Q. 3 show that the shape factor for minimum spherical aberration is given by \( \sigma=-\frac{2\left(n^{2}-1\right)}{n+2}\left(\frac{s^{\prime}-s}{s^{\prime}+s}\right) \), where \( \mathrm{n} \) is the refractive index of lens, \( s^{\prime} \) and \( s \) are the image and object distances respectively.
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Show that Eq. $(6.36),$ relating the object and image distances measured from the vertices of a lens, reduces to Gauss's Formula [Eq. $(5.17)]$ for thin lenses. Remember that when $s_{o}>0, d_{10}<0$ and when $s_{i}>0, d_{l 2}>0.$
Derive the lens-makers' equation as follows. Consider an object in vacuum at $p_{1}=\infty$ from a first refracting surface of radius of curvature $R_{1}$. Locate its image. Use this image as the object for the second refracting surface, which has nearly the same location as the first because the lens is thin Locate the final image, proving it is at the image distance $q_{2}$ given by $$\frac{1}{q_{2}}=(n-1)\left(\frac{1}{R_{1}}-\frac{1}{R_{2}}\right)$$
Going back to Section $5.2 .3,$ prove that for a thin lens immersed in a medium of index $n_{m}$ $$\frac{1}{f}=\frac{\left(n_{l}-n_{m}\right)}{n_{m}}\left(\frac{1}{R_{1}}-\frac{1}{R_{2}}\right)$$ That done, imagine a double-concave air lens surrounded by water; determine if it's converging or diverging.
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