One planer concave lenses and planer convex lenses are combined if n1& n2 is index then f is

When combining a plano-concave lens and a plano-convex lens, we can determine the effective focal length (f) of the system using the lensmaker's equation. This equation takes into account the refractive indices of the materials involved, as well as the radii of curvature of the lenses. Let’s break this down step by step.

Understanding the Lens Types

A plano-concave lens has one flat surface and one inward-curving surface, while a plano-convex lens has one flat surface and one outward-curving surface. The focal lengths of these lenses are influenced by their shapes and the materials they are made from, which is where the refractive index (n) comes into play.

Lensmaker's Equation

The lensmaker's equation is given by:

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

In this equation:

• f is the focal length of the lens.
• n is the refractive index of the lens material.
• R1 is the radius of curvature of the first surface (positive for convex, negative for concave).
• R2 is the radius of curvature of the second surface (again, positive for convex, negative for concave).

Combining the Lenses

When you combine a plano-concave lens and a plano-convex lens, you need to consider their individual focal lengths. Let’s denote:

• f1 for the plano-concave lens
• f2 for the plano-convex lens

The effective focal length (f) of the combination can be found using the formula:

1/f = 1/f1 + 1/f2

Calculating Individual Focal Lengths

For the plano-concave lens:

f1 = -R/n1

For the plano-convex lens:

f2 = R/n2

Here, R is the radius of curvature (positive for the convex lens and negative for the concave lens). The negative sign for the plano-concave lens indicates that it diverges light.

Final Calculation

Substituting these values into the effective focal length equation gives:

1/f = -n1/R + n2/R

Combining these terms leads to:

1/f = (n2 - n1)/R

From this, you can find the effective focal length:

f = R/(n2 - n1)

Example

Let’s say you have a plano-concave lens with a refractive index of 1.5 and a plano-convex lens with a refractive index of 1.6, both having a radius of curvature of 10 cm. Plugging these values into our formula:

f = 10/(1.6 - 1.5) = 10/0.1 = 100 cm

This means the effective focal length of the combined lens system is 100 cm.

In summary, by understanding the properties of each lens and applying the lensmaker's equation, you can effectively calculate the focal length of a combined lens system. This approach is fundamental in optics and is widely used in designing optical instruments.