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6.1.7. Spherical Mirrors

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Curved Mirrors That Form Images

A spherical mirror is a mirror whose reflecting surface is part of a sphere. Unlike a plane mirror, which is flat, a spherical mirror is curved, so it can make light rays come together or spread apart. Because of this, spherical mirrors can form images that may be larger, smaller, upright, or inverted.

There are two main types of spherical mirrors. A concave mirror curves inward, like the inside of a spoon. A convex mirror curves outward, like the back of a spoon. Their different shapes make light behave differently after reflection.

Basic Parts of a Spherical Mirror

To describe image formation with spherical mirrors, several special points are used. The pole is the center point of the mirror surface. The principal axis is the straight line through the pole and the center of curvature. The center of curvature is the center of the sphere of which the mirror is a part. The radius of curvature is the distance from the pole to the center of curvature.

Another very important point is the focal point, or focus. For rays near the principal axis, parallel rays reflect so that they pass through the focus for a concave mirror, or appear to come from the focus for a convex mirror.

The focal length is the distance from the pole to the focus. For a spherical mirror, the focal length is half the radius of curvature:

$$
f = \frac{R}{2}
$$

For spherical mirrors, the focal length and radius of curvature are related by
$$
f = \frac{R}{2}
$$
This is one of the most important geometric facts about spherical mirrors.

Concave and convex spherical mirrors

Concave and Convex Mirrors

A concave mirror is also called a converging mirror because it reflects parallel rays inward toward a point. This focusing effect allows a concave mirror to form real images when the object is far enough from the mirror. It can also form virtual images when the object is very close.

A convex mirror is called a diverging mirror because reflected parallel rays spread out. A convex mirror always forms a virtual, upright, and reduced image. This is why convex mirrors are often used for rear-view and security mirrors, because they provide a wide field of view.

Principal Rays for Image Construction

To locate an image formed by a spherical mirror, a few standard rays are especially useful. These are called principal rays. Their exact use belongs to ray diagrams, but the main mirror behaviors are important here.

For a concave mirror, a ray parallel to the principal axis reflects through the focus. A ray passing through the focus reflects parallel to the principal axis. A ray passing through the center of curvature reflects back on itself, because it strikes the mirror normally.

For a convex mirror, a ray parallel to the principal axis reflects as if it came from the focus behind the mirror. A ray directed toward the focus reflects parallel to the axis. A ray directed toward the center of curvature reflects back along its path.

Principal ray rules for spherical mirrors:
A ray parallel to the principal axis reflects through the focus, for a concave mirror.
A ray parallel to the principal axis reflects as if it came from the focus, for a convex mirror.
A ray through the center of curvature reflects back on itself.

Parallel rays reflecting from a concave mirror
Parallel rays reflecting from a convex mirror

Image Formation by Concave Mirrors

A concave mirror can produce different kinds of images depending on the object position.

If the object is beyond the center of curvature, the image is real, inverted, and smaller than the object. If the object is at the center of curvature, the image is real, inverted, and the same size. If the object is between the center of curvature and the focus, the image is real, inverted, and magnified. If the object is at the focus, reflected rays are parallel, so the image is formed very far away. If the object is between the focus and the mirror, the image becomes virtual, upright, and magnified.

These cases are very important because they explain how shaving mirrors and makeup mirrors enlarge nearby objects, while telescopes and other instruments can use concave mirrors to form real images.

Image Formation by Convex Mirrors

A convex mirror behaves more simply. No matter where the object is placed in front of it, the image is always virtual, upright, and smaller than the object. The image appears behind the mirror, between the pole and the focus.

This consistent behavior makes convex mirrors useful when a wide field of view matters more than magnification.

Mirror Equation

The positions of object and image are related by the mirror equation. If $d_o$ is the object distance, $d_i$ is the image distance, and $f$ is the focal length, then

$$
\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}
$$

This equation works for both concave and convex mirrors, as long as a consistent sign convention is used.

Mirror equation:
$$
\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}
$$
You must use a consistent sign convention when applying it.

Magnification

The size and orientation of the image are described by the magnification:

$$
m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}
$$

Here, $h_o$ is the object height and $h_i$ is the image height. If $m$ is positive, the image is upright. If $m$ is negative, the image is inverted. If $|m| > 1$, the image is magnified. If $|m| < 1$, the image is reduced.

Magnification formula:
$$
m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}
$$
Positive magnification means upright image.
Negative magnification means inverted image.

Sign Convention

A common sign convention is used in geometrical optics. Object distances are usually taken as positive when the object is in front of the mirror. For mirrors, a real image in front of the mirror has positive image distance, while a virtual image behind the mirror has negative image distance. A concave mirror has positive focal length, and a convex mirror has negative focal length.

This gives a practical way to interpret the equations.

QuantityConcave mirrorConvex mirror
Focal length $f$PositiveNegative
Radius of curvature $R$PositiveNegative
Real image distance $d_i$PositiveNot formed for real objects
Virtual image distance $d_i$Negative, if object inside focusNegative

Summary of Image Characteristics

The overall behavior of spherical mirrors can be summarized clearly.

Mirror typeImage typeOrientationSize
Concave, object beyond focusRealInvertedMay be larger, equal, or smaller
Concave, object inside focusVirtualUprightLarger
ConvexVirtualUprightSmaller

Example

Suppose a concave mirror has focal length $f = 10 \text{ cm}$ and an object is placed at $d_o = 30 \text{ cm}$.

Using the mirror equation,

$$
\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}
$$

$$
\frac{1}{10} = \frac{1}{30} + \frac{1}{d_i}
$$

$$
\frac{1}{d_i} = \frac{1}{10} - \frac{1}{30} = \frac{2}{30} = \frac{1}{15}
$$

So,

$$
d_i = 15 \text{ cm}
$$

The image is real because $d_i$ is positive.

Now find magnification:

$$
m = -\frac{d_i}{d_o} = -\frac{15}{30} = -0.5
$$

So the image is inverted and half the size of the object.

Spherical Mirror Approximation

The mirror rules given here work best for rays close to the principal axis. This is called the paraxial approximation. For rays far from the axis, spherical mirrors do not bring all rays to exactly the same point. This effect is called spherical aberration. At the beginner level, it is enough to know that the simple mirror equation assumes the ideal paraxial case.

Everyday Uses

Concave mirrors are used when magnification or focusing is needed. Examples include makeup mirrors, shaving mirrors, reflecting telescopes, and some headlights.

Convex mirrors are used when a wider view is needed. Examples include vehicle side mirrors, hallway safety mirrors, and store security mirrors.

Final Ideas

Spherical mirrors are curved reflecting surfaces that can change the direction of light in useful ways. Concave mirrors can converge light and form either real or virtual images, depending on where the object is placed. Convex mirrors always diverge light and form virtual, upright, reduced images. Their behavior is described by the focal length relation, the mirror equation, and the magnification formula. These ideas are the foundation for understanding how curved mirrors produce images.

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