Physics Topics cover a broad range of concepts that are essential to understanding the natural world.

## Power of a Lens – Definition, Formula, Units, Calculation

A convex lens converges the light rays falling on it whereas a concave lens diverges the light rays falling on it. The power of a lens is a measure of the degree of convergence or divergence of light rays falling on it. If a convex lens converges a beam of parallel light rays more strongly by focussing them closer to the optical centre, it is said to have greater power (than another convex lens which focusses the same parallel light rays at a greater distance from the optical centre). Similarly, a concave lens which diverges a parallel beam of light rays more strongly is said to have a greater diverging power than another concave lens which diverges the light rays less strongly. The power of a lens depends on its focal length. We can define the power of a lens as follows :

The power of a lens is defined as the reciprocal of its focal length in metres.

Thus, Power of a lens \(=\frac{1}{\text { focal length of the lens (in metres) }}\)

or P = \(\frac{1}{f}\)

where P = Power of the lens

and f = focal length of the lens (in metres)

Since the power of a lens is inversely proportional to its focal length, therefore, a lens of short focal length has more power whereas a lens of long focal length has less power. For example, a lens of 5 cm focal length will have more power than a lens of 20 cm focal length. A more powerful lens is one that bends the light rays more ; and has a shorter focal length (see Figure).

The unit of the power of a lens is dioptre, which is denoted by the letter D. One dioptre is the power of a lens whose focal length is 1 metre. The power of a lens can be measured directly by using an instrument called dioptremeter. It is used by opticians to measure the power of spectacle lenses (see Figure). A convex lens has a positive focal length, so the power of a convex lens is positive (and written with a

sign). A concave lens has a negative focal length, so the power of a concave lens is negative [and written with a minus (-) sign]. Since a convex lens is also known as a converging lens and a concave lens is also known as a diverging lens, we can also say that the power of a converging lens is positive whereas the power of a diverging lens is negative. When an optician, after testing the eyes of a person, prescribes the corrective lenses of say, + 2.0 D and + 1.5 D for the left eye and right eye, respectively, he actually refers to the power of convex lenses required for making eye-glasses or spectacles to make him see clearly.

In order to calculate the power of a lens, we need its focal length in metres. In many problems, the focal length of a lens is usually given to us in “centimetres’. So, to calculate the power of such a lens, we should first convert the focal length of the lens into ‘metres’ by dividing the given ‘centimetre’ value by 100. If, however, the focal length is already in metres, then there is no need to change it. We will now solve some problems based on the calculation of power of lenses.

**Example Problem 1.**

A convex lens is of focal length 10 cm. What is its power ?

**Solution.**

Here, the focal length of the lens is given in ‘centimetres’ so to calculate the power of this lens, we should first convert the focal length into ‘metres’ because our formula uses the focal length in ‘metres’.

Now, 10 cm = \(\frac{10}{100}\) m

= 0.1 m

So, Focal length, f = 0.1 m (A convex lens has positive focal length)

Now, putting this value of focal length in the formula for the power of a lens :

P = \(=\frac{1}{f(\text { in metres })}\)

we get: P = \(\frac{1}{0.1}\)

or, P = \(\frac{1 \times 10}{1}\)

Thus, Power, P = +10 dioptres (or +10 D)

Thus, the power of this convex lens is +10 dioptres which is also written as +10 D. The plus sign with the power indicates that it is a converging lens or convex lens.

**Sample Problem 2.**

A person having a myopic eye uses a concave lens of focal length 50 cm. What is the power of the lens ?

**Solution.**

Here we have a concave lens. Now, the focal length of a concave lens is considered negative, so it is to be written with a minus sign.

Thus, Focal length, f = – 50 cm

= \(-\frac{50}{100}\)m

= – 0.5 m

Now, Power, P = \(\frac{1}{f \text { (in metres) }}\)

P = \(\frac{1}{-0.5}\)

P = \(-\frac{1 \times 10}{5}\)

P = -2 dioptres (or -2 D)

Thus, the power of this concave lens is, – 2 dioptres which can also be written as, – 2D. The minus sign with the power indicates that it is a diverging lens or concave lens.

**Example Problem 3.**

A thin lens has a focal length of, – 25 cm. What is the power of the lens and what is its nature ?

**Solution.**

Since the focal length is negative, it is a concave lens or diverging lens. Calculate the power yourself as shown in the above question. The power will be, – 4 D.

**Example Problem 4.**

The power of a lens is + 2.5 D. What kind of lens it is and what is its focal length ?

