Silvering Of Lens

Silvered Lens

A silvered lens is a lens that has been coated with a reflective layer, usually of silver or similar material, to transform it into a reflective optical element. This technique combines the features of both lenses and mirrors, making it valuable in various optical applications. Silvering a surface has the effect of converting the lens into a mirror.

As we have learned earlier in a lens, a ray of light undergoes refraction and emerges on the side opposite to the side of the object. In the case of a silvered lens, after refraction, a ray of light is reflected on the silvered surface and the ray emerges on the same side as the object.

If we silvered a convex lens, then that silvered side will act as a concave mirror and similarly, if we silvered the convex lens then the silvered side will act as a convex mirror.

Our objective is to find the effective focal length of this silvered lens. Let us take an example of a silvered convex lens as shown in the given figure.

Now we use the principle of superposition to find the focal length of the silvered lens. See the image given below which shows we are separating the lens and the mirror

See the image given below and see the arrangement. In this arrangement, a ray of light is first refraction by lens L, then it is reflected at the curved mirror M and finally refracted once again at the lens L. Let the object O be located in front of the lens. Let the image from the lens $I_1$ be formed at $v_1$ .

Then, from the lens-makers formula, (Assume the focal length of the lens f_L1) we have

$\frac{1}{v_1}-\frac{1}{u}=\frac{1}{f_{L_1}}$

Now the image $I_1$ formed by the lens will act as an object for the mirror having focal length $\mathrm{f}_{\mathrm{m}}$ Let $\mathrm{I}_2$ be the image formed by the mirror at a distance of $\mathrm{v}_2$. Again applying the formula

_{$\frac{1}{v_2}+\frac{1}{v_1}=\frac{1}{f_m}$}

Now, $I_2$ will be the object for the final refraction at lens L. If $I_3$ be the final image formed at $v$ from the centre of the lens, then we

$
\frac{1}{v}-\frac{1}{v_2}=\frac{1}{f_{L_2}}
$

Now, $f_{L_1}=f_L \quad$ then $\quad f_{L_2}=f_L$

So the above equation becomes

$\begin{aligned} & \frac{1}{v_1}-\frac{1}{u}=\frac{1}{f_L} \\ & \frac{1}{v_2}+\frac{1}{v_1}=\frac{1}{f_m} \\ & \frac{1}{v}-\frac{1}{v_2}=-\frac{1}{f_L}\end{aligned}$

By manipulating the above equation we get,

$\frac{1}{v}+\frac{1}{u}=\frac{1}{f_m}-\frac{2}{f_L}$

So the equivalent focal length will be equal to

$\frac{1}{f_e}=\frac{1}{f_m}-\frac{2}{f_L}$

Silvering of a Lens

In all cases, there will be two refractions and one reflection. On silvering, each lens behaves as a mirror.
The focal length of the equivalent mirror is given by

$\frac{1}{f}=\frac{1}{\left|f_l\right|}+\frac{1}{\left|f_m\right|}+\frac{1}{\left|f_l\right|}=\frac{2}{\left|f_l\right|}+\frac{1}{\left|f_m\right|}$

where

• $f=$ focal length of the equivalent mirror
• $f_l=$ focal length of the lens
• $f_m=$ focal length of the mirror (i.e., silvered face)

Case (a) : The plane surface of a plano-convex lens is silvered.
$f_m=\infty, f_l=\frac{R}{\mu-1}$, where $R$ is the radius of curvature of curved surface.

$
\begin{array}{ll}
\therefore & \frac{1}{f}=\frac{2}{f_l}+\frac{1}{\infty} \\
\Rightarrow & f=\frac{f_l}{2}=\frac{R}{2(\mu-1)}
\end{array}
$
It behaves as a concave mirror.

Case (b) : The spherical surface of a plano-convex lens is silvered.

$
\begin{aligned}
& f_l=\frac{R}{\mu-1}, f_m=\frac{R}{2} \\
\therefore & \frac{1}{f}=\frac{2(\mu-1)}{R}+\frac{2}{R}=\frac{2 \mu}{R}
\end{aligned}
$

Hence, $f=\frac{R}{2 \mu}$.
This behaves as a concave mirror.

Case (c) : The plane surface of a plano-concave lens is silvered.

$
\begin{aligned}
& f_m=\infty,\left|f_l\right|=\frac{R}{\mu-1} \\
\therefore & \frac{1}{f}=\frac{2(\mu-1)}{R} \\
\Rightarrow & f=\frac{R}{2(\mu-1)} .
\end{aligned}
$

This behaves as a convex mirror.

Case (d) : The spherical surface of a plano-concave lens is silvered.

$
\begin{array}{ll}
& \left|f_l\right|=\frac{R}{\mu-1}, f_m=\frac{R}{2} \\
\therefore & \frac{1}{f}=\frac{2(\mu-1)}{R}+\frac{2}{R}=\frac{2 \mu}{R} \\
\text { or, } & f=\frac{R}{2 \mu} .
\end{array}
$

This behaves as a convex mirror.

Case (e) : One surface of a double-convex lens has been silvered.

