- Curved reflectors are obtained from hollow spheres, cones or cylinders.
- The surfaces of these hollow solids are then highly polished from the inside or outside depending on the type of curved reflector being made.
- Concave and convex mirrors are also called spherical mirrors.
Types of Curved Reflectors
1. Concave Reflector/ Mirror
- This is a reflector whose reflecting surface curves inwards.
- It is obtained by highly polishing the outside of a glass sphere portion.
2. Convex Reflector/Mirror
- This is a reflector whose reflecting surface curves outwards.
- It is obtained by highly polishing the inside of a glass sphere portion.
3. Parabolic Reflector/ Mirror
- It is a curved reflector whose reflecting surface curves more inwards than that of the concave reflector.
- It is obtained by highly polishing the outside of a glass cone.
Terms Associated with Curved Reflectors
- Consider the diagrams below;
- This is the width of the mirror.
- This is the geometrical centre of the mirror.
c. Centre of curvature, C
- This is the centre of the sphere of which the mirror forms a part.
- The centre of curvature of a concave mirror is infront while that of a convex mirror is behind the mirror.
d. Radius of curvature, r
- It is the radius of a sphere of which the curved mirror forms a part.
e. Principal/main axis
- This is the line passing through the pole and the centre of curvature of the curved mirror.
f. Paraxial rays and marginal rays
- These are rays which are close and parallel to the principal axis while marginal rays are those that are parallel but not close to the principal axis.
g. Principal focus, F, of a concave mirror
- It is the point at which all the rays parallel and close to the principal axis converge after reflection.
- Principal focus of a concave mirror is real because reflected rays actually pass through it.
h. Principal focus, F, of a convex mirror
- This is the point at which all rays parallel and close to the principal axis appear to emerge/diverge from after reflection.
- The principal focus of a convex mirror is virtual because reflected rays only appear to pass through it.
i. Focal plane
- This is a plane perpendicular to the principal axis and passes through the principal focus.
- For a concave mirror, parallel rays which are not parallel to the principal axis converge at a point on the focal plane after reflection.
- For a convex mirror, parallel rays which are not parallel to the principal axis appear to emerge from a point on focal plane after reflection.
j. Focal length, f
- This is the distance from the pole of the mirror to its principal focus.
Note: In optics full lines represent real rays and objects while dotted lines represent virtual rays and images.
The figure below shows two parallel rays incident on a concave mirror. F is the focal point of the mirror.
Sketch on the same diagram the path of the rays after striking the mirror.
Relationship between Radius of Curvature and Focal Length
It can be shown through geometry that the radius of curvature is twice the focal length i.e.
Laws of Reflection in Curved Mirrors
- Reflection at curved surfaces also obeys laws of reflection:
- The angle of incidence is equal to the angle of reflection at the point of incidence.
- The incident ray, the normal and the reflected ray all lie on the same plane at the point of incidence.
- Ray diagrams are used to show and explain how images are formed by curved mirrors and the characteristics of these images.
Four Major Cases in the Construction of Ray Diagrams
- A ray close and parallel to principal axis passes through principal focus (for concave mirror) or appears to emerge from the principal focus (for convex mirror) after reflection.
- A ray through the principal focus of a concave or appearing to be directed to the principal focus of convex mirror is reflected parallel to the principal axis.
- A ray through the centre of curvature(for concave) or appearing to pass through centre of curvature (for convex mirror) is reflected along the same path.
- A ray at an angle to the principal axis and incident at the pole is reflected in such a way that the angle of incidence is equal to the angle of reflection.
Characteristics of Images Formed by Curved Reflectors
A. Concave mirrors
i. Object at infinity
- Image is real, formed at F, inverted and smaller than the object.
ii. Object at c
- Image formed is at C, real, inverted and same size as the object.
iii. Object beyond C
- Image formed is between C and F, real, inverted and smaller than the object.
iv. Object between C and F
- The image formed is beyond C, real, inverted, larger than the object (magnified)
v. Object at F
- The image is formed at infinity because the rays emerge parallel after reflection.
vi. Object between F and P
- The image formed is behind the mirror, virtual, upright and larger than the object.
B. Convex mirror
- Irrespective of the distance of the object in front of the mirror, images formed by convex mirrors are always upright, smaller than the object and between P and F.
A lady holds a large concave mirror of focal length 1m, 80cm from her face.
- Using suitable construction illustrate how her image is formed.
- State two characteristics of her image in the mirror.
- Image upright
- Image magnified
- The figure below shows a ray of light incident on a convex mirror. Using suitable construction, determine the radius of curvature of the mirror.
- The figure below shows parabolic surface with a source of light placed at its focal point F.
Draw rays to show reflection from the surface when rays from the source strike the surface at points A,B,C and D
- The figure below shows a vertical object,O, placed in front of a convex mirror.
On the same diagram draw the appropriate rays and locate the image formed.
- The figure below shows a bright behind a screen which has a hole covered with wire gauze. A concave mirror of focal length 25cm is placed in front of the screen. The position of the mirror is adjusted until a sharp image of the gauze is formed on the screen.
Determine the distance between the screen and the mirror.
Graphical Construction of Ray Diagrams
- Images are drawn to scale in a ray diagram and this is best done on graph paper.
- It is the ratio of the image height to the object height.
- It can also be defined as the ratio of the image distance to the object distance.
