For the electromagnetic field described by the equations below, show (a) under what conditions of ω and k does the field satisfy maxwell’s equations? and (b), suppose that ω = 1010 s−1 and e0 = 10 kv/m. what is the wavelength? what is the energy density in joules per cubic meter, averaged over a large region? from this calculate the power density, the energy flow in joules per square meter per second.

a) w / k = c , b)    λ  = 1,887 10⁻¹ m ,   =  4.42 10⁻⁴ J / m³

Explanation:

a) Maxwell's equations when solved for an electromagnetic wave result in

              E = E₀ cos (kx - wt)

             B = B₀ cos (kx -wt)

Where k is the vector ce wave and w the angular velocity

            k = 2π / λ

            w = 2π f

Let's divide the two equations

           w / k = f λ  

          w / k = c

Therefore, for w and k to be a solution to Maxwell's equations, their relationship must be equal to the speed of light.

b) If w = 10¹⁰ s⁻¹

      w = 2π f

       f = w / 2π

       f = 10¹⁰ / 2π

       f = 1.59 10⁹ Hz

The speed of light is

       c = λ  f

       λ  = c / f

       λ  = 3 10⁸ / 1.59 10⁹

       λ  = 1,887 10⁻¹ m

Energy density is

      = ½ ε₀ E₀²

      = ½ 8.85 10⁻¹² (10 10³)²

      =  4.42 10⁻⁴ J / m³

Power is energy per unit of time

          P =   / t

We calculate for every second

         P = 4.42 10⁻⁴ W / m³

The flow or intensity of energy is

           I = S = c u

           I = 3. 108 4.42 10⁻⁴

           I = 1.33 10⁵ W / m2

jupiter is the largest planet in our solar system, having a mass and radius that are respectively, 318 and 11.2 times that of earth. suppose that an object falls from rest near the surface of each planet and that the acceleration due to gravity remains constant during the fall. each object falls the same distance before striking the ground. determine the ratio of the time of fall on jupiter to that on earth.