Reflection by Spherical Mirrors

TL;DR

Spherical mirrors are curved mirrors that can focus or diverge light, forming images in predictable ways. Concave mirrors curve inward and can form both real and virtual images, while convex mirrors curve outward and always form virtual, diminished images. We use ray diagrams and the mirror formula to locate and characterize these images.

1. The Mental Model

Imagine a tiny part of a giant shiny sphere. If you look into the inside of that sphere, that's a concave mirror. If you look at the outside, that's a convex mirror. These curves change how light bounces off them, either bringing rays together or spreading them out.

2. The Core Material

When light hits a spherical mirror, it reflects in a predictable way. The shape of the mirror dictates how the light rays behave, which in turn determines the characteristics of the image formed (e.g., its size, orientation, and whether it's real or virtual).

Mirror Terminology

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Before diving into reflection, let's define some key terms:
* Pole (P): The geometric center of the mirror's reflecting surface.
* Center of Curvature (C): The center of the sphere from which the mirror is a part.
* Radius of Curvature (R): The distance between the pole (P) and the center of curvature (C). $R = PC$.
* Principal Axis: The straight line passing through the pole (P) and the center of curvature (C).
* Principal Focus (F): A point on the principal axis where rays parallel to the principal axis converge after reflection (concave mirror) or appear to diverge from after reflection (convex mirror).
* Focal Length (f): The distance between the pole (P) and the principal focus (F). For spherical mirrors, $f = R/2$.

Types of Spherical Mirrors

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Concave Mirror

A concave mirror is a converging mirror. It curves inward, like the inside of a spoon. When parallel rays of light strike a concave mirror, they reflect and converge at its principal focus (F).

Convex Mirror

A convex mirror is a diverging mirror. It curves outward, like the back of a spoon. When parallel rays of light strike a convex mirror, they reflect and appear to diverge from its principal focus (F) behind the mirror.

Ray Tracing for Image Formation

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To find where an image forms and what it looks like, we typically trace at least two special rays:

1. Ray parallel to the principal axis: After reflection, it passes through the principal focus (F) for a concave mirror, or appears to diverge from F for a convex mirror.
2. Ray passing through the principal focus (F): After reflection, it becomes parallel to the principal axis. (For convex, a ray directed towards F becomes parallel).
3. Ray passing through the center of curvature (C): After reflection, it retraces its path (reflects back along the same line) because it strikes the mirror perpendicularly.
4. Ray striking the pole (P): It reflects symmetrically about the principal axis.

The point where the reflected rays intersect (or appear to intersect) is where the image is formed.

Image Characteristics

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Images can be:
* Real: Formed by actual intersection of reflected rays; can be projected onto a screen. Always inverted.
* Virtual: Formed by apparent intersection of reflected rays; cannot be projected. Always erect.
* Erect: Upright, in the same orientation as the object.
* Inverted: Upside down, opposite orientation to the object.
* Magnified: Larger than the object.
* Diminished: Smaller than the object.
* Same size: Same size as the object.

Here's a summary of image formation for different mirror types and object positions:

graph TD
    A["Object Position"] --> B{Mirror Type};
    B --> C["Concave Mirror"];
    B --> D["Convex Mirror"];

    C --> C1{"Object at Infinity"};
    C --> C2{"Object Beyond C"};
    C --> C3{"Object at C"};
    C --> C4{"Object Between C and F"};
    C --> C5{"Object at F"};
    C --> C6{"Object Between F and P"};

    C1 --> C1R["Image at F, Real, Inverted, Diminished"];
    C2 --> C2R["Image Between F and C, Real, Inverted, Diminished"];
    C3 --> C3R["Image at C, Real, Inverted, Same Size"];
    C4 --> C4R["Image Beyond C, Real, Inverted, Magnified"];
    C5 --> C5R["Image at Infinity, Real, Inverted, Highly Magnified"];
    C6 --> C6R["Image Behind Mirror, Virtual, Erect, Magnified"];

    D --> D1{"Anywhere (except Infinity)"};
    D1 --> D1R["Image Between P and F (behind mirror), Virtual, Erect, Diminished"];

The Mirror Formula

This formula relates the object distance ($u$), image distance ($v$), and focal length ($f$) of a spherical mirror:

$\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$

Where:
* $u$ = object distance from the pole (P)
* $v$ = image distance from the pole (P)
* $f$ = focal length of the mirror ($f = R/2$)

Sign Convention (New Cartesian Sign Convention):
* All distances are measured from the pole (P).
* Distances measured in the direction of incident light are taken as positive.
* Distances measured opposite to the direction of incident light are taken as negative.
* Heights measured upward and perpendicular to the principal axis are positive.
* Heights measured downward and perpendicular to the principal axis are negative.

