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Volume charge density

  1. Jan 31, 2006 #1
    In a particular region of the earth's atmosphere, the electric field above the earth's surface has been measured to be 155 N/C downward at an altitude of 271 m and 175.5 N/C downward at an altitude of 409 m. The permittivity of free space is 8.85 x 10^-12 C^2/Nm^2. Neglecting the curvature of the earth, calculate the volume charge density of the atmosphere assuming it to be uniform between 271 m and 409 m. Answer in units of C/m^3.

    First I figured out the 2 equations for flux and set them equal to each other. [tex] \Phi = q_e_n_c / E_o [/tex] and [tex] \Phi = E cos \theta * A [/tex] Since theta= 90, cos 90=1, so it's just * A.
    To find the area of the sphere, I took the area of the larger sphere and subtracted the smaller one.
    so [tex] A= (E_R * 4\pi R^2)- (E_r * 4\pi r^2) [/tex]
    So then [tex] q_e_n_c/ E_o = (E_RA- E_rA) [/tex]
    q_enc= (E_RA- E_rA)*E_o
    since it's charge per volume, you divide each side by the volume and solve for the charge.
    so [tex] q_e_n_c = (E_RA - E_rA)*E_o / 4/3 \pi (R^3-r^3) [/tex]
    i was a little unsure of what each variable was in the problem, but I think
    E_r = 155 N/C
    E_R = 175.5 N/C
    r= 271 m
    R= 409
    So plugging in gives (155* 4pi (271)^2)=1.43 x 10^8 for E_rA
    E_RA = (175.5 * 4pi (409)^2) = 3.69 x 10^8
    so 2.26x 10^8/ (4/3)pi (409^3 -271^3)) =1.11 C/m^3
    Can someone tell me what I'm doing wrong?
  2. jcsd
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