Reflection of Light¶

by Plane and Spherical Mirrors¶

Introduction¶

When light shines onto the surface of an object, some of the light is reflected, while the rest is either absorbed or transmitted. We can imagine the light consisting of many narrow beams that travel in straight-line paths called rays. The light rays that strike the surface are called the incident rays. The light rays that reflect off the surface are called the reflected rays. This model of light is called the ray model, and it can be used to describe many aspects of light, including the reflection and formation of images by plane and spherical mirrors.

Law of Reflection¶

To measure the angles of the incident and reflected rays, we first draw the normal, which is the line perpendicular to the surface. The angle of incidence, \(\theta_{i}\), is the angle between the incident ray and the normal. Likewise, the angle of reflection, \(\theta_{r}\), is the angle between the reflected ray and the normal. The incident ray, the reflected ray, and the normal to the reflecting surface all lie within the same plane. This is shown in the figure above. Notice that the angle of reflection is equal to the angle of incidence. This is known as the law of reflection, and it can be expressed by the following equation:

\[\theta_{r} = \theta_{i}\]

Use the slider below to change the angle of incidence. This changes the angle between the incident ray and the normal. Notice how the angle of reflection also changes when the slider is moved.

interactive_plot = widgets.interactive(f, Angle=widgets.IntSlider(value=45,min=0,max=90,step=15,continuous_update=False))
output = interactive_plot.children[-1]
output.layout.height = '280px'
interactive_plot

Question: When the angle of incidence increases, what happens to the angle of reflection?

#Assign each multiple choice to these four variables
#Option_1 contains the answer
option_1 = "The angle of reflection increases." 
option_2 = "The angle of reflection decreases."
option_3 = "The angle of reflection remains constant."
option_4 = "The angle of reflection equals zero."

multiple_choice(option_1, option_2, option_3, option_4)

(A) The angle of reflection decreases.
(B) The angle of reflection remains constant.
(C) The angle of reflection equals zero.
(D) The angle of reflection increases.

Specular and Diffuse Reflections¶

For a very smooth surface, such as a mirror, almost all of the light is reflected to produce a specular reflection. In a specular reflection, the reflected light rays are parallel to one another and point in the same direction. This allows specular reflections to form images. If the surface is not very smooth, then the light may bounce off of the surface in various directions. This produces a diffuse reflection. Diffuse reflections cannot form images.

../../../_images/specular_diffuse_reflections.svg

Note: The law of reflection still applies to diffuse reflections, even though the reflected rays are pointing in various directions. We can imagine that each small section of the rough surface is like a flat plane orientated differently than the sections around it. Since each of these sections is orientated differently, the angle of incidence is different at each section. This causes the reflected rays to scatter.

Question: Which of the following is an example of a specular reflection?

#Assign each multiple choice to these four variables
#Option_1 contains the answer
option_1 = "The reflection off a clean window." 
option_2 = "The reflection off a wooden deck."
option_3 = "The reflection off a carpet floor."
option_4 = "The reflection off a table cloth."

multiple_choice(option_1, option_2, option_3, option_4)

(A) The reflection off a wooden deck.
(B) The reflection off a clean window.
(C) The reflection off a carpet floor.
(D) The reflection off a table cloth.

Question: Which of the following is an example of a diffuse reflection?

#Assign each multiple choice to these four variables
#Option_1 contains the answer
option_1 = "The reflection off a concrete sidewalk." 
option_2 = "The reflection off a mirror."
option_3 = "The reflection off the surface of a still lake."
option_4 = "The reflection off a polished sheet of metal."

multiple_choice(option_1, option_2, option_3, option_4)

(A) The reflection off the surface of a still lake.
(B) The reflection off a polished sheet of metal.
(C) The reflection off a concrete sidewalk.
(D) The reflection off a mirror.

Image Formation by Plane Mirrors¶

A plane mirror is simply a mirror made from a flat (or planar) surface. These types of mirrors are commonly found in bedroom or bathroom fixtures. When an object is reflected in a plane mirror, the image of the object appears to be located behind the mirror. This is because our brains interpret the reflected light rays entering our eyes as having travelled in straight-line paths. The light rays entering our eyes simply do not contain enough information for our brains to differentiate between a straight-line path and a path that changed direction due to a reflection.

