Question

Why do goggles (sunglasses) have zero power even though their surfaces are curved?

Answer 1

Hint:
Recall what contributes to the power of the lens. In other words, the power of a lens arises from the difference in the radii of curvature of the two spherical surfaces of the lens. If that is the case then arriving at how zero power is obtained is pretty straightforward.
Formula Used:
Power of the lens: $P = (\mu -1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$, where $\mu$ is the refractive index of the lens, $R_1$ is the radius of curvature of one side of the lens and $R_2$ is the radius of curvature of the other side of the lens.

Complete answer:
Let us first quantitatively derive the reason and then justify it with an explanation.
We know that power of the lens is given by the expression:
$P = (\mu -1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$
We are given that sunglasses have zero power.
$\Rightarrow 0 = (\mu -1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$
We know that $(\mu-1) \neq 0$ since the refractive index of lens material is always greater than free space.
$\Rightarrow \left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right) =0 \Rightarrow \dfrac{R_2 -R_1}{R_1 R_2} = 0 \Rightarrow R_2-R_1 = 0 \Rightarrow R_1 = R_2$
This means that zero power of the lens is achieved when the radius of curvature of the two sides of the lens are equal.

This means that the outer and inner surfaces of the sunglasses have the same radius of curvature. This indicates that the two surfaces run parallel to each other and the magnification produced by the outer convex surface gets neutralized by the magnification produced by the inner concave surface and hence produces a net magnification or minification of zero, and the light passes through it and emerges out parallel to the incident ray. Since the power of a lens arises from the difference in the radii of curvature of either spherical surfaces, we can say that the power of the sunglasses are zero for this reason.

Note:
Remember that the purpose of sunglasses is to protect the eye from harmful rays and glare and not magnify or minify objects. Thus, most sunglasses are polarized glasses wherein, they allow vertically polarized light to pass through but obstruct the horizontally polarized light, preventing eye-damage from subsequent glare. The film or vinyl of the sunglasses are responsible for blocking the UV-A and UV-B rays from entering the eye and damaging the eye’s cornea and lens.