Calculate the energy density of the Earth's atmosphere

[phystudent515]

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

Homework Statement

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.

Homework Equations

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

We also have that $$\kappa = \frac{E_0}{E}, E = \frac{F}{q_0}$$.

The Attempt at a Solution

I assume that I could just obtain the energy density directly from using $$(1/2)\kappa \epsilon_0 E^2$$ directly? I know the correct answers should be $$u_1 = 4.4\cdot 10^{-8} J, u_2 = 4.4\cdot 10^{-4} J$$ However I don't know how to proceed to obtain the variables.

 

[tiny-tim]

hi phystudent515!

i find the question very confusing

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

have you given us the whole question?​

 

[phystudent515]

hi phystudent515!

i find the question very confusing

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

have you given us the whole question?​

Forgive me, the unit for the correct answers are $$J/m^3$$, 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):

"Due to lightning discharges a separation of charge will be created between the atmosphere and the surface of the Earth. The result of which is a vertical electric field in the atmosphere, pointing down towards the Earth. In the lower part of the atmosphere the field strength is typically 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 fields."

That is the entire problem text with nothing omitted.

I have tried the following to obtain the same solutions:

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

For the magnetic field: I use $$(1/2) \epsilon_0 E^2 + \frac{B^2}{2\mu_0}$$ = 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.

 

[tiny-tim]

hi phystudent515!

yes, that second 4.4 is clearly a misprint …

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

1/2 B²/µ_o is the correct formula

btw, J/m³ and Pa are the same, see eg http://en.wikipedia.org/wiki/Energy_densityhttp://en.wikipedia.org/wiki/Energy_density …

Energy per unit volume has the same physical units as pressure, and in many circumstances is an exact synonym: for example, the energy density of the magnetic field may be expressed as (and behaves as) a physical pressure …

In short, pressure is a measure of volumetric enthalpy of a system.

 

[asteriatic]

To calculate the energy density of the Earth's atmosphere, we will use the formula:

Energy density = (1/2) * kappa * epsilon_0 * E^2

First, we need to obtain the value of kappa, which is the dielectric constant. We can do this by using the formula:

kappa = E_0 / E

where E_0 is the permittivity of space (8.85*10^-12) and E is the electric field strength.

In the lower layer of the atmosphere, the electric field strength is given as 100 V/m. Therefore, kappa = (8.85*10^-12) / (100 V/m) = 8.85*10^-14.

Next, we can calculate the energy density in the lower layer by using the formula:

Energy density = (1/2) * 8.85*10^-14 * (100 V/m)^2 = 4.4*10^-8 J

In the upper layer of the atmosphere, the electric field strength is unknown. However, we can use the fact that E = F/q_0, where F is the force and q_0 is the charge. Since the Earth's magnetic field is approximately 50*10^-6 T, we can use the formula F = q_0 * v * B (where v is the velocity and B is the magnetic field) to calculate the force.

Assuming a velocity of 10 m/s, we get F = (1.6*10^-19 C) * (10 m/s) * (50*10^-6 T) = 8*10^-25 N.

Therefore, the electric field strength in the upper layer is E = F/q_0 = (8*10^-25 N) / (1.6*10^-19 C) = 5*10^-6 V/m.

Plugging this value into the formula for energy density, we get:

Energy density = (1/2) * 8.85*10^-14 * (5*10^-6 V/m)^2 = 4.4*10^-4 J

Therefore, the energy density in the upper layer of the Earth's atmosphere is 4.4*10^-4 J and in the lower layer it is 4.4*10^-8 J.