../../../_images/plane_mirror_reflection.svg

Notice in the figure above that the light rays do not actually converge at the location where the image appears to be formed (behind the mirror). Since the light rays do not actually go behind the mirror, they are represented as projections using dashed lines. If a film were placed at the image location behind the mirror, it would not be able to capture the image. As a result, this type of image is called a virtual image.

For objects reflected in a plane mirror, the distance of the image from the mirror, \(d_{i}\), is always equal to the distance of the object from the mirror, \(d_{o}\). If the object is moved toward the mirror, the image of the object will also move toward the mirror such that the object and the image are always equidistant from the surface of the mirror.

Use the slider below to change the object distance. Notice how the image distance also changes when the slider is moved.

interactive_plot = widgets.interactive(f,Distance=widgets.IntSlider(value=30,min=10,max=50,step=10,continuous_update=False))
output = interactive_plot.children[-1]
output.layout.height = '280px'
interactive_plot

#Print question
distance = round(random.uniform(5,10),1)
print("If you stand " + str(distance) + " m in front of a plane mirror, how many metres behind the mirror is your virtual image?")

#Answer calculation
answer = distance
    
#Assign each multiple choice to these four variables
#Option_1 contains the answer
option_1 = str(round((answer),1)) + " m" 
option_2 = str(round((answer * 2),1)) + " m"
option_3 = str(round((answer / 2),1)) + " m"
option_4 = str(round((answer / 4),1)) + " m"

multiple_choice(option_1, option_2, option_3, option_4)

If you stand 6.0 m in front of a plane mirror, how many metres behind the mirror is your virtual image?
(A) 12.0 m
(B) 1.5 m
(C) 6.0 m
(D) 3.0 m

#Print question
distance = round(random.uniform(5,10),1)
print("If you stand " + str(distance) + " m in front of a plane mirror, how many metres will separate you from your virtual image?")

#Answer calculation
answer = (distance * 2)
    
#Assign each multiple choice to these four variables
#Option_1 contains the answer
option_1 = str(round((answer),1)) + " m" 
option_2 = str(round((answer * 2),1)) + " m"
option_3 = str(round((answer / 2),1)) + " m"
option_4 = str(round((answer / 4),1)) + " m"

multiple_choice(option_1, option_2, option_3, option_4)

If you stand 6.8 m in front of a plane mirror, how many metres will separate you from your virtual image?
(A) 13.6 m
(B) 6.8 m
(C) 27.2 m
(D) 3.4 m

Spherical Mirrors¶

Two common types of curved mirror are formed from a section of a sphere. If the reflection takes place on the inside of the spherical section, then the mirror is called a concave mirror. The reflecting surface of a concave mirror curves inward and away from the viewer. If the reflection takes place on the outside of the spherical section, then the mirror is called a convex mirror. The reflecting surface of a convex mirror curves outward and toward the viewer.

../../../_images/concave_convex_mirrors.svg

The centre of curvature, \(C\), is the point located at the centre of the sphere used to create the mirror. The vertex, \(V\), is the point located at the geometric centre of the mirror itself. The focus, \(F\), is the point located midway between the centre of curvature and the vertex. The line passing through the centre of curvature and the vertex is called the principal axis. Notice that the focus also lies on the principal axis.

When an incident ray parallel to the principle axis strikes the mirror, the reflected ray always passes through the focus. When an incident ray passes through the focus and strikes the mirror, the reflected ray is always parallel to the principle axis. (In the above diagrams, reverse the arrow directions to see this case). These properties make the focus particularly useful when examining spherical mirrors.

Note: The distance from the centre of curvature to the vertex is equal to the radius, \(R\), of the sphere used to create the mirror. Any straight line drawn from the centre to any point on the surface of a spherical mirror will have a length equal to the radius. The distance from the vertex to the focus is called the focal length, \(f\). This distance is equal to half the distance of the radius.

\[f = \frac{R}{2}\]

#Print question
radius = round(random.uniform(10,30),1)
print("If the radius of a curved mirror is " + str(radius) + " cm, how many centimetres is the focal length?")

