Question

For concave mirror, paraxial rays are focused at distance $\frac{R}{2}$ from pole and marginal rays are focused at distance $x$ from pole, then

• A. $x=\frac{R}{2}$
• B. $x=-f$
• C. $x>\frac{R}{2}$
• D. $x<\frac{R}{2}$

Answer 1

Answer: (B)

$x=-f$

$\tan \theta =\frac{PN}{PC}=\frac{PN}{RC}-\left(i\right)$

$\tan 2\theta =\frac{PN}{PF}=\frac{PN}{RF}-\left(ii\right)$

$\begin{array}{c}\tan \theta =\theta \{\text{for paraxial rays}}\\ \tan 2\theta =2\theta \{\text{for paraxial rays}}\end{array}$

$\begin{array}{c}\text{So on dividing (i)}/\left(ii\right)\\ \frac{\theta }{2\theta }=\frac{PF}{PC}=\frac{RF}{RC}\\ \frac{1}{2}=\frac{f}{R}\\ R=2f\end{array}$

$\begin{array}{c}\Rightarrow f=R/2\\ \because \text{focus is on left side}\\ \text{of marginal axis so}\\ \text{it will be}-f\text{.}\\ x=-f\end{array}$