Unit 14: Reflection at Curved Mirror

Introduction to Curved Mirrors

Curved mirrors are reflecting surfaces that are either concave (curving inward) or convex (curving outward). Unlike plane mirrors, they can produce images that are either real or virtual, magnified or diminished, and inverted or upright depending on the object's position relative to the mirror's focal point and centre of curvature. Understanding these mirrors is essential for applications ranging from telescopes and shaving mirrors to automobile rear‑view mirrors and solar concentrators.

1. Real and Virtual Images

Definitions

• Real Image: Formed when light rays actually converge at a point after reflection. It can be projected onto a screen and is always inverted relative to the object.
• Virtual Image: Formed when light rays appear to diverge from a point behind the mirror. It cannot be projected onto a screen and is upright (erect) relative to the object.

Behaviour of Concave Mirrors

A concave mirror can produce both real and virtual images depending on where the object is placed:

1. Object beyond the centre of curvature (C): Real, inverted, diminished image formed between C and focus (F).
2. Object at C: Real, inverted, same‑size image formed at C.
3. Object between C and F: Real, inverted, magnified image formed beyond C.
4. Object at F: No image formed (rays emerge parallel).
5. Object between F and the pole (P): Virtual, erect, magnified image formed behind the mirror.

Behaviour of Convex Mirrors

A convex mirror always forms a virtual, erect, and diminished image irrespective of the object's position. The reflected rays diverge, and their extensions appear to meet behind the mirror.

2. Mirror Formula and Sign Convention

Derivation of the Mirror Formula

Consider a concave mirror with pole P, focus F, and centre of curvature C. Let an object AB be placed at a distance u from the pole, and its image A'B' be formed at a distance v. Using the geometry of similar triangles formed by the incident and reflected rays, we obtain:

\(\frac{AB}{A'B'} = \frac{PB}{PB'} = \frac{PC}{PF}\)

Substituting the distances with sign conventions (discussed later) and simplifying yields the mirror formula:

1/f = 1/v + 1/u

The same expression holds for convex mirrors when the appropriate signs are applied.

Sign Convention (New Cartesian Sign Convention)

1. All distances are measured from the pole (P) of the mirror.
2. Distances measured in the direction of incident light (i.e., from the object towards the mirror) are taken as positive.
3. Distances measured opposite to the direction of incident light are taken as negative.
4. Heights measured upward (perpendicular to the principal axis) are positive; heights measured downward are negative.
5. For a concave mirror, the focal length (f) and radius of curvature (R) are negative because the focus lies in front of the mirror (against the incident direction). For a convex mirror, f and R are positive as the focus lies behind the mirror.

Applying this convention ensures that the mirror formula works uniformly for both mirror types.

Magnification

Linear magnification (m) relates the size of the image to the size of the object:

m = -v/u = h_i / h_o

where:

• h_i = height of the image
• h_o = height of the object
• The negative sign indicates that a real image (formed by actual convergence) is inverted relative to the object.

If |m| > 1, the image is magnified; if |m| < 1, it is diminished; if |m| = 1, the image is the same size as the object. The sign of m also tells us about orientation: negative → inverted, positive → erect.

3. Ray Diagram Construction

Ray diagrams are a graphical method to locate the image formed by a spherical mirror. Three principal rays are typically used:

1. Ray parallel to the principal axis: After reflection, it passes through the focus (F) for a concave mirror, or appears to diverge from the focus for a convex mirror.
2. Ray passing through the centre of curvature (C): It strikes the mirror normally and reflects back along the same path.
3. Ray passing through the focus (F): After reflection, it emerges parallel to the principal axis.

For a convex mirror, the second and third rays are considered as if they appear to come from C and F located behind the mirror.

Step‑by‑Step Procedure

1. Draw the principal axis (a horizontal line). Mark the pole (P) at the origin.
2. Locate the focus (F) and centre of curvature (C) at distances f and 2f from P, respectively, on the appropriate side (in front for concave, behind for convex).
3. Draw the object AB as a vertical arrow perpendicular to the principal axis at the given object distance u.
4. Trace the three principal rays from the tip of the object (point A) as described above.
5. For concave mirrors, extend the reflected rays forward until they meet; for convex mirrors, extend the reflected rays backward (behind the mirror) until they appear to meet.
6. The point of intersection (or apparent intersection) gives the tip of the image A'. Drop a perpendicular to the principal axis to locate the base B'.
7. Measure distances PV (image distance) and heights to determine nature, orientation, and size.

Examples of Ray Diagrams

Concave Mirror – Object beyond C

Ray Diagram Description: Draw the principal axis, mark pole (P), focus (F), and centre of curvature (C). Place the object beyond C. Draw the three rays as described; they converge at a point between C and F, giving a real, inverted, diminished image.

