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`

Step-by-step 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 = (d)/(2)`

=>
`r = (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 = (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 =  (3,0*10^8 *  8.85*10^(-12) *  (4 *10^6)^2 * 0.00154)/( 2)`

`P = 3.270960 *10^(6) \  W`