Ray Optics Fundamentals
Spherical Mirrors
Master the physics of curved reflection. Explore how concave makeup mirrors enlarge details, how convex security mirrors expand fields of view, and how optics lab benches project real images. Drag objects dynamically to verify focal equations.
Spherical Mirrors Lab
Interact with magnifying makeup mirrors, security mirrors, optical lab benches, or side mirror safety view models.
Live Telemetry
Formula: 1/f = 1/u + 1/v- Mirror Type
- Concave
- Object Distance (u)
- -120.0 cm
- Image Distance (v)
- -60.0 cm
- Focal Length (f)
- -40.0 cm
- Radius (R)
- -80.0 cm
- Magnification (m)
- -0.50
Fundamentals of Spherical Mirrors
A spherical mirror is a reflecting surface whose curved shape forms a portion of a sphere. Based on which side of the sphere is polished to reflect light, they are categorized into two types:
1. Concave Mirror (Converging)
The inner surface curves inward like a cave. When parallel light rays strike a concave mirror, they reflect and converge (meet) at a single focal point in front of the mirror. Depending on how close the object is to the focal point, it can form both real, inverted images and virtual, upright, magnified images.
2. Convex Mirror (Diverging)
The outer surface curves outward. When parallel light rays strike a convex mirror, they reflect and diverge (spread apart). When traced backward behind the mirror, these reflected rays appear to originate from a virtual focus. A convex mirror always forms a virtual, upright, and diminished image, providing a much wider field of view.
Key Terms and Parameters
- Pole (P): The geometric center of the spherical mirror surface.
- Center of Curvature (C): The center of the sphere of which the mirror is a part.
- Radius of Curvature (R): The radius of the sphere. It is the linear distance from the pole to the center of curvature ($PC = R$).
- Principal Axis: The straight line passing through the pole and the center of curvature.
- Principal Focus (F): For a concave mirror, the point on the principal axis where reflected rays actually converge. For a convex mirror, the point from which diverging reflected rays appear to originate.
- Focal Length (f): The distance between the pole and the focus ($PF = f$). It is mathematically equal to half of the radius of curvature:
The Mirror Formula and Magnification
The relationship between the object distance (\(u\)), the image distance (\(v\)), and the focal length (\(f\)) is given by the Mirror Formula:
The ratio of the image height (\(h_i\)) to the object height (\(h_o\)) represents the Linear Magnification (\(m\)):
Cartesian Sign Convention
To solve numerical problems accurately, we must follow the Cartesian sign convention rules:
- All distances are measured starting from the mirror's pole (P) as the origin (0,0).
- Distances measured in the direction of the incident light (behind the mirror) are positive.
- Distances measured against the direction of the incident light (in front of the mirror) are negative.
- Heights measured vertically upward (above the principal axis) are positive.
- Heights measured vertically downward (below the principal axis) are negative.
[!IMPORTANT] Key Signs to Remember:
- Object distance (\(u\)) is always negative.
- Focal length (\(f\)) and Radius (\(R\)) are negative for a concave mirror, and positive for a convex mirror.
- For real images (formed in front of the mirror), \(v\) is negative and magnification \(m\) is negative.
- For virtual images (formed behind the mirror), \(v\) is positive and magnification \(m\) is positive.
Concave Mirror Image Formation Cases
The position and nature of the image formed by a concave mirror depend entirely on the object's position relative to the center of curvature (C) and the focus (F):
| Object Position | Image Position | Size of Image | Nature of Image |
|---|---|---|---|
| At Infinity | At Focus (F) | Highly Diminished (Point size) | Real and Inverted |
| Beyond C | Between F and C | Diminished | Real and Inverted |
| At C | At C | Same Size | Real and Inverted |
| Between C and F | Beyond C | Magnified | Real and Inverted |
| At Focus (F) | At Infinity | Highly Magnified | Real and Inverted |
| Between F and P | Behind Mirror | Magnified | Virtual and Upright |
Solved Examples
Example 1
A candle is placed at a distance of 30.0 cm in front of a concave mirror of focal length 15.0 cm. Find the position, magnification, and properties of the image formed.
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Final Answer:
Example 2
A toy car of height 5.0 cm is placed 20.0 cm in front of a convex mirror of radius of curvature 30.0 cm. Find the focal length, image position, size of the image, and magnification.
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Final Answer:
Example 3
A concave shaving mirror has a focal length of 20.0 cm. At what distance should a person place their face in order to see an upright virtual image that is magnified by 2.0 times?
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Final Answer:
Self-Check Questions
Question 1
Explain the sign conventions used for spherical mirrors. Why is the object distance always negative?
Show Answer & Explanation
Under the Cartesian sign convention: (1) All distances are measured from the pole (P) of the mirror, which serves as the origin (0,0). (2) Distances measured in the direction of the incident light are positive (behind the mirror), while distances opposite to the incident light are negative (in front of the mirror). (3) The object is always placed in front of the mirror where light originates. Thus, measuring from the pole to the object moves against the incident light direction, making the object distance (u) always negative.
Question 2
Why does a convex mirror always produce a virtual, diminished, and upright image, regardless of the object distance?
Show Answer & Explanation
A convex mirror curves outward, causing parallel incident rays to diverge away from the principal axis upon reflection. When these diverging reflected rays are extended backward behind the mirror, they always intersect between the pole (P) and the principal focus (F). Since the intersection occurs behind the mirror, no actual light rays meet, making the image virtual. Because the reflected rays always diverge more as they travel outward, the virtual intersection point is always compressed vertically, resulting in a diminished and upright image for all real object positions.
Question 3
What is the difference between a real image and a virtual image formed by spherical mirrors? Can they both be caught on a screen?
Show Answer & Explanation
A real image is formed when reflected light rays physically converge and intersect at a point in front of the mirror. It is always inverted and can be projected onto a screen placed at the intersection point. A virtual image is formed when reflected light rays diverge and only appear to intersect when traced backward behind the mirror. It is always upright and cannot be caught on a screen, as no light rays actually pass through the image point.
Question 4
Under what condition does a concave mirror form a virtual image? How does this make it useful as a makeup/shaving mirror?
Show Answer & Explanation
A concave mirror forms a virtual image only when the object is placed inside the focal length (between the pole P and focus F, i.e., u < f). In this case, the reflected rays diverge, and their virtual extensions behind the mirror create an upright, enlarged (magnified) image. This magnifying property makes it ideal as a makeup or shaving mirror, as it allows people to see a detailed, upright close-up of their face.