Reflection at Spherical Surfaces

Law of Reflection at Spherical Surfaces

To begin our discussion of reflection at spherical surfaces, let's review the law of reflection. The law of reflection says that the reflected angle of a light beam incident on a smooth surface between two media will be equal to the incident angle with respect to the normal (perpendicular) of the surface where it hits. The incident ray, reflected ray, and normal to the surface all lie in the same plane.

This law is described by the equation:

\[\theta_i=\theta_r.\]

Fig. 2 - Light incident on a spherical surface obeys the law of reflection.

The law of reflection applies to any surface, including spherical surfaces. According to this law, when the light from an object hits a surface, the light beam reflects at the same angle as the incident angle on the other side of the normal. For a spherical surface, the normal to it points in a different direction at every point along the surface. This is a consequence of the curvature of the spherical surface. The above image shows two light beams hitting the spherical surface at different locations, causing the reflected light rays to be reflected at different angles. Notice how the normal is different at those points but in both cases, the reflected angles are the same as the incident angles on the other side of their respective normal.

Formula for Reflection at a Spherical Surface

The law of reflection is what allows us to see an image of ourselves when we look in a mirror. The location at which the reflected light rays come to a point, or can be traced to a point, is where the image is located. This is called the image location.

An important property of these images is that a real image can be projected onto a screen but a virtual image cannot.

The way light reflects off of a surface according to the law of reflection determines the size and location of the image produced. A spherical mirror has a curved surface whose center of curvature is the radius of the circle that the mirror would make, \(R.\) When light rays that are parallel to the axis of symmetry hit a spherical mirror, they get reflected as if they were coming from a point behind the mirror. This point is called the focus or focal point, of the mirror and it is located halfway between the center of the imaginary circle the mirror would make. So, the distance to the focus is given by \(f=\frac{R}{2}.\) We calculate the image location using the mirror equation:

\[\frac{1}{s_o}+\frac{1}{s_i}=\frac{1}{f},\]

where \(s_o\) is the object location, \(s_i\) is the image location, and \(f\) is the distance to the focus all in the same units of lengths, for example in meters, \(\mathrm{m}.\)

We determine the size of the image and whether it is upright or inverted using the magnification equation:

\[M=\frac{h_i}{h_o}=-\frac{s_i}{s_o},\]

where \(h_i\) and \(h_o\) are the heights of the image and object, respectively, both in the same units of lengths, for example in meters, \(\mathrm{m}.\) If the magnification of an object is less than one, the image is smaller than the object, and if it is greater than one, the image is larger than the object. A negative magnification indicates that the image is inverted, while a positive magnification indicates that it's upright.

Reflection on Different Types of Spherical Surfaces

We will discuss reflection off of the two types of spherical mirrors: concave and convex. The reflective surface of a concave mirror curves inward, causing the reflected light to converge to the focus. On the other hand, the reflective surface of a convex mirror curves outward, causing the reflected light to diverge. The diverging rays reflected off of a convex mirror can be traced back to the virtual focus. The image below shows how light rays reflect off of concave and convex mirrors on the left and right, respectively.

Fig. 3 - Light rays parallel to the principal axis converge after reflection off of a concave mirror and diverge after reflection off of a convex mirror.

We can use ray diagrams to help us determine the size and location of the image formed from a concave or convex mirror. A ray diagram traces incident rays from the object to the reflective surface, and the direction the rays go after reflection.

Reflection of Light at Concave Mirror

The image created by reflection of light from a concave mirror can be real or virtual, depending on the location of the object with respect to the mirror. The image of an object located beyond the center of curvature of the mirror will be real, inverted, and reduced in size, as we can see by tracing the rays from the object. This is shown in the figure below.

Fig. 4 - Image formed from an object far from the focus of a concave mirror.

The smallest image possible is formed when an object is infinitely far from the concave mirror; in this case, the real, inverted image is located at the focus. If we move the object so that it is located at the center of curvature, the image will still be real and inverted, but will also now be the same size as the object. Moving the object so that it is between the center of curvature and the focus will increase the size of the real, inverted image, making it larger than the object, as shown below.

Fig. 5 - Image formed from an object close to the focus of a concave mirror.

The largest real, inverted image possible is achieved when the object is located at the focus. The image, in this case, will be formed at infinity. Moving the object in front of the focus will finally produce a virtual, upright image that is located behind the surface of the mirror, shown below.

Fig. 6 - Image formed from an object in front of the focus of a concave mirror.

The table below summarizes the images formed at different object locations for a concave mirror.

Table 1. - Image formation for a concave mirror.

Reflection of Light at a Convex Mirror

A convex mirror always produces upright, virtual images. For an object that is infinitely far away, the image location is at the focus and is the smallest image possible. An object placed at any distance between infinity and the mirror's surface will produce an image that is still reduced in size and is located between the mirror and the focus.

Fig. 7 - Image formed from an object in front of the focus of a convex mirror.

The table below summarizes the images formed at different locations for a convex mirror.

Table 2 - Image formation for a convex mirror.

Examples of Reflection at Spherical Surfaces

Let's do a couple of examples to get some practice with reflection in spherical surfaces!