Question

What is the maximum power that can be delivered by a 1.4-cm-diameter laser beam propagating through air

Answer 1

Complete Question

The maximum electric field strength in air is 4.0 MV/m. Stronger electric fields ionize the air and create a spark. What is the maximum power that can be delivered by a 1.4-cm-diameter laser beam propagating through air

Answer:

The  value  is  $P = 3.270960 *10^{6} \  W$

Explanation:

From the question we are told that

   The  electric field strength is  $E = 4.0 \ M \ V/m = 4.0 *10^6 \ V/m$

    The  diameter is  $d = 1.4 \ cm = 0.014 \ m$

Generally the radius is mathematically represented as

        $r = \frac{d}{2}$

=>     $r = \frac{0.014}{2}$

=>    $r = 0.007 \ m$

Generally the cross-sectional area is mathematically represented as

        $A =  \pi r^2$

        $A =  3.142 *  (0.007)^2$

       $A = 0.000154 \ m^2$

Generally the maximum power that can be delivered is mathematically represented as

        $P = \frac{c *  \epsilon_o  *  E^2 *  A }{2}$

Here c is the speed of light with value  $c =  3.0*10^{8} \  m/s$

        $\epsilon_o$ is the permittivity of free space with value  $\epsilon_o  =  8.85 *10^{-12}  \ m^{-3} \cdot kg^{-1}\cdot  s^4 \cdot A^2$

     $P =  \frac{3,0*10^8 *  8.85*10^{-12} *  (4 *10^6)^2 * 0.00154}{ 2}$

       $P = 3.270960 *10^{6} \  W$