Power of a Lens

The ability of a lens to bend the light falling on it is called the power of a lens. Since the lens of shorter focal length will bend the light rays more will have more power. A convex lens converges the light rays towards the principal axis whereas a concave lens diverges the light rays away from the principal axis. 

Here,

\[P=\frac{1}{F}\]      

The power of a lens is defined as the inverse of its focal length (f) in meters (m).

Power of a Lens Formula Definition

The power of a lens is specified as \[P=\frac{1}{F}\], where f is the focal length.

The S.I. unit of power of a lens is \[m^{-1}\].  This is also known as diopter.

The focal length (f) of a converging lens is considered positive and that of a diverging lens is considered negative. Thus, the power of a converging lens is positive and that of the diverging lens is negative.

Lens Formula in Terms of Power

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Fig.1 shows two lenses L1 and L2 placed in contact. The focal lengths of the lenses are f1 and f2, respectively. Let P be the point where the optical centres of the lenses coincide (lenses being thin). 

Now, let us place an object ‘O’ beyond the focus of lens L1 such that OP = u (object distance) on the common principal axis (coaxially). 

Here, the first lens L1 alone forms an image at I1 where PI1 = v1 (image distance). 

Also, this point I1 works as the virtual object for the second lens L2 and the final image is formed at I, at a distance PI = v. The ray diagram (Fig.1) formed by the combination of two convex lenses has the following attributes:

u  = Object-distance for the first lens

v =  final image-distance for the second lens

v1 = image-distance for the first image I1 for the first lens. As the lenses are pretended to be thin, v1 is also the object distance for the second lens.

The lens formula for the image I1 formed by lens L1 will be

\[\frac{1}{v_1}-\frac{1}{u}=\frac{1}{F}\].....(1) 

The equation for the image formation for the second lens L2:

\[\frac{1}{v}-\frac{1}{v_1}=\frac{1}{f_2}\].....(2) 

Adding eq (1) and (2):

\[\frac{1}{v_1}-\frac{1}{u}+\frac{1}{v}-\frac{1}{v_1}=\frac{1}{F_1}+\frac{1}{f_2}\]

\[\frac{1}{v}-\frac{1}{u}=\frac{1}{F_1}+\frac{1}{f_2}\].....(3) 

The focal length of the combined lens is given by-

If the combination is replaced by a single lens of focal length F such that it forms the image of O at the position I,

1/v - 1/ u =  1/ F……(4)

This type of lens is called the equivalent lens for the combination.

Combining (3) and (4),

1/F = 1 / f1 + 1/ f2……(5)

Here, F is the focal length of the equivalent lens for the combination. As the power of a lens is P =  1/ F, eq (5) immediately gives, 

The power of any number of lenses in contact is equal to the algebraic sum of the power of two individual lenses. This is true for any situation involving two thin lenses in contact.

How to Find the Power of the Lens Using the Focal Length?

The power of a lens is measured as the reciprocal of the focal length of the lens.

Relation with focal length: A lens of less focal length produces more converging or diverging and is said to have more power.

I.e.,

According to the lens maker’s formula,

Since, P = 1/ F 

We get,

Here, 

v = refractive index of the material

R1 = Radius of curvature of the first surface of the lens

R2 =  Radius of curvature of the second surface of the lens

For a converging lens, power is taken as positive and for a diverging lens, power is taken as negative.

Definition for the Power of Lens Unit

The S.I. the unit of power is dioptre (D).

When f = 1 meter, P = 1/ f = 1/ 1 = 1 dioptre

Hence, one dioptre is the power of a lens of focal length one meter.

When f is in 1 cm, P = 1/ f / 100 = 100/ f

So, we get the formulas to describe the relationship between P and f,

Optical Power (Lens Power)

Optical power is defined as the degree to which a lens, mirror, or other optical system converges or diverges the light. Optical power is also referred to as dioptric power, convergence power, refractive power, or refractive power. It is equal to P = 1/ f.

Dioptre Formula 

The Dioptre formula is used to calculate the optical power of a lens or curved mirror. The dioptre is the unit for a measure of the refractive index of a  lens. The power of a lens is specified as the inverse of the focal length in meters, or D =1/ f, where D is the power in dioptres.

Power of Lens Calculation: Solved Example

1. Find the power of a plano-convex lens, when the radius of a curved surface is 15 cm and v =1.5.

Solution: Given  R1 = ∞, R2 = - 15 cm, v= 1.5 cm

P = \[\frac{1}{f}=(v-1)(\frac{1}{R_1}-\frac{1}{R_2})\]

   = (1.5-1)\[(\frac{1}{\infty }+\frac{1}{0.15})\]

   = 0.5\[\times \frac{1}{0.15}=3.33\]

From the above formula for the power of the lens, we understand that the power of a lens is the reciprocal of the focal length (which we calculate in metres). Lens power is measured in dioptres (D), which is also equal to 1/m. 

Converging (convex) lenses have positive focal lengths, so they also have positive power values. However, diverging (concave) lenses have negative focal lengths, so they also have negative power values.