Geometrical Optics: Plane Mirrors, Concave Mirrors, and Convex Mirrors
Zsmart.id - Geometrical optics is a branch of physics that studies the behavior of light as it interacts with reflective and refractive surfaces. It uses the concept of light rays to explain phenomena such as image formation, reflection, and refraction. In this chapter, we will focus on reflection and the properties of three types of mirrors: plane mirrors, concave mirrors, and convex mirrors.
1. Plane Mirrors
1.1. Definition
A plane mirror is a flat, smooth surface that reflects light. It is the simplest type of mirror and is commonly used in everyday life.
1.2. Properties of Plane Mirrors
1.2.1. Image Formation:
- The image formed by a plane mirror is virtual, meaning it cannot be projected onto a screen.
- The image is upright and the same size as the object.
- The image is laterally inverted (left and right are swapped).
1.2.2. Distance Relationship:
- The distance of the image (\(d_i\)) from the mirror is equal to the distance of the object (\(d_o\)) from the mirror:
\[d_i = d_o\]
1.3. Magnification:
- The magnification (\(M\)) of a plane mirror is 1, indicating that the image size is the same as the object size:
\[M = \frac{h_i}{h_o} = 1\]
1.4. Applications of Plane Mirrors
- Household mirrors.
- Periscopes in submarines.
- Optical instruments like kaleidoscopes.
2. Concave Mirrors
2.1. Definition
A concave mirror is a spherical mirror with a reflecting surface that curves inward. It is also known as a converging mirror because it converges parallel rays of light to a focal point.
2.2. Key Terms
- Center of Curvature (\(C\)): The center of the sphere from which the mirror is cut.
- Radius of Curvature (\(R\)): The distance between the mirror's surface and the center of curvature.
- Focal Point (\(F\)): The point where parallel rays converge after reflection. The focal length (\(f\)) is half the radius of curvature:
\[f = \frac{R}{2}\]
2.3. Image Formation in Concave Mirrors
The type of image formed by a concave mirror depends on the position of the object relative to the mirror:
2.3.1. Object Beyond \(C\):
- Image is real, inverted, and smaller than the object.
2.3.2. Object at \(C\):
- Image is real, inverted, and the same size as the object.
2.3.3. Object Between \(C\) and \(F\):
- Image is real, inverted, and larger than the object.
2.3.4. Object at \(F\):
- No image is formed (rays are parallel).
2.3.5. Object Between \(F\) and the Mirror:
- Image is virtual, upright, and larger than the object.
2.4. Mirror Equation
The relationship between the object distance (\(d_o\)), image distance (\(d_i\)), and focal length (\(f\)) is given by:
\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]
2.5. Magnification
The magnification (\(M\)) is defined as the ratio of the image height (\(h_i\)) to the object height (\(h_o\)):
\[M = \frac{h_i}{h_o} = -\frac{d_i}{d_o}\]
- If \(M\) is negative, the image is inverted.
- If \(M\) is positive, the image is upright.
2.6. Applications of Concave Mirrors
- Shaving and makeup mirrors.
- Telescope mirrors.
- Solar concentrators.
3. Convex Mirrors
3.1. Definition
A convex mirror is a spherical mirror with a reflecting surface that curves outward. It is also known as a diverging mirror because it diverges parallel rays of light, making them appear to come from a virtual focal point.
3.2. Key Terms
3.2.1. Focal Point (\(F\)):
The point from which diverging rays appear to originate. The focal length (\(f\)) is negative for convex mirrors:
\[f = -\frac{R}{2}\]
3.2.2. Image Formation in Convex Mirrors
- The image formed by a convex mirror is always virtual, upright, and smaller than the object, regardless of the object's position.
3.3. Mirror Equation
The same mirror equation applies, but the focal length is negative:
\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]
3.4. Magnification
The magnification formula is the same as for concave mirrors:
\[M = \frac{h_i}{h_o} = -\frac{d_i}{d_o}\]
Since \(d_i\) is negative for convex mirrors, \(M\) is positive and less than 1.
3.5. Applications of Convex Mirrors
- Rear-view mirrors in vehicles.
- Security mirrors in stores.
- Wide-angle mirrors in parking lots.
4. Summary of Mirror Properties
5. Practice Questions
5.1. Plane Mirror Problem
An object is placed 12 cm in front of a plane mirror. Determine the image distance and describe the characteristics of the image.
Solution:
For a plane mirror, the image distance (\(d_i\)) is equal to the object distance (\(d_o\)):
\[d_i = d_o = 12 \, \text{cm} \]
The image formed by a plane mirror is:
- Virtual (cannot be projected onto a screen).
- Upright (same orientation as the object).
- Same size as the object.
- Laterally inverted (left and right are swapped).
Answer:
Image distance: \(d_i = 12 \, \text{cm}\).
Image characteristics: Virtual, upright, same size, laterally inverted.
2. Concave Mirror Problem
An object is placed 20 cm in front of a concave mirror with a focal length of 10 cm. Calculate the image distance and magnification. Describe the image.
Solution:
Given: Object distance, \(d_o = 20 \, \text{cm}\), Focal length, \(f = 10 \, \text{cm}\).
Use the mirror equation:
\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]
Substitute the values:
\[\frac{1}{10} = \frac{1}{20} + \frac{1}{d_i}\]
Solve for \(d_i\):
\[\frac{1}{d_i} = \frac{1}{10} - \frac{1}{20} = \frac{2 - 1}{20} = \frac{1}{20} \]
\[d_i = 20 \, \text{cm}\]
Calculate magnification (\(M\)):
\[M = -\frac{d_i}{d_o} = -\frac{20}{20} = -1\]
The negative magnification indicates the image is inverted.
Since \(d_i\) is positive, the image is real.
Answer:
Image distance: \(d_i = 20 \, \text{cm}\).
Magnification: \(M = -1\).
Image characteristics: Real, inverted, same size.
3. Convex Mirror Problem
An object is placed 30 cm in front of a convex mirror with a focal length of -15 cm. Find the image distance and magnification. What are the characteristics of the image?
Solution:
Given: Object distance, \(d_o = 30 \, \text{cm}\), Focal length, \(f = -15 \, \text{cm}\) (negative for convex mirrors).
Use the mirror equation:
\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]
Substitute the values:
\[\frac{1}{-15} = \frac{1}{30} + \frac{1}{d_i}\]
Solve for \(d_i\):
\[\frac{1}{d_i} = \frac{1}{-15} - \frac{1}{30} = -\frac{2}{30} - \frac{1}{30} = -\frac{3}{30} = -\frac{1}{10} \]
\[d_i = -10 \, \text{cm}\]
Calculate magnification (\(M\)):
\[M = -\frac{d_i}{d_o} = -\frac{-10}{30} = \frac{1}{3} \approx 0.33\]
The positive magnification indicates the image is upright.
Since \(d_i\) is negative, the image is virtual.
Answer:
Image distance: \(d_i = -10 \, \text{cm}\).
Magnification: \(M = 0.33\).
Image characteristics: Virtual, upright, smaller.
6. Problems
6.1. An object is placed 8 cm in front of a plane mirror. Determine:
1. The image distance.
2. The characteristics of the image.
6.2. An object is placed 15 cm in front of a concave mirror with a focal length of 10 cm. Calculate:
1. The image distance.
2. The magnification.
3. Draw and describe the image (real/virtual, upright/inverted, larger/smaller).
6.3. An object is placed 25 cm in front of a convex mirror with a focal length of -10 cm. Find:
1. The image distance.
2. The magnification.
3. Draw and describe the image (real/virtual, upright/inverted, larger/smaller).
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