# Spherical Lenses

Have you heard of Sherlock Holmes, the greatest detective in the world? How did he become so good at his work? Look closely at the image and you’d find the answer! That’s right! His “magnifying glass” was his greatest tool. But what’s so special about it? It’s because of the kind of spherical lenses used in it. Let us find out more about spherical lenses ahead.

Source: Moziru.com

## Spherical Lenses

A lens is a part of a transparent thick glass which is bounded by two spherical surfaces. It is an optical device through which the rays of light converge or diverge before transmitting. Thus spherical lenses are of two major kinds called “Convex or Convergent” lenses and “Concave or Divergent” lenses. The point from which these rays converge or appear to diverge is called the “Focus” or “Focal point” and is denoted by the letter ‘f’.

## Types of Lenses

As discussed already spherical lenses are of two kinds :

a. Convex or Convergent Lenses
b. Concave or Divergent Lenses

### a. Convex or Convergent Lenses

A convex lens is thicker in the middle and thinner at the edges. A convex lens is also known as a “biconvex lens” because of two spherical surfaces bulging outwards.

Source: Amazon.in

Source: Wikipedia

Convex lenses include lenses that are plano-convex (i.e. these lenses are flat on one side and bulged outward on the other), and convex meniscus (i.e. these lenses are curved inward on one side and on the outer side it’s curved more strongly).

### b. Concave or Divergent Lenses

A concave lens is thicker at the edges and thinner in the middle. A concave lens is also known as a “biconcave lens” because of two spherical surfaces bulging inwards.

Source: Indiamart.com

Source: Wikipedia

Concave lenses include lenses like plano-concave (i.e. these lenses are flat on one side and curved inward on the other), and concave meniscus (i.e. these lenses are curved inward on one side and on the outer side it’s curved less strongly).

## Sign Conventions for Spherical Lenses

a. All of the distances that are to be measured are measured from optical center of the lens.

b. Distances that are in the direction of the incident rays are taken to be positive and those in opposite direction are taken to be negative.

c. Above the principal axis, measurement of height is taken as positive and below the principal axis measurement of height is taken as negative.

d. A convex lens has a positive focal length and a concave lens has a negative focal length.

e. Object distance is positive, real image distance is positive and virtual image distance is negative.

f. Magnification (m) = h1/h2, wherein for a virtual image it’s positive and negative for a real image. Also, h1 ( height of the object) is positive and h2 ( height of the image which varies depending on the type of the image ); i.e. h2 for a real image is negative and h2 for a virtual image is positive.

## Ray Diagrams for Spherical Lenses

While drawing ray diagrams for spherical lenses, one needs to know the basic and important set of rules of refraction that state that:

Any incident ray parallel to the principal axis of a convex lens shall refract through the lens and travel on the opposite side of the lens through the focal point.

Any incident ray which travels through the focal point of the lens will refract through the lens and then travel parallelly to the principal axis.

Any incident ray in effect shall continue to travel in the same direction that it had when it entered the lens when it passes through the center of the lens.

### Ray Diagrams for a Convex Lens

For a convex lens, there are six possible positions where the object can be positioned and an image is formed:

#### a. The object is positioned at infinity

When the object is placed at infinity, light rays MX and NY which run parallel to the principal axis are refracted at point X and Y respectively and at the principal focus, they intersect each other. Hence, here the image of M’N’ is formed at the focus, the image is highly diminished ( i.e.point-sized ), real and inverted.

#### b. The object is positioned beyond 2F

An object MN is placed beyond the centre of curvature (2F) then a light ray MX running parallel to the principal axis, passes through the focus (F) after refraction in along the direction XF. Another ray MO passes through the optical center (O) and without any deviation goes straight. At point M’, between the focus (F) and 2F of the lens on the other side, the two refracted rays of light intersect each other. Therefore, the image M’N’ that is formed is diminished, real and inverted is formed.

#### c. The object is positioned at 2F

Object MN is placed at 2F of a convex lens and the ray of light MY parallel to the principal axis passes through the focus (F) in the direction YF after refraction. Another ray MO which passes through the optical center O travels straight without any deviation. At 2F, on the other side of the lens, the two refracted rays of light intersect each other at point M’. Hence, the image M’N’ formed is at 2F, the image is of the same size as the object, and is real and inverted.

#### d. The object is positioned between the focus and the center of curvature

Object MN is positioned between the focus (F) and the center of curvature (2F) of a convex lens. Then the light ray MX  running parallel to the principal axis after refraction passes through focus (F) in the direction XF. Another ray MO passing through the optical center (O) travels straight without any deviation. Beyond the center of curvature (2F) on the other side of the lens, the two refracted rays intersect each other at point M’. Therefore, the image M’N’ formed is larger than the object and is real and inverted.

