Calculation of Change in Focal Length of a Convex Lens Immersed in Water

Introduction: The focal length of a convex lens changes when it is immersed in a medium of different refractive index. This is because the speed of light changes when it passes from one medium to another with a different refractive index. In this case, we will calculate the change in focal length of a convex lens of focal length 20 cm when it is immersed in water.

Given:
Focal length of convex lens, f = 20 cm
Refractive index of glass, ng = 3/2
Refractive index of water, nw = 4/3

Formula:
The formula for calculating the focal length of a lens in a medium of refractive index n is:

1/f = (n - 1) * (1/R1 - 1/R2)

Where, R1 and R2 are the radii of curvature of the two surfaces of the lens.

Calculation:
The radii of curvature of the two surfaces of the lens are not given. Hence, we will assume that the lens is a thin lens and use the formula for thin lens:

1/f = (ng - nw) * (1/R1 - 1/R2)

Substituting the values, we get:

1/20 = (3/2 - 4/3) * (1/R1 - 1/R2)

Simplifying the equation, we get:

1/20 = (-1/6) * (1/R1 - 1/R2)

Multiplying both sides by -6, we get:

-3/10 = 1/R1 - 1/R2

Adding 1/R1 to both sides, we get:

1/R1 - 3/10 = 1/R2

Now, let us assume that the lens is made of a material of refractive index ng and is in air. Hence, its focal length is:

f1 = 1/((ng - 1) * (1/R1 - 1/R2))

Substituting the values, we get:

f1 = 1/((3/2 - 1) * (1/R1 - 1/R2))

Simplifying the equation, we get:

f1 = 2R1/3

Now, let us assume that the lens is made of a material of refractive index nw and is in water. Hence, its focal length is:

f2 = 1/((nw - 1) * (1/R1 - 1/R2))

Substituting the values, we get:

f2 = 1/((4/3 - 1) * (1/R1 - 1/R2))

Simplifying the equation, we get:

f2 = 3R1/4

The change in focal length, Δf, is given by:

Δf = f2 - f1

Substituting the values, we get:

Δf = 3R1/4 - 2R1/3

Simplifying the equation, we get:

Δf = -R1/12

Hence, the change in focal length of the convex lens when it is immersed in water is -R1/12 or approximately