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2.8C: Power of a Mirror

  • Page ID
    9096
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    In Figure II.13 shows a reflecting surface of radius of curvature \(r\) submerged in a medium of index \(n\). I show a real object at O, a virtual image at I and the centre of curvature at C. We see that \(h = \alpha p = \beta q = \gamma r\). By Euclid, \(\theta = \alpha + \gamma\) and \(2\theta = \alpha + \beta\). Remember again that all angles are supposed to be small (even \(\beta\)!), in spite of the drawing. From these we obtain

    \[ \frac{1}{q} = \frac{1}{p} + \frac{2}{r}. \label{eq:2.8.4} \]

    On multiplying this by \(−n\), we find that the power is \(−2n/r\). Again the reader should try this for other situations, such a concave mirror, or a real image, and so on. The same result will always be obtained.

    FIGURE 11.13 .png


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