Chapter 35: Q.43 (page 1019)

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 $5.0 c m$ 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 $5.0 - μ m - d i a m e t e r$ spot in order to have sufficient power to make the cut. what is the minimum diameter of lens you must install?

Expert verified

The minimum diameter of the lens is $2.58 c m$ .

Step by step solution

We have given that:

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

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

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

we need to find the diameter of lens to install.

Let us use this formula,

$ω = \frac{2.44 λ f}{D}$

Here, $f$ is the focal length, $λ$ is wavelength and $ω$ is the value for laser must be focused.

Now multiplying both sides by D,

localid="1649142735572" $D \times ω = D \times \frac{2.44 λ f}{D}$

localid="1649142750541" $D \times ω = 2.44 λ f$

Dividing both the sides by $ω$ ,

$\frac{D \times ω}{ω} = \frac{2.44 λ f}{ω}$

$D = \frac{2.44 λ f}{ω}$

Let us substitute values in the equation,

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

$D = 0.0258 m \to 2.58 c m .$

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

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