radius of curvature \( R \) can be found using its diameter. Given the diameter is 6.00 cm, the radius is \( R = \frac{6.00}{2} = 3.00 \) cm. The focal length \( f \) of a spherical mirror is \( \frac{R}{2} \). For a convex mirror, \( f = -\frac{R}{2} = -1.50 \) cm.

Step 3: Apply the Mirror Equation

Use the mirror equation: \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f = -1.50 \) cm, \( d_o = 18.0 \) cm, and \( d_i \) is the image distance we need to find. Solve for \( \frac{1}{d_i} \): \[ \frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} = \frac{1}{-1.50} - \frac{1}{18.0} \]

Step 4: Calculate the Image Distance

Compute \( \frac{1}{d_i} \):\[ \frac{1}{d_i} = -\frac{2}{3} - \frac{1}{18} = -\frac{12}{18} - \frac{1}{18} = -\frac{13}{18} \]Thus, \( d_i = -\frac{18}{13} \approx -1.38 \) cm. The negative sign indicates the image is virtual and located behind the mirror.

Step 5: Calculate the Magnification

The magnification \( m \) is given by \( m = \frac{-d_i}{d_o} \). Substitute \( d_i = -1.38 \) cm and \( d_o = 18.0 \) cm to get:\[ m = \frac{-(-1.38)}{18.0} = \frac{1.38}{18.0} \approx 0.077 \]The positive magnification indicates the image is upright.

Key Concepts

Convex Mirror

A convex mirror is a type of spherical mirror where the reflecting surface is curved outward. This shape causes reflected light to spread out, which is why convex mirrors are known as diverging mirrors. Convex mirrors create images that have some unique properties which are very useful in everyday life.

• They always form virtual images. This means the reflected light appears to originate from a point behind the mirror.
• The images are diminished, meaning they are smaller than the actual object.
• These images are typically upright.

Convex mirrors are commonly used in places where a wide field of view is needed, like in vehicle side mirrors and security mirrors. Their ability to give a wider perspective is due to the way they reflect light.

Mirror Equation

The mirror equation is a fundamental formula used to relate the object distance (\(d_o\)), the image distance (\(d_i\)), and the focal length (\(f\)) of a spherical mirror. The equation is:\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]This equation is crucial as it helps determine where the image will form relative to the mirror. For convex mirrors, the focal length is always negative because the focus is behind the mirror.

• The equation provides a systematic way to find unknown values, like the image distance when the object distance and focal length are known.
• Using this equation, students can predict whether an image is real or virtual based on the signs of the image distance.

Mastering the usage of the mirror equation is essential for anyone studying optics, as it builds the foundation for understanding how mirrors and lenses work.

Virtual Image

A virtual image, unlike a real image, cannot be projected onto a screen. This is because the light rays that form a virtual image do not actually meet; they only appear to diverge from a common point behind the mirror. In the case of convex mirrors, all the images are virtual.

• Virtual images are always the same orientation as the object — they are upright.
• The size of a virtual image in a convex mirror is always smaller than the object.
• These images appear to be located behind the mirror surface.

Understanding virtual images is important for applications in various optical devices. Since these images are non-projectable, they are often used in applications where direct sight or reflection is the goal.

Magnification Calculation

Magnification is a measure of how much larger or smaller the image is compared to the object itself. The formula to determine magnification (\(m\)) in mirrors is:\[ m = \frac{-d_i}{d_o} \]Where \(-d_i\) is the image distance, and \(d_o\) is the object distance.

• A positive magnification indicates the image is upright compared to the object.
• A magnification of less than 1 means the image is smaller than the object, which is always true for convex mirrors.
• Conversely, a negative magnification would indicate an inverted image, although not relevant for convex mirrors since their images are upright.

Through magnification calculations, students can quantify how convex mirrors affect the size of images, which is instrumental in designing optical systems like cameras or security devices.