sign convention rules and image formation by a single curved surface

 

SIGN CONVENTION RULES FOR LENSES:

1) All distances are measured from the pole (or optic centre).

2) Distances measured along the direction of the incident light ray are taken as positive

3) Distances measured opposite to the direction of the incident light ray are taken as negative

4) The heights measured vertically above from the points on axis are taken as positive.

5) The heights measured vertically down from points on axis are taken as negative.

IMAGE FORMATION BY SINGLE CURVED SURFACE:

Consider a curved surface separating two media of refractive indices

n₁ and n₂ . A point object is placed on the principal axis at point

O. The ray, which travels along the principal axis passes through the pole

undeviated. The second ray, which forms an angle α with principal axis,

meets the interface (surface) at A. The angle of incidence is q₁. The ray

bends and passes through the second medium along the line AI. The angle

of refraction is q₂. The two refracted rays

meet at I and the image is formed there.

Let the angle made by the second

refracted ray with principal axis be g and

the angle between the normal and

principal axis be β.

PO is the object distance which we

denote as ‘u’

PI is image distance which we denote

as ‘v’

PC is radius of curvature which we denote as ‘R’

n1, n2 are refractive indices of two media.

 

In the triangle ACO, q₁ = α + β

and in the triangle ACI, β= q₂ + g => β – g = q₂

According to Snell’s law, we know

n₁ sin q₁ = n₂ sin q₂

substituting the values of q₁ and q₂, we get,

n₁ sin(α+ β) = n₂ sin(β- g) ................. (1)

If the rays move very close to the principal axis, the rays can be treated

as parallel and are called paraxial rays. Then the angles α,β and g become

very small. This approximation is called paraxial approximation.

sin (α+ β) = α+β and sin (β- g) = β- g     [since for very smaller angles sinq=q]

Substituting in equation (1)

n₁ (α+ β) = n₂ (β- g)

ð  n₁α + n₁ β = n₂ β – n₂ g ................(2)

since all angles are small, we can write

tan α = AN/NO = α            [since for very smaller angles tanq=q]

tan β = AN/NC = β

tan g = AN/NI = g

Substitute these in equation (2), we get,

n₁  (AN/NO) + n₁ (AN/NC) = n₂ (AN/NC) – n₂ (AN/NI) ............... (3)

As the rays move very close to the principal axis, the point N coincides

with pole of the interface (P). Therefore NI, NO, NC can be replaced by PI, PO and PC respectively.

After substituting these values in equation (3), we get,

n₁/PO + n₁/PC = n₂/PC – n₂/PI

n₁/PO + n₂/PI = (n₂- n₁) /PC ................(4)

Equation (4) shows the relation between refractive indices of two media, object distance, image distance and radius of curvature.

The above equation is true for the case we considered.

We can generalize equation (4) if we use the following sign convention.

For all purposes of applications of refraction at curved surfaces and through lenses following conventions are used. (recall sign convention rules for lenses)

Here PO is called the object distance (u)

PI is called the image distance (v)

PC is called radius of curvature (R)

According to sign convention mentioned above, we have

PO = -u ; PI = v ; PC = R

Substituting these values in equation (4) we get,

 

(n₂/v) – (n₁/u) = (n₂-n₁ ) /R ..........(5)

 

This formula can also be used for plane surfaces. In the case of a plane

surface, radius of curvature (R) approaches infinity. Hence 1/R becomes

zero. Substituting this in equation 5, we get formula for the plane surfaces

n₂/v - n₁/u = 0 => n₂/v = n₁/u

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