1. \(\frac{1}{f}=\Big(\frac{n_2}{n_1}-1\Big)\Big(\frac{1}{R_1}-\frac{1}{R_2}\Big)\)

**2. Refraction by a lens:**

Lens Maker’s Formula (for a thin lens): Consider a thin lens of refractive index n_{2} formed by the spherical surfaces ABC and ADC. Let the lens is kept in a medium of refractive index n_{1} . Let an object ‘O’ is placed in the medium of refractive index n_{1} . Hence the incident ray OM is in the medium of refractive index n_{1} and the refracted ray MN is in the medium of refractive index n_{2} .

The spherical surface ABC (radius of curvature R,) forms the image at l_{1} . Let ‘u’ be the object distance and ‘v_{1} ’ be the image distance.

Then we can write,

\(\frac{n_2}{v_1}-\frac{n_1}{u}=\frac{n_2-n_1}{R_1}\) ......(1)

Adding eq (1) and eq (2) we get

\(\frac{n_2}{v_1}-\frac{n_1}{u}+\frac{n_1}{v}-\frac{n_2}{v_1}\) = \(\frac{n_2-n_1}{R_1}+\frac{n_1-n_2}{R_2}\)

\(\frac{n_1}{v}-\frac{n_1}{u}=(n_2-n_1)\)\(\Big(\frac{1}{R_1}-\frac{1}{R_2}\Big)\)

Dividing throughout by n_{1} , we get

\(\frac{1}{v}-\frac{1}{u}=\Big(\frac{n_2}{n_1}-1\Big)\)\(\Big(\frac{1}{R_1}-\frac{1}{R_2}\Big)\)

if the lens is kept in air ,\(\frac{n_2}{n_1}\) = n

So the above equation can be written as,

\(\frac{1}{v}-\frac{1}{u}\)\((n-1)\Big(\frac{1}{R_1}-\frac{1}{R_2}\Big)\) ......(4)

From the definition of the lens, we can take, when u = ∞, f = v

Substituting these values in the eq (3), we get

∴ \(\frac{1}{f}-\frac{1}{∞}\)\(=(n-1)\Big(\frac{1}{R_1}-\frac{1}{R_2}\Big)\)

This is lens maker’s formula

\(\frac{1}{f}=(n-1)\Big(\frac{1}{R_1}-\frac{1}{R_2}\Big)\) ......(5)

For convex lens.

f = +ve, R = +ve, R = – ve

\(\frac{1}{f}=(n-1)\Big(\frac{1}{R_1}+\frac{1}{R_2}\Big)\)

For concave lens,

f = -ve, R = -ve, R = +ve

\(\frac{1}{f}=(n-1)\Big(\frac{1}{R_1}-\frac{1}{R_2}\Big)\)

Lens formula

From eq(4),

\(\frac{1}{v}-\frac{1}{u}=(n-1)\Big(\frac{1}{R_1}-\frac{1}{R_2}\Big)\)

From eq(5)

\(\frac{1}{f}=(n-1)\Big(\frac{1}{R_1}-\frac{1}{R_2}\Big)\)

From these two equations, we get

\(\frac{1}{f}=\frac{1}{v}-\frac{1}{u}\)

**Linear magnification :**

If h is the height of the object and h_{o} is the height of the image, then linear magnification.

\(m=\frac{h_1}{h_o}=\frac{v}{u}\)

3. a. R_{1} = R, R_{2} = +R

∴ \(\frac{1}{f}=\Big(\frac{n_2}{n_1}-1\Big)\Big(\frac{1}{R}-\frac{1}{R}\Big)\)

power of lens, P = 0

b. We know

\(\frac{1}{f}=\Big(\frac{n_2}{n_1}-1\Big)\Big(\frac{1}{R_1}-\frac{1}{R_2}\Big)\)

The above equation shows when n_{1} increases f decreases the refractive index of water is greater than air. Hence when we place a lens in water, focul length decreases.