Calculate the energy density of the Earth's atmosphere

  1. Not really homework, just practice for a midterm, I also have the correct answers; but I guess this is the correct section.

    1. The problem statement, all variables and given/known data

    In the Earth's atmosphere we have an electric field with a vertical direction down towards the earth. The lower part of the atmosphere has a typical field strength of 100 V/m. The strength of the earth's magnetic field is approximately 50*10^(-6) T.

    Find the energy density in each of the two layers.

    I assume the "two layers" are the upper and lower layers.

    2. Relevant equations

    [tex] \mbox{Energy density} = \frac{\mbox{Electric energy}}{\mbox{Volume}} = (1/2)\kappa \epsilon_0 E^2[/tex], where k is the Dielectric constant, e_0 is the permittivity of the space (8.85*10^(-12))

    We also have that [tex]\kappa = \frac{E_0}{E}, E = \frac{F}{q_0}[/tex].

    3. The attempt at a solution

    I assume that I could just obtain the energy density directly from using [tex](1/2)\kappa \epsilon_0 E^2[/tex] directly? I know the correct answers should be [tex]u_1 = 4.4\cdot 10^{-8} J, u_2 = 4.4\cdot 10^{-4} J[/tex] However I don't know how to proceed to obtain the variables.
    Last edited: Mar 17, 2012
  2. jcsd
  3. tiny-tim

    tiny-tim 26,016
    Science Advisor
    Homework Helper

    hi phystudent515! :smile:

    i find the question very confusing :redface:

    also the answer … energy density isn't in J, it's in Pa (pascals) :confused:

    have you given us the whole question? ​
  4. Forgive me, the unit for the correct answers are [tex]J/m^3[/tex], I simply misread. However I don't know if this seems more correct or not.

    I did omit some text due to it being translated by hand. I'll try to restate the problem text somewhat better worded (due to translation):

    That is the entire problem text with nothing omitted.

    I have tried the following to obtain the same solutions:

    For the electric field: [tex](1/2)\kappa \epsilon_0 E^2[/tex] = 1/2 * 1 * 8.85*10^(-12) * 100^2 = 4.4 * 10^(-8), which is correct.

    For the magnetic field: I use [tex](1/2) \epsilon_0 E^2 + \frac{B^2}{2\mu_0}[/tex] = 1/2 * 8.85*10^(-12) * 100^2 + (50*10^(-6))^2/(2*4*pi*10^(-7)) = 9.94*10^(-4), which is not correct. I observe however that it is roughly twice that of the result I'm looking for.
  5. tiny-tim

    tiny-tim 26,016
    Science Advisor
    Homework Helper

    hi phystudent515! :smile:

    yes, that second 4.4 is clearly a misprint …

    the typesetter has got bored and typed the number twice! :zzz:

    1/2 B2o is the correct formula

    btw, J/m3 and Pa are the same, see eg
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