A laser produces a heat spot on a flat metal sheet with intensity profile I(x,y) =…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Mathematical Methods for Physics, Ch Multiple integrals and change of variables

Problem

A laser produces a heat spot on a flat metal sheet with intensity profile

I(x,y) = I_0\,e^{-(x^2+y^2)/b^2},

where

I_0 = 500\,\mathrm{W\,m^{-2}}

b = 0.020\,\mathrm{m}

, and

x, y

are measured in metres from the centre of the spot. The intensity is negligible beyond a few multiples of

b

, so the total power delivered to the sheet may be found by integrating

I

over the whole plane. Using

\int_{-\infty}^{\infty} e^{-a x^2}\,\mathrm{d}x = \sqrt{\pi/a}

, find the total power.

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