Answer:
The value is [tex]P_{max} = 2.11 * 10^{6} \ W[/tex]
Explanation:
From the question we are told that
The maximized electric field strength is [tex]E = 3.0 \ MV/m = 3.0 *10^{6} \ V/m[/tex]
The diameter is [tex]d = 1.5\ cm = 0.015 \ m[/tex]
Generally the cross -sectional area is mathematically represented as
=> [tex]A = \frac{\pi * d^2 }{4}[/tex]
=> [tex]A = \frac{3.142 * 0.015^2 }{4}[/tex]
=> [tex]A = 0.0001767 \ m^2[/tex]
Generally the maximum power is mathematically represented as
[tex]P_{max} = \frac{E_{max}^2}{ 2 * \mu_o * c } * A[/tex]
Here c is the speed of light with value [tex]c = 3.0 *10^{8} \ m/s[/tex]
[tex]\mu_o[/tex] is the permeability of free space with value [tex]\mu_o = 4 \pi *10^{-7} \ H/m[/tex]
[tex]P_{max} = \frac{ (3.0 *10^{6})^2}{ 2 * (4 \pi *10^{-7}) * 3.0 *10^{8} } * 0.0001767[/tex]
=> [tex]P_{max} = 2.11 * 10^{6} \ W[/tex]
The maximum power that can be delivered by a laser beam propagating through air is equal to [tex]2.11 \times 10^6\;Watts[/tex]
Given the following data:
Radius = [tex]\frac{Diameter}{2} = \frac{0.015}{2} = 0.0075\;meter[/tex]
Scientific data:
To determine the maximum power that can be delivered by a laser beam propagating through air:
First of all, we would solve for the area of the laser beam as follows:
[tex]Area = \pi r^2\\\\Area = 3.142 \times 0.0075^2\\\\Area = 3.142 \times 0.00005625\\\\Area = 1.77 \times 10^{-4}\;m^2[/tex]
Now, we can determine the maximum power that can be delivered by a laser beam:
Mathematically, maximum power is given by the formula:
[tex]P_{max} = \frac{AE_{max}^2}{2 \mu_o c}[/tex]
Substituting the given parameters into the formula, we have;
[tex]P_{max} = \frac{1.77 \times 10^{-4} \;\times \;(3 \times 10^6)^2}{2 \;\times \;1.25663706 \times 10^{-6} \; \times \;3 \times 10^8}\\\\P_{max} = \frac{1.77 \times 10^{-4} \;\times \;9 \times 10^{12}}{753.982236}\\\\P_{max} = \frac{1593000000}{753.982236}\\\\P_{max} = 2.11 \times 10^6\;Watts[/tex]
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