#Answer calculation
answer = radius/2
    
#Assign each multiple choice to these four variables
#Option_1 contains the answer
option_1 = str(round((answer),1)) + " cm" 
option_2 = str(round((answer * 2),1)) + " cm"
option_3 = str(round((answer / 2),1)) + " cm"
option_4 = str(round((answer * 4),1)) + " cm"

multiple_choice(option_1, option_2, option_3, option_4)

If the radius of a curved mirror is 26.6 cm, how many centimetres is the focal length?
(A) 53.2 cm
(B) 6.7 cm
(C) 13.3 cm
(D) 26.6 cm

#Print question
focal_length = round(random.uniform(5,15),1)
print("If the focal length of a curved mirror is " + str(focal_length) + " cm, how many centimetres is the radius of curvature?")

#Answer calculation
answer = focal_length*2
    
#Assign each multiple choice to these four variables
#Option_1 contains the answer
option_1 = str(round((answer),1)) + " cm" 
option_2 = str(round((answer * 2),1)) + " cm"
option_3 = str(round((answer / 2),1)) + " cm"
option_4 = str(round((answer / 4),1)) + " cm"

multiple_choice(option_1, option_2, option_3, option_4)

If the focal length of a curved mirror is 6.1 cm, how many centimetres is the radius of curvature?
(A) 3.0 cm
(B) 24.4 cm
(C) 6.1 cm
(D) 12.2 cm

Image Formation by Spherical Mirrors¶

A simple way to determine the position and characteristics of an image formed by the rays reflected from a spherical mirror is to construct a ray diagram. A ray diagram is used to shown the path taken by light rays as they reflect from an object or mirror. This was used to find the image created by a plane mirror in the previous section. When constructing a ray diagram, we need only consider ourselves with finding the location of a single point on the reflected image. To do this, any point on the object may be chosen, but for consistency, we will choose the topmost point for the diagrams shown below. Any rays may be chosen, but there are three particular rays that are easy to draw:

Ray 1: This ray is drawn parallel to the principle axis from a point on the object to the surface of the mirror. Since the incident ray is parallel to the principle axis, the reflected ray must pass through the focus.
Ray 2: This ray is drawn from a point on the object and through the focus. Since the incident ray passes through the focus, the reflected ray must be parallel to the principle axis.
Ray 3: This ray is drawn from a point on the object and through the centre of curvature. This ray is therefore perpendicular to the mirror’s surface (incident angle = 0). As such, the reflected ray must return along the same path and pass through the centre of curvature.

The point at which any two of these three rays converge can be used to find the location and characteristics of the reflected image.

Concave Mirrors¶

The characteristics of an image formed in a concave mirror depend on the position of the object. There are essentially five cases. Each of these five cases are demonstrated below:

Case 1: Object Located at a Distance Greater than \(C\)¶

In the first case, the distance of the object from the mirror is greater than the radius used to define the centre of curvature. In other words, the object is further away from the mirror than the centre of curvature. In this example, we can draw any two of the three rays mentioned above to find the image of the reflected object.

Note: You only need to draw two of the three rays to find the image of the reflected object.

output_case_1 = widgets.Output()
frame_case_1 = 1

#Toggle images
def show_svg_case_1():
    global frame_case_1
    if frame_case_1 == 0:
        display(SVG("Images/case_1_0.svg"))
        frame_case_1 = 1
    elif frame_case_1 == 1:
        display(SVG("Images/case_1_1.svg"))
        frame_case_1 = 2
    elif frame_case_1 == 2:
        display(SVG("Images/case_1_2.svg"))
        frame_case_1 = 3
    elif frame_case_1 == 3:
        display(SVG("Images/case_1_3.svg"))
        frame_case_1 = 0
        
button_case_1 = widgets.Button(description="Toggle rays", button_style = 'success')
display(button_case_1)

def on_submit_button_case_1_clicked(b):
    with output_case_1:
        clear_output(wait=True)
        show_svg_case_1()

with output_case_1:
    display(SVG("Images/case_1_0.svg"))
    
button_case_1.on_click(on_submit_button_case_1_clicked)
display(output_case_1)