**Solution.**

The power of this lens has positive sign, so it is a convex lens. Now,

Power, P = \(=\frac{1}{f \text { (in metres) }}\)

So, +2.5 = \(\frac{1}{f}\)

and f = \(\frac{1}{2.5}\) m

= \(\frac{1}{2.5}\) × 100 cm

So, Focal length, f = 40 cm (or + 40 cm)

**Example Problem 5.**

A lens has a power of, – 2.5 D. What is the focal length and nature of the lens ?

**Solution.**

The power of this lens has minus sign, so it is a concave lens. Calculate the focal length yourself as shown in the above question. The focal length will be, – 40 cm.

**Example Problem 6.**

Find the power of a concave lens of focal length 2 m. (NCERT Book Question)

**Solution.**

A concave lens has negative focal length, so it is to be written with a minus sign. Thus,

Focal length, f = – 2 m (It is in metres)

Now, Power, P = \(=\frac{1}{f(\text { in metres })}\)

P = \(\frac{1}{-2}\)

P = -0.5D

Thus, the power of this concave lens is, – 0.5 dioptre.

**Example Problem 7.**

A convex lens forms a real and inverted image of a needle at a distance of 50 cm from the lens. If the image is of the same size as the needle, where is the needle placed in front of the lens? Also, find the power of the lens. (NCERT Book Question)

**Solution.**

(i) In this case needle is the object. Since the image is real, inverted and of same size as the needle (or object), the needle must be at the same distance (50 cm) in front of lens, as the image is behind the lens. Thus, the needle is placed at a distance of 50 cm from lens in the front.

(ii) When the image formed by a convex lens is of the same size as the needle (or object), then the distance of needle from the lens is 2f (twice the focal length). In this case :

2f = 50 cm

So, f = \(\frac{50}{2}\) cm

f = 25 cm

Thus, the focal length of this convex lens is +25 cm. This is equal to \(\frac{+25}{100}\)m or + 0.25 m. Now,

Power, P = \(\frac{1}{f \text { (in metres) }}\)

= \(\frac{1}{+0.25}\) = + 4.0 D

So, the power of this convex lens is + 4.0 dioptres.

### Power of a Combination of Lenses

If a number of lenses are placed in close contact, then the power of the combination of lenses is equal to the algebraic sum of the powers of individual lenses. Thus, if two lenses of powers p1 and p2 are placed in contact with each other, then their resultant power P is given by :

P = p_{1} + p_{2}

For example, if a convex lens of power, + 4 D and a concave lens of power, -10 D are placed in contact with each other, then their resultant power will be :

P = p_{1} + p_{2}

= + 4 + (-10)

= 4 – 10

= -6D

This shows that a combination of convex lens of power + 4 D and a concave lens of power, – 10 D has a resultant power of , – 6 D. So, this combination of convex lens and concave lens behaves like a concave lens (of power, – 6 dioptres). This is shown clearly in the Figure given below :

In general, if a number of thin lenses having powers p_{1}, p_{2}, p_{3},…… etc., are placed in close contact with

one another, then their resultant power P is given by :

P = p_{1} + p_{2 } + p_{3} + ……

Please note that the individual powers p_{1}, p_{2}, p_{3}, etc., of the lenses should be put in the above formula with their proper signs.

The use of powers of lenses (instead of their focal lengths) makes the work of opticians very simple and straightforward. For example, when an optician places two convex lenses of powers +2.0 D and +0.25 D in front of a person’s eye during eye-testing, he immediately knows that this convex lens combination is equivalent to a single convex lens of power +2.25 D.

The lens systems consisting of several lenses in contact are used in designing the optical instruments like cameras, microscopes and telescopes, etc. The use of a combination of lenses increases the sharpness of the image. The image produced by using a combination of lenses is also free from many defects which otherwise occur while using a single lens. We will now solve a problem based on combination of lenses.

**Example Problem.**

Two thin lenses of power, + 3.5 D and, – 2.5 D are placed in contact. Find the power and focal length of the lens combination. (NCERT Book Question)

**Solution.**

We know that :

Power of combination of lenses, P = p_{1} + p_{2}

So, P = +3.5 + (-2.5)

P = +3.5 – 2.5

P = +1.0 D

Thus, the power of this combination of lenses is, +1.0 dioptre.

We will now calculate the focal length of this combination of lenses. We know that :

Power, P = \(\frac{1}{f}\)

or +1 = \(\frac{1}{f}\)

And, f = +1 m

So, the focal length of this combination of lenses is, +1 metre.