# Calculate the energy density of the Earth's atmosphere

• phystudent515
In summary: Energy per unit volume is a measure of the power available to do work (or heat or electrical current) inside a unit volume. This is why energy density is often reported in J/m^3, Pa, or Watts/m^3.
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.

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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?​

tiny-tim said:
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.

hi phystudent515!

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 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.

Last edited by a moderator:

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.

## 1. What is energy density?

Energy density is a measure of the amount of energy contained in a given space or volume. It is typically measured in joules per cubic meter (J/m3).

## 2. How do you calculate the energy density of the Earth's atmosphere?

The energy density of the Earth's atmosphere can be calculated by dividing the total amount of energy contained in the atmosphere by its volume. This can be done using various methods, such as measuring the temperature and pressure of the atmosphere and using equations to determine the energy contained in the gas molecules.

## 3. What factors affect the energy density of the Earth's atmosphere?

Several factors can affect the energy density of the Earth's atmosphere, including temperature, pressure, and the composition of gases. For example, warmer temperatures and higher pressures generally result in a higher energy density, while gases with low molecular weights, such as helium, have a higher energy density than heavier gases like nitrogen or oxygen.

## 4. What is the typical energy density of the Earth's atmosphere?

The energy density of the Earth's atmosphere varies depending on location and altitude, but on average, it is around 5.3 Joules per cubic meter (J/m3). This value can also change over time due to fluctuations in temperature, pressure, and other factors.

## 5. Why is it important to calculate the energy density of the Earth's atmosphere?

Calculating the energy density of the Earth's atmosphere is important for understanding the behavior and dynamics of our planet's