Question

High-power lasers are used to cut and weld materials by focusing the laser beam to a very small spot. This is like using a magnifying lens to focus the sun light to a small spot that can burn things. As an engineer you have designed a laser cutting device in which the material to be cut is placed $5.0cm$behind the lens. you have selected a high-power laser with a wavelength of

your calculation indicates that the laser must be focused to a $5.0-\mu m-diameter$spot in order to have sufficient power to make the cut. what is the minimum diameter of lens you must install?

Short answer

The minimum diameter of the lens is$2.58cm$.

Step-by-step solution

Given information

We have given that:

High power laser wavelength $\lambda =1.06x{10}^{-6}m$

Laser cutting device$f=5cm\to 5.0x{10}^{-2}m$

sufficient power to cut laser $\omega =5\mu m\to 5.0x{10}^{-6}m$

we need to find the diameter of lens to install.

Simplification

Let us use this formula,

$\omega =\frac{2.44\lambda f}{D}$

Here, $f$is the focal length, $\lambda$is wavelength and $\omega$ is the value for laser must be focused.

Now multiplying both sides by D,

localid="1649142735572" $D\times \omega =\cancel{D}\times \frac{2.44\lambda f}{\cancel{D}}$

localid="1649142750541" $D\times \omega =2.44\lambda f$

Dividing both the sides by $\omega$,

$\frac{D\times \cancel{\omega }}{\cancel{\omega }}=\frac{2.44\lambda f}{\omega }$

$D=\frac{2.44\lambda f}{\omega }$

Let us substitute values in the equation,

$D=\frac{2.44(1.06x{10}^{-6}m)(5.0x{10}^{-2}m)}{5.0x{10}^{-6}m}$

$D=0.0258m\to 2.58cm.$

Here,$D$is the diameter of the lens.