- An object 3 cm high is placed 6cm infront of a concave mirror of radius of curvature 10cm. By scale drawing determine the:
- Position of the image
- Size of the image
- Nature of the image
- Image position is 13.6X2=27.2cm from P in front of the mirror.
- Size of the image is 6.6X2=13.2cm
- Nature of the image: the image is inverted, real and magnified
- A convex mirror of focal length 15cm produces an image 10cm away from the mirror. If the image is 3 cm high, determine by scale drawing.
- The object distance
- Object height/size
- The object distance is 6X5=30cm
- Object height/size is 1.4X5=7cm
- A concave mirror of focal length 10cm forms a sharp image at 40cm from the mirror. Determine graphically the position of the object and magnification of the image.
- A concave mirror of focal length 20cm forms a real image two times the size of the object. If the object height is 10 cm, determine by scale drawing:
- The object distance
- The image distance
The Mirror Formula
The object distance u, the focal length f, and the image distance v related by the mirror formula:
1/f = 1/u + 1/v
- This is a sign convention used with the mirror formula in order to determine the position and nature of the image formed by a curved mirror. According to the real-is-positive sign convention:
- All distances are measured from the mirror as the origin.
- Distances of real objects and images are considered positive e.g. focal length of concave mirrors.
- Distances of virtual objects and images are considered negative e.g. focal length of convex mirror.
- An object is placed 10cm in front of a concave mirror of focal length 20cm. Determine the position and nature of the image.
The image is virtual(because v is negative), upright and magnified(because v is greater than u)
- An object is placed 10cm in front of a convex mirror of focal length 20cm. Determine the position and nature of the image.
f is negative (-20 cm) according to real-is-positive convention
The image is virtual(because v is negative), upright and diminished (because v is smaller than u)
- A concave mirror with radius of curvature 10 cm produces an inverted image two times the size of an object placed in front of it and perpendicular to the principal axis. Determine the position of:
- The object
- The image
- The distance between an erect image and the object is 40cm. The image is twice as tall as the object. Determine:
- The object distance.
- The radius of curvature.
- A vertical object 10cm high is placed 20cm away from a convex mirror of radius of curvature 30cm.
- The image distance.
- The height of the image.
- The magnification of the image.
- The distance between an object and its magnified real image produced by a concave mirror is 40cm when the object is placed 20cm from the pole of the mirror. Determine the:
- Transverse magnification of the image.
- The focal length of the mirror.
Graphical Analysis of the Mirror Formula
1. Graph of 1/u against 1/v
- It is a straight line graph with a negative gradient, implying that the image is inverted relative to the object.
- The 1/u -intercept or the 1/v -intercept gives 1/f
2. Graph of uv against u+v
- From the mirror formula 1/f = 1/v + 1/u
- Therefore, a graph of uv against u+v is a straight line through the origin whose gradient is positive. The gradient of the graph gives f
3. Graph of m against v
- From the mirror formula 1/f = 1/v + 1/u
multiplying all through by v gives
- Therefore, a graph of m against v is a straight line with a gradient of 1/f and is -1. Also the v-intercept gives the focal length, f
A concave mirror and an illuminated object are used to produce a sharp image of the object on a screen. The object distances and image distances are given below.
- Complete the table
Object distance, u(cm)
80.0 26.7 22.4 20.6 19.6 Image distance, v(cm) 20.0 40.0 56.0 72.0 88 u + v (cm) uv(cm2) Magnification, m
- Using suitable values:
- Plot a graph 1/u of against 1/v
- Determine the radius of curvature, f from the graph
- Plot a graph of uv against u+v and use it to find the radius of curvature, f
- Plot a graph of magnification, m against v and use it to find the radius of curvature, f
Applications of Curved Mirrors
A. Concave Mirrors
- Used as shaving mirrors because they produce magnified and upright images when the object is between principal focus, F and the pole,P.
- Used by dentist when examining teeth they produce magnified and upright images when the object is between principal focus,F and the pole, P.
- Used as reflector behind projector lamp to reflect light travelling away from the projector. The lamp is placed at the centre of curvature of the concave mirror.
- Used in telescopes to bring distant objects (objects at infinity) like stars into focus at the focal point.
- Used as solar concentrators to bring light energy into focus.
B. Convex Mirrors
- Used as car and motorcycle side mirrors because they form upright images and have a wide field of view
- Used in supermarkets to monitor movement of customers because they form upright images and have a wide field of view
Note: The defect of spherical mirrors in which marginal rays are not brought into focus at the principal focus resulting in blurred images is called spherical aberration.
Disadvantage of Convex Mirror
- Convex mirror forms diminished images giving an impression that the vehicles behind are farther away than they actually are.
C. Parabolic Mirrors
- Used for propagation of parallel light beams of high intensity in hand torches, search lights and car head lights.
Advantage of Parabolic Mirrors over Concave Mirrors
- Unlike concave mirrors in which marginal rays are not converged at principal focus, parabolic mirror converges all rays parallel to principal axis and incident on its surface at its principal focus.
- With the aid of a well labeled diagram, explain the wide field of view of a convex mirror.
- State one application of each of the following
- Convex mirror.
- Parabolic mirror
- The figure below which is drawn to a scale of 1:5 represents an object O and its image I formed by a convex mirror.
By drawing suitable rays, locate and mark on the figure the position of the principal focus, F of the mirror. Determine the focal length, f.
- The figure below shows a point object O placed infront of a convex mirror.
Draw appropriate rays to locate the image of the object.
- State the advantage of parabolic mirror over concave mirror.