The focal lengths of a thin lens for the red and violet colours are f R and f v respectively. The focal length f of the lens and the dispersive power of its material for the mean colour are :

Question

The focal lengths of a thin lens for the red and violet colours are f R and f v respectively. The focal length f of the lens and the dispersive power of its material for the mean colour are : (A) (fR+fv/2) ; (fR-fv/f) (B) √fR fv ; (fR+fv/2) (C) √fR fv ; (fR-fv/2) (D) (fR+fv/2) ; √fR fv

Tardigrade Admin · Accepted Answer

Focal length of the lens for mean colour (yellow)

f = \frac{f _{R} + f _{v}}{2}

Dispersive power of the material of lens,

ω = \frac{μ _{v} - μ _{R}}{μ - 1}

Using lens formula,

\frac{1}{f} = (μ - 1) [\frac{1}{R _{1}} - \frac{1}{R _{2}}] \dots

(i)

(μ_{v} - 1) = \frac{1}{f _{v} \cdot x}

where,

x = \frac{1}{R _{1}} - \frac{1}{R _{2}}

and

(μ_{R} - 1) = \frac{1}{f _{R} \cdot x}

Subtracting,

μ_{v} - μ_{R} = (\frac{1}{f _{v}} - \frac{1}{f _{R}}) \frac{1}{x} = (\frac{f _{R} - f _{v}}{f _{R} f _{v}}) \frac{1}{x}

μ_{v} - μ_{R} = \frac{f _{R} - f _{V}}{f ^{2} \cdot x}

where

f = f_{R} f_{v}

From eq. (i),

\frac{1}{f} = (μ - 1) [\frac{1}{R _{1}} - \frac{1}{R _{2}}] = (μ - 1) x

Putting the value of

\frac{1}{f}

in eqn. (ii), we get

μ_{v} - μ_{R} = \frac{( f _{R} - f _{V} ) ( μ - 1 ) x}{f x}

or

\frac{μ _{v} - μ _{R}}{μ - 1} = \frac{f _{R} - f _{v}}{f} = ω

Q. The focal lengths of a thin lens for the red and violet colours are $f_{R}$ and $f_{v}$ respectively. The focal length $f$ of the lens and the dispersive power of its material for the mean colour are :

$\frac{f _{R} + f _{v}}{2}; \frac{f _{R} - f _{v}}{f}$

$f_{R} f_{v}; \frac{f _{R} + f _{v}}{2}$

$f_{R} f_{v}; \frac{f _{R} - f _{v}}{2}$

$\frac{f _{R} + f _{v}}{2}; f_{R} f_{v}$

Q. The focal lengths of a thin lens for the red and violet colours are fR​ and fv​ respectively. The focal length f of the lens and the dispersive power of its material for the mean colour are :

2fR​+fv​​;ffR​−fv​​

fR​fv​​;2fR​+fv​​

fR​fv​​;2fR​−fv​​

2fR​+fv​​;fR​fv​​

Q. The focal lengths of a thin lens for the red and violet colours are $f_{R}$ and $f_{v}$ respectively. The focal length $f$ of the lens and the dispersive power of its material for the mean colour are :

$\frac{f _{R} + f _{v}}{2}; \frac{f _{R} - f _{v}}{f}$

$f_{R} f_{v}; \frac{f _{R} + f _{v}}{2}$

$f_{R} f_{v}; \frac{f _{R} - f _{v}}{2}$

$\frac{f _{R} + f _{v}}{2}; f_{R} f_{v}$