Reflection of Light

A. Laws of Reflection:

1. The incident ray, the reflected ray, and the normal to the reflecting surface at the point of incidence all lie in the same plane.
2. The angle of incidence is strictly equal to the angle of reflection: $$\mathbf{\angle i = \angle r}$$

B. Characteristics of Image Formed by a Plane Mirror:

• Virtual and Erect: Formed behind the mirror by diverging rays traced backward.
• Equal Size: Size of image equals size of object ($h_i = h_o \implies m = +1$).
• Equal Distance: Object distance equals image distance ($u = v$).
• Laterally Inverted: Left and right are reversed.

C. Images in Two Inclined Plane Mirrors:

Let two plane mirrors be inclined at an angle $\theta$:

• Calculate ratio: $m = \frac{360^\circ}{\theta}$.
• If $m$ is an even integer: $\mathbf{n = m - 1 = \frac{360^\circ}{\theta} - 1}$ (for all object positions).
• If $m$ is an odd integer:
  • Object on angle bisector (symmetric): $\mathbf{n = \frac{360^\circ}{\theta} - 1}$.
  • Object off bisector (asymmetric): $\mathbf{n = \frac{360^\circ}{\theta}}$.
• Parallel mirrors ($\theta = 0^\circ$): $\mathbf{n = \infty}$ (infinite series of images).

A. Terminology:

• Concave Mirror (Converging): Silvered on outer convex surface; reflecting surface curves inward.
• Convex Mirror (Diverging): Silvered on inner concave surface; reflecting surface bulges outward.
• Pole ($P$): Geometric center of the spherical reflecting surface.
• Center of Curvature ($C$): Center of the hollow sphere of which the mirror is a part.
• Radius of Curvature ($R$): Radius $PC = R$.
• Principal Focus ($F$): Point on principal axis where rays parallel to axis converge (concave) or appear to diverge from (convex).
• Focal Length ($f$): Distance $PF = f = \mathbf{\frac{R}{2}}$.

B. The Three Principal Construction Rays:

1. A ray parallel to the principal axis passes through (or appears to diverge from) the principal focus $F$ after reflection.
2. A ray passing through (or directed toward) the principal focus $F$ reflects parallel to the principal axis.
3. A ray passing through the center of curvature $C$ strikes normally ($\angle i = 0$) and reflects back along its own path.

A. New Cartesian Sign Convention:

• All distances are measured from the Pole ($P$) as the origin.
• Distances measured in the direction of incident light are positive ($+$); against incident light are negative ($-$).
• Heights above principal axis are positive ($+$); below are negative ($-$).
• Focal Lengths: Concave mirror: $\mathbf{f < 0}$ (negative); Convex mirror: $\mathbf{f > 0}$ (positive).
• Object distance $u$ is virtually always negative ($-$).

B. Mirror Formula and Magnification:

• If $m < 0$: Image is Real and Inverted.
• If $m > 0$: Image is Virtual and Erect.
• If $|m| > 1$: Magnified; if $|m| < 1$: Diminished; if $|m| = 1$: Same size.

Problem 1: An object of height $4\text{ cm}$ is placed at a distance of $30\text{ cm}$ in front of a concave mirror of focal length $20\text{ cm}$. Find: (i) The position of the image, (ii) The size and nature of the image.

Solution:

By sign convention: $u = -30\text{ cm}$, $f = -20\text{ cm}$ (concave), $h_o = +4\text{ cm}$.

1. Find image distance $v$ using the mirror formula:

The image is formed $60\text{ cm}$ in front of the mirror (on the same side as the object).

2. Find magnification and height of image:

Nature: Since $v$ and $m$ are negative, the image is Real, Inverted, and Magnified (twice the size).