Refraction of Light
5.0 Lens makers formula & Other Functions of lens.
5.1 Thin Lens Formula
5.2 Magnification and Power of lens
5.3 Combination of lenses
5.4 Displacement method to find focal length.
5.5 Silvering of lens
5.0 Lens makers formula & Other Functions of lens.
5.2 Magnification and Power of lens
5.3 Combination of lenses
5.4 Displacement method to find focal length.
5.5 Silvering of lens
Lens makers formula is given by,
$$\frac{1}{f} = \left( {\mu - 1} \right)\left( {\frac{1}{{{R_1}}} - \frac{1}{{{R_2}}}} \right)$$
where,
$f$: Focal length of the lens
$\mu $: Refractive index of the material of the lens
${{R_1}}$: Radius of curvature of the $1^{st}$ spherical surface of the lens
${{R_2}}$: Radius of curvature of the $2^{nd}$ spherical surface of the lens
Derivation of lens makers formula
Before deriving lens makers formula, we need to understand than in lens refraction takes place from two surfaces i.e. surface 1 and surface 2.
Object $O$ acts as an object for spherical surface 1 and produces image $I_1$.
This image $I_1$ now acts as a object for spherical surface 2 and produces an image $I$.
Let $\ mu$ be the refractive index of the lens.
For spherical surface 1
As we know,
$$\frac{{{\mu _2}}}{v} - \frac{{{\mu _1}}}{u} = \frac{{{\mu _2} - {\mu _1}}}{{{R_1}}}\quad ...(i)$$
For spherical surface 2
As we know,
$$\frac{{{\mu _2}}}{v} - \frac{{{\mu _1}}}{u} = \frac{{{\mu _2} - {\mu _1}}}{R}$$
Substituting for the above situation,
$\begin{equation} \begin{aligned} {\mu _2} \to {\mu _1} \\ {\mu _1} \to {\mu _2} \\ v \to v \\ u \to {v_1} \\ R \to {R_2} \\\end{aligned} \end{equation} $
So, $$\frac{{{\mu _1}}}{v} - \frac{{{\mu _2}}}{{{v_1}}} = \frac{{{\mu _1} - {\mu _2}}}{{{R_2}}}\quad ...(ii)$$
Adding equation $(i)$ and $(ii)$ we get,
$$\begin{equation} \begin{aligned} \left( {\frac{{{\mu _1}}}{v} - \frac{{{\mu _2}}}{{{v_1}}}} \right) + \left( {\frac{{{\mu _2}}}{v} - \frac{{{\mu _1}}}{u}} \right) = \left( {\frac{{{\mu _1} - {\mu _2}}}{{{R_2}}}} \right) + \left( {\frac{{{\mu _2} - {\mu _1}}}{{{R_1}}}} \right) \\ \frac{{{\mu _1}}}{v} - \frac{{{\mu _1}}}{u} = \left( {\frac{{{\mu _2} - {\mu _1}}}{{{R_1}}}} \right) - \left( {\frac{{{\mu _2} - {\mu _1}}}{{{R_2}}}} \right) \\\end{aligned} \end{equation} $$
$$\frac{1}{v} - \frac{1}{u} = \left( {\frac{{{\mu _2}}}{{{\mu _1}}} - 1} \right)\left( {\frac{1}{{{R_1}}} - \frac{1}{{{R_2}}}} \right)$$
The above equation can also be written as,
$$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$$
The above equation is known as thin lens formula.
Also, $$\frac{1}{f} = \left( {\frac{{{\mu _2}}}{{{\mu _1}}} - 1} \right)\left( {\frac{1}{{{R_1}}} - \frac{1}{{{R_2}}}} \right)$$
The above equation is known as lens makers formula.
Note:
- This formula is valid for thin lenses. It is valid for both convex and concave lenses.
- Sign conventions
| Type of lens | $R_1$ | $R_2$ | $f$ (using lens maker's formula) | Ray Diagram |
| Convex lens | $+\ ve$ | $-\ ve$ | $+\ ve$ |
|
| Concave lens | $-\ ve$ | $+\ ve$ | $-\ ve$ |
|
- When the refractive index of the material of the lens is greater than that of surrounding, then biconvex lens acts as converging lens and a biconcave lens acts as a diverging lens.
- When the refractive index of the material of the lens is smaller than that of the surrounding medium, then biconvex lens acts as a diverging lens and a biconcave lens acts as a converging lens.
Proof:
| Convex lens | Concave lens |
$f > 0$ (converging lens) $$\frac{1}{f} = \left( {\frac{{{\mu _2}}}{{{\mu _1}}} - 1} \right)\left( {\frac{1}{{{R_1}}} - \frac{1}{{{R_2}}}} \right) > 0$$ If ${\mu _1} > {\mu _2}$ then, $$\frac{1}{f} < 0$$ So, $$f < 0$$ So, the converging lens becomes diverging. | $f < 0$ (diverging lens) $$\frac{1}{f} = \left( {\frac{{{\mu _2}}}{{{\mu _1}}} - 1} \right)\left( {\frac{1}{{{R_1}}} - \frac{1}{{{R_2}}}} \right) < 0$$ If ${\mu _1} > {\mu _2}$ then, $$\frac{1}{f} > 0$$ So, $$f > 0$$ So, the diverging lens becomes converging. |