For spherical mirrors:
* Focal length ($f$) of a concave mirror is negative.
* Focal length ($f$) of a convex mirror is positive.
* Object distance ($u$) is almost always negative (object usually placed in front of mirror).
* Real image distance ($v$) is negative (formed in front of mirror).
* Virtual image distance ($v$) is positive (formed behind mirror).

Magnification

Magnification ($m$) describes how much larger or smaller the image is compared to the object, and whether it's erect or inverted.

$m = \frac{\text{Height of image} (h_i)}{\text{Height of object} (h_o)} = -\frac{v}{u}$

• If $m > 0$ (positive), the image is erect (and virtual).
• If $m < 0$ (negative), the image is inverted (and real).
• If $|m| > 1$, the image is magnified.
• If $|m| < 1$, the image is diminished.
• If $|m| = 1$, the image is the same size as the object.

3. Worked Example

Problem: An object 4.0 cm high is placed at a distance of 25.0 cm in front of a concave mirror of focal length 15.0 cm. Find the position, nature, and size of the image.

Given:
* Object height, $h_o = +4.0 \text{ cm}$
* Object distance, $u = -25.0 \text{ cm}$ (always negative as per convention)
* Focal length, $f = -15.0 \text{ cm}$ (concave mirror, so negative)

To find: $v$, $h_i$, nature of image.

1. Find image distance ($v$) using the mirror formula:
$\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$
$\frac{1}{-15} = \frac{1}{v} + \frac{1}{-25}$
$\frac{1}{v} = \frac{1}{-15} - \frac{1}{-25}$
$\frac{1}{v} = \frac{1}{25} - \frac{1}{15}$
$\frac{1}{v} = \frac{3 - 5}{75}$
$\frac{1}{v} = \frac{-2}{75}$
$v = -\frac{75}{2} = -37.5 \text{ cm}$

Interpretation: The image is formed at 37.5 cm in front of the mirror. Since $v$ is negative, the image is real.

2. Find image height ($h_i$) using magnification formula:
$m = \frac{h_i}{h_o} = -\frac{v}{u}$
$h_i = -h_o \frac{v}{u}$
$h_i = -(4.0 \text{ cm}) \frac{(-37.5 \text{ cm})}{(-25.0 \text{ cm})}$
$h_i = -(4.0 \text{ cm}) \times (1.5)$
$h_i = -6.0 \text{ cm}$

Interpretation: The image is 6.0 cm high. Since $h_i$ is negative, the image is inverted. Since $|h_i| > |h_o|$, the image is magnified.

Summary: The image is formed 37.5 cm in front of the mirror, it is real, inverted, and magnified (6.0 cm tall).

4. Key Takeaways

• Concave mirrors converge light rays and can form both real/inverted and virtual/erect images depending on object position.
• Convex mirrors always diverge light rays and always form virtual, erect, and diminished images.
• The focal length ($f$) is half the radius of curvature ($R$) for spherical mirrors ($f = R/2$).
• The mirror formula $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$ mathematically relates object distance, image distance, and focal length.
• Magnification $m = -v/u$ tells you the image's size relative to the object and its orientation.
• Always use the New Cartesian Sign Convention consistently for $u, v, f, h_o, h_i$ to get correct results.
• Real images are always inverted and form in front of the mirror; virtual images are always erect and form behind the mirror.

Common Mistakes to Avoid:
- Forgetting to apply the correct sign convention for $f$, $u$, and $v$. Concave $f$ is negative, convex $f$ is positive.
- Mixing up concave and convex mirror characteristics (e.g., thinking a convex mirror can form a real image).
- Not correctly interpreting the sign of $v$ or $m$ to determine if an image is real/virtual or erect/inverted.
- Doing algebraic calculations incorrectly, especially with fractions and negative signs in the mirror formula.