$
\begin{aligned}
& & f_l=\frac{R}{2(\mu-1)}, f_m=\frac{R}{2} \\
& \therefore & \frac{1}{f}=\frac{4(\mu-1)}{R}+\frac{2}{R}=\frac{4 \mu-2}{R} \\
& \therefore & f=\frac{R}{4 \mu-2} .
\end{aligned}
$

This behaves as a concave mirror.

Case (f) : One surface of a double-concave lens has been silvered

$
\begin{aligned}
& \left|f_l\right|=\frac{R}{2(\mu-1)}, \quad f_m=\frac{R}{2} \\
& \therefore \quad \frac{1}{f}=\frac{4(\mu-1)}{R}+\frac{2}{R}=\frac{4 \mu-2}{R} \\
& \therefore f=\frac{R}{4 \mu-2} .
\end{aligned}
$

This behaves as a convex mirror.

Applications of Silvered Lenses

1. Reflecting Telescopes: Silvered lenses are used in reflecting telescopes where they serve as mirrors to gather and focus light. This design helps in achieving high magnification and resolution for astronomical observations.
2. Vehicle Mirrors: Rear-view mirrors in vehicles often use silvered lenses to reflect light from behind the vehicle, providing drivers with a clearer view of the road.
3. Optical Instruments: In microscopes and cameras, silvered lenses can be used to direct light paths more effectively, enhancing image quality and precision.
4. Scientific Instruments: Various scientific tools employ silvered lenses to manipulate light in specific ways, critical for experiments and measurements requiring accurate optical control.

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Solved Example on Silvering of Lens

Example 1:

The radius of curvature of the face of a plano-convex lens silvered from plane surface is 12 cm and its refractive index is 1.5. At what distance from the lens will parallel rays incident on the convex face converge?

Solution :
Such lens acts as a concave mirror of focal length $f$ given by $\frac{1}{f}=\frac{2}{f_l}+\frac{1}{f_m}$.

Here

$
\begin{aligned}
& \frac{1}{f_l}=\frac{\mu-1}{R} \\
& ==\frac{1.5-1}{12 \mathrm{~cm}}=\frac{1}{24 \mathrm{~cm}}=\frac{1}{24 \mathrm{~cm}}
\end{aligned}
$

and $\frac{1}{f_{\mathrm{m}}}=\frac{1}{\infty}=0$

$
\therefore \quad \frac{1}{f}=2 \times \frac{1}{24 \mathrm{~cm}}=\frac{1}{12 \mathrm{~cm}}
$

or $\quad f=12 \mathrm{~cm}$
$\therefore \quad$ Parallel rays will converge at 12 cm from the lens.

Example 2:

A biconcave lens is made of glass with a refractive index of 1.5 and has radii of curvature of 20 cm and 30 cm . If the 20 cm surface is silvered then the effective focal length of the mirror formed is:

A $\frac{-60}{11} \mathrm{~cm}$

B $\frac{-60}{9} \mathrm{~cm}$

C $\frac{-50}{11 \mathrm{~cm}}$

D $\quad \frac{-50}{9} \mathrm{~cm}$

Solution:
For silvered lens - $\frac{1}{f_e}=\frac{1}{f_m}-\frac{2}{f_l}$

Therefore, focal length of lens $\frac{1}{f_e}=(1.5-1)\left(\frac{1}{30}-\frac{1}{-20}\right)$ or $f_e=24 \mathrm{~cm}$

Focal length of the mirror, $f_m=\frac{R_2}{2}=\frac{-20}{2}=-10 \mathrm{~cm}$

Effective focal length of silverd lens is, $\frac{1}{f_e}=\frac{1}{f_m}-\frac{2}{f_e} \Rightarrow f_e=\frac{-60}{11} \mathrm{~cm}$

The silvered convex lens behaves as a concave mirror of focal length $\frac{60}{11} \mathrm{~cm}$.

Example 3:

For a plano convex lens $(\mu=1.5)$ has radius of curvature 10 cm . It is silvered on its plane surface. Find focal length after silvering.
(a) 10 cm
(b) 20 cm
(c) 15 cm
(d) 25 cm

Solution:

$
\begin{aligned}
& \frac{1}{f}=(\mu-1)\left[\frac{1}{R_1}-\frac{1}{R_2}\right] \\
= & (1.5-1)\left[\frac{1}{\infty}-\frac{1}{(-10)}\right] \\
= & 0.5\left[\frac{1}{10}\right] \Rightarrow f=20 \mathrm{~cm}
\end{aligned}
$

When plane surface is silvered $F=f / 2=20 / 2=10 \mathrm{~cm}$

Summary

Silvering a lens involves coating its surface with a reflective layer, transforming it into a reflective optical element. This process combines the features of lenses and mirrors, making silvered lenses valuable in applications like reflecting telescopes, vehicle mirrors, and scientific instruments. Converting a lens into a mirror, it enhances light collection and focusing capabilities, providing clearer visibility and improved image quality. In reflecting telescopes, for example, silvered lenses enable high-resolution observations, while in vehicles, they offer a wider and clearer rear view.