AP Physics B- Optics: Equations to be Remembered in Geometric Optics

The topics included under Optics in the AP Physics B Exam are the following

(a) Physical Optics: Interference and diffraction, Dispersion of light and the electromagnetic spectrum.

(b) Geometric optics: Reflection and refraction, Mirrors, Lenses.

These topics carry 10% of the total points (5% for Physical Optics and 5% for Geometric Optics )

You have to remember the following equations in Geometric optics:

(1) When a plane mirror is rotated through an angle θ, the reflected ray rotates through an angle 2θ.

(2) The minimum size (length) of a plane mirror for enabling a person of height ‘h’ to see his full image is h/2. You can easily prove this using the law of reflection (angle of incidence = angle of reflection).

(3) The radius of curvature (R) of a spherical mirror is related to its focal length (f) as

R = 2f

(4) The distance of object (s_o), distance of image (s_i) and the focal length (f) of a spherical mirror are related by the law of distances as

1/f = 1/s_o + 1/s_i

[The above equation is often written as 1/f = 1/u + 1/v where ‘u’ and ‘v’ stand for the distance of object and the distance of image respectively].

When you deal with reflection and refraction at spherical surfaces, you have to adopt a suitable sign convention. Let us quote from the post dated 22^nd August 2006 in entrancephysics.blogspot.com in this context:

“The sign convention adopted widely in Optics is the Cartesian convention. The ray incident on the curved surface is to be considered as proceeding in the positive X-direction and you have to measure all distances from the pole, which is supposed to be the origin.

Occasionally, you may be given a ray diagram in which the incident ray may be proceeding from right to left. To avoid confusion, imagine that the direction of the incident ray is still in the positive X-direction. You can even redraw the diagram to make the incident ray proceed from left to right if you want. The signs given to the distances are as in the Cartesian coordinate system: pole to right positive and pole to left negative. Distances measured upwards are positive and those measured downwards are negative but you will mostly encounter problems with leftward and rightward measurements.

While solving problems, you should apply the signs to all known quantities. The unknown quantities are left as they are in the formulae. You will be able to arrive at conclusions by interpreting the sign of the unknown quantity you finally arrive at as the answer. For instance, if the distance of an image is obtained as negative, you will immediately understand that the image is on the same side of the curved surface as the object is”.

According to the Cartesian sign convention, the radius of curvature (R) and the focal length (f) of a concave mirror are negative since you have to measure these distances leftwards (opposite to the direction of the incident ray), from the pole P (fig.).

In the case of a convex mirror, R and f are positive since you have to measure these distances rightwards (in the same direction as that of the incident ray), from the pole P. Distance of a real object will be always negative since it is to be measured from pole to the object (towards the left and opposite to the direction of the incident light).

You have to use the Cartesian sign convention in problems involving reflection as well as refraction at spherical surfaces. But, don’t be scared of the Cartesian sign convention. It’s simple. You will get accustomed to this sign convention when we discus problems involving spherical surfaces.

(5) Magnification (M) produced by a mirror is the ratio of the size of image to the size of object and is given by

M = h_i/h_o = –s_i/s_o

When we apply the sign convention, for the upright (and virtual) image formed by a convex or concave mirror, the magnification is positive. If the image is inverted (and real), the magnification is negative.

(6) n₁ sin θ₁ = n₂ sin θ₂ where n₁ is the refractive index of medium 1, n₂ is the refractive index of medium 2, θ₁ is the angle of incidence (in medium 1) and θ₂ is the angle of refraction (in medium 2).

Refractive index(n) of medium 2 with respect to medium 1 is given by

n = n₂/ n₁ = sin θ₁/ sin θ₂

If medium1 is air or free space, the refractive index of medium 2 is given by

n₂ = n = sin θ₁/ sin θ₂ = sin i/sin r, on putting θ₁ = i and θ₂ = r

Note that when you say ‘refractive index of a medium’ without mentioning a reference nedium, you mean the refractive index with respect to free space (or air). The refractive index of free space (or air) is unity. [When air is compressed, its refractive index is increased]

[Refractive index(n) of medium 2 with respect to medium 1 is also given given by

n = n₂/ n₁ = λ₁/ λ₂ = v₁/v₂ where λ₁ and λ₂ are the wave lengths and v₁ and v₂ are the velocities of light in medium 1 and medium 2 respectively].

(7) When an object in medium 2 is viewed from medium 1, normally through the plane surface separating the two media, the apparent distance of the object (which is the distance of the refracted image) from the surface is related to the real distance as

Apparent distance = (Real distance)/n

where ‘n’ is the refractive index of medium 2 with respect to medium 1.

In the figure, O is the object and I is the image formed by normal refraction at the plane surface AB separating the two media.. PI is the apparent distance and PO is the real distance of the object O.