#### e. The object is positioned at focus F

An object MN is placed at the principal focus (F), then a ray of light MY running parallel to the principal axis passes through the focus(F) on the other side of the lens after refraction, in the direction YQ. Another ray MO travels straight without any deviation in the direction OP while passing through the optical center(O). The refracted rays: YQ and OP are parallel to each other, and hence cannot intersect each other, as shown in the diagram given above, therefore the image formed will be at infinity. So, the image M’N’ which that will be formed will be highly enlarged, real and inverted.

#### f. The object is positioned between the focus (F ) and the optical center (O)

Object MN is positioned between the principal focus (F) and the optical center (O), a ray of light MX  parallel to the principal axis after refraction passes through the focus (F) on the other side of the lens in the direction XD. Another ray MO travels straight without any deviation in the direction OE while passing through the optical center (O). The two refracted light rays  XD and OE diverge away from one another, and hence cannot intersect each other; due to this, a real image cannot be formed. The refracted rays XD and OE are, therefore, extended backwards by dotted lines. In doing so, they appear to intersect at point M’. So, the image M’N’ formed here is a virtual, erect and highly enlarged that is formed on the same side of the convex lens behind the object.

### Ray Diagrams formed by Concave Lens

For a concave lens, there are two possible positions where the object can be positioned and an image is formed:

#### a. The object is positioned at infinity

When an object is placed at infinity, the rays PM and QN parallel to the principal axis are refracted at points M and N respectively and diverge in the directions MR and NS respectively. The diverged rays MR and NS after being extended backwards by dotted lines appear to intersect each other at the principal focus(F) of the lens. Hence, when the object is placed at infinity in case of concave lens the image formed is at the principal focus (F), the image is highly diminished (point-sized), virtual and erect.

#### b. The object is positioned between infinity and the optical center (O)

Object MN is positioned between infinity and the optical center (O) of the lens, a ray MD parallel to the principal axis after refraction gets diverged in the direction DP and appears to come from the principal focus (F) in the direction DF. Another ray MO travels straight without any deviation through the optical center (O) of the lens in the direction OQ. Both the refracted rays DP and OQ are diverging in nature and hence they appear to intersect each other at point M’ on the same side of the lens on extending backwards as shown in the diagram given above. Therefore, the image M’N’ formed is virtual, diminished and erect image and is formed between the optical centre (O) and the focus on the same side of the lens.

## Terms Related to Spherical Lenses

#### a. Optical Centre

The centre point of a lens which lies on its principal axis is known as its optical center. The optical centre is denoted by letter O.

#### b. Principal Axis

The principal axis of a lens is defined as a straight line passing through the optical center and the centre of curvature.

#### c. Principal Focus

The principal focus of a lens is a point on its principal axis wherein the rays of light parallel to it and after passing through it converge (for a convex lens) or appear to diverge (for a concave lens). The principal focus of a lens is denoted by the letter F.

#### d. Focal Length

The distance between the optical center and the principal focus of a spherical lens is termed as the “Focal Length”. The focal length of a spherical lens is denoted by the letter f.

Focal length of a spherical lens can also be defined as half of the radius of curvature.

2f = R or f = R/2

This is also the reason that the center of curvature is usually denoted as 2F for a spherical lens instead of C.

“Radius of curvature” of a spherical lens is defined as the distance between its optical center and the center of curvature. The radius of curvature is denoted is by the letter R.

#### f. Centre of curvature

The centre of curvature of the lens is defined as the center of sphere of a part of which a spherical lens is formed.

## Lens Formula and Magnification

Lens formula is an expression which gives the relationship between the image distance (v), object distance (u) and the focal length (f) of a lens.
$$\frac{1}{v} – \frac{1}{u} = \frac{1}{f}$$

Note:

a. One should be really careful regarding sign convention while using lens formula.

b. Focal length for a convex lens is taken as positive and for a concave lens it’s taken as negative.

Magnification formula remains the same for spherical lenses as well i.e.

Magnification (m) = h1/h2 = v/u

Where h1 is the height of the object taken to be positive
h2 is the height of the image taken to be negative
v is the image distance
u is the object distance

## Power

Power of a spherical lens can be defined as it’s ability to converge or diverge a ray of light. It’s expressed in terms of its Power(P). The SI unit of power of lens is Dioptre. It is represented by the letter “D”.

P=1/f

Therefore, power can also be defined as the reciprocal of focal length of a lens.

## Solved Example for You

Q. A needle of length 5 cm, placed 45 cm from a spherical lens form an image on a screen placed 90 cm on the other side of the lens. The type of lens and its focal length are :

a. Convex, 30 cm

b. Concave, 30 cm

c. Convex, 60 cm

d. Concave, 60 cm

Sol:

1/v – 1/4 = 1/f

1/90 – 1/-45 = 1/f

Therefore, f = 30cm (+ve). Since f is positive, is a convex lens.

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