Concave Mirror – Object between F and P

Ray Diagram Description: Place the object between F and P. The reflected rays diverge; their extensions meet behind the mirror, yielding a virtual, erect, magnified image.

Convex Mirror – Any Object Position

Ray Diagram Description: For any object placed in front of a convex mirror, the reflected rays diverge. Extending these rays backward shows they appear to originate from a point behind the mirror, giving a virtual, erect, diminished image.

4. Problem Solving Using the Mirror Formula

Example 1: Concave Mirror

Question: An object of height 4 cm is placed 30 cm in front of a concave mirror of focal length 12 cm. Find the position, nature, and size of the image.

Solution:

1. Given: u = -30 cm (object distance negative), f = -12 cm (concave mirror focal length negative).
2. Apply mirror formula: 1/f = 1/v + 1/u → 1/(-12) = 1/v + 1/(-30).
3. Solve for v: 1/v = 1/(-12) - 1/(-30) = -1/12 + 1/30 = (-5 + 2)/60 = -3/60 = -1/20. Thus, v = -20 cm.
4. Negative v indicates the image is formed in front of the mirror (real).
5. Magnification: m = -v/u = -(-20)/(-30) = 20/(-30) = -2/3 ≈ -0.67.
6. Image height: h_i = m × h_o = (-0.67) × 4 cm ≈ -2.7 cm. The negative sign shows the image is inverted.
7. Conclusion: The image is real, inverted, diminished, located 20 cm in front of the mirror.

Example 2: Convex Mirror

Question: An object is placed 15 cm in front of a convex mirror of focal length 10 cm. Determine the image characteristics.

Solution:

1. Given: u = -15 cm, for a convex mirror f = +10 cm (focal length positive).
2. Mirror formula: 1/10 = 1/v + 1/(-15) → 1/v = 1/10 + 1/15 = (3 + 2)/30 = 5/30 = 1/6.
3. Thus, v = +6 cm.
4. Positive v indicates the image is formed behind the mirror (virtual).
5. Magnification: m = -v/u = -(6)/(-15) = 6/15 = 0.4.
6. Image height: h_i = m × h_o (if object height is, say, 5 cm, then h_i = 0.4 × 5 = 2 cm). Positive magnification indicates erect image.
7. Conclusion: The image is virtual, erect, diminished, located 6 cm behind the mirror.

Example 3: Object at the Focus of a Concave Mirror

Question: A candle of height 3 cm is placed exactly at the focus of a concave mirror of focal length 8 cm. What is the nature of the image formed?

Solution:

1. Given: u = -8 cm, f = -8 cm.
2. Mirror formula: 1/(-8) = 1/v + 1/(-8) → 1/v = 0 → v → ∞.
3. The image is formed at infinity; the reflected rays are parallel to the principal axis.
4. Thus, no finite image is formed; the mirror acts as a collimator.

5. Applications of Curved Mirrors

• Concave Mirrors: Used in reflecting telescopes, shaving/makeup mirrors (magnified virtual image), headlights (to produce a parallel beam), solar furnaces (to concentrate sunlight), and ophthalmoscopes.
• Convex Mirrors: Employed as rear‑view mirrors in vehicles (wide field of view), security mirrors in shops, and as safety mirrors at road bends.

6. Summary Table

7. Key Points to Remember

• The mirror formula 1/f = 1/v + 1/u is valid for all spherical mirrors when the sign convention is applied correctly.
• Focal length is half the radius of curvature: f = R/2 (sign follows mirror type).
• Real images can be caught on a screen; virtual images cannot.
• Magnification sign tells you about orientation: negative → inverted, positive → erect.
• Drawing accurate ray diagrams helps visualize image formation and verify results obtained from the formula.
• Spherical mirrors suffer from spherical aberration; parabolic mirrors are used when a perfect focus is required.

8. Practice Questions

1. An object 2 cm tall is placed 25 cm in front of a concave mirror of focal length 10 cm. Calculate the image distance, magnification, and image height. State whether the image is real or virtual.
2. A convex mirror produces an image that is 0.3 times the size of the object and located 6 cm behind the mirror. Find the focal length of the mirror and the object distance.
3. Draw a ray diagram for a concave mirror when the object is placed exactly at the centre of curvature. What are the characteristics of the image formed?
4. A concave mirror forms a real, inverted image that is twice the size of the object. If the object distance is 18 cm, find the focal length of the mirror.
5. Explain why a convex mirror is preferred as a rear‑view mirror in vehicles, using the concepts of image size and field of view.