The apparent depth (d_app) of a liquid in a container when viewed normally from above is given by

d_app = d_real /n

where d_real is the real depth and ‘n’ is the refractive index of the liquid.

If there are many layers of liquids of refractive indices n₁, n₂, n₃,….etc, with real thickness t₁, t₂, t₃ …etc, the apparent thickness (t_app) of the entire liquid column is given by

t_app = t₁/n₁ + t₂/n₂ + t₃/n₃ +….etc

Evidently, the above equation is true in the case of parallel slabs of transparent media piled up to make a composite slab.

(8) sin θ_c = n₂/n₁ where θ_c is the critical angle (for total internal reflection) of the medium of higher refractive index n₁ with respect to the medium of lower refractive index n₂.

If light is proceeding from a (denser) medium of refractive index ‘n’ to air or free space, the critical angle of the medium is given by

sin θ_c = 1/n

(9) The angle of minimum deviation ‘D’ produced by a prism is related to the refractive index ‘n’ of the material of the prism as

n = sin[(A+D)/2] / sin(A/2) where A is the angle of the prism.

(10) When refraction occurs at a spherical surface of radius of curvature R, separating two media of refractive indices n₁ and n₂, the object distance s₀ and the image distance s_i are related to R, n₁ and n₂ by the equation,

n₂/s_i – n₁/s_o = (n₂ – n₁) /R

[This is often written as n₂/v – n₁/u = (n₂ – n₁) /R, using symbols ‘v’ and ‘u’ for the image distance and the object distance respectively].

(11) Lens maker’s equation is

1/f = [(n₂/n₁) – 1] (1/R₁ – 1/R₂) where ‘f’ is the focal length of the lens, n₂ is the refractive index of the material of the lens and R₁ and R₂ are the radii of curvature of its surfaces. n₁ is the refractive index of the medium in which the lens is placed.

Note that if the lens is placed in free space (or air at ordinary pressures), the refractive index n₁ = 1. Putting n₂ = n, the lens maker’s equation becomes

1/f = (n– 1) (1/R₁ – 1/R₂)

In the case of a biconvex lens, the sign of R₁ will be positive and the sign of R₂ will be negative so that the above equation will become

1/f = [(n₂/n₁) – 1] (1/R₁ + 1/R₂)

In the case of a biconcave lens, the sign of R₁ will be negative and the sign of R₂ will be positive so that the above equation will become

1/f = [(n₂/n₁) – 1] (–1/R₁ – 1/R₂) or,

1/f = – [(n₂/n₁) – 1] (1/R₁ + 1/R₂)

The focal length of a convex lens is thus positive while that of a concave lens is negative [if the medium in which the lens is placed is rarer (n₁<>2)].

(12) Law of distances for a lens is

1/f = 1/s_i – 1/s_o

[This equation is often written as 1/f = 1/v – 1/u where the distance of object s_o and the distance of image s_i in the above equation are denoted by the symbols ‘u’ and ‘v’ respectively].

In the case of a converging lens forming a real image, on applying the signs in accordance with the Cartesian convention, the above equation will become

1/f = 1/s_i + 1/s_o since the sign of s_o is negative and the signs of s_i and f are positive.

(13) Magnification (M) produced by a lens is the ratio of the size of image to the size of object and is given by

M = h_i/h_o = s_i/s_o

When we apply the sign convention, for the upright (and virtual) image formed by a converging or diverging lens, the magnification is positive. If the image is inverted (and real), the magnification is negative.

(14) Focal power (or simply, power) of a lens is the reciprocal of focal length in metre. Therefore, power (P) is given by

P = 1/f

Unit of focal power is dioptre. The sign of ‘f’ is to be considered when you express the power. Therefore, converging lenses have positive power while diverging lenses have negative power.

(15) The effective focal length (F) of two or more thin lenses (of focal lengths f₁, f₂, f₃, …etc.) in contact is given by the reciprocal relation

1/F = 1/f₁ + 1/f₂ + 1/f₃ + …..etc.

This is really a power equation which says that the effective power is the (algebraic) sum of individual powers.

You can obtain informations such as the position of the image, nature of the image etc., by resorting to the method of ray tracing in many situations. The important things you must remember in this context are:

(i) A ray proceeding parallel to the principal axis will pass through the principal focus in the case convex lenses (to be more general, converging lenses) and concave mirrors. In the case of concave lenses (diverging lenses) and convex mirrors, the ray will appear to diverge from the principal focus.

(ii) A ray that passes through or appears to pass through the centre of curvature of a mirror will retrace the path after reflection at the mirror. A ray passing through the optic centre of a lens (converging as well as diverging) will proceed without deviation.

(iii) The law of reflection which states that the angle of incidence is equal to the angle of reflection is true for all surfaces (plane as well as spherical).

In the next post we will consider questions on geometric optics. Solution of problems based on the method of ray tracing as well by the application of the formulas mentioned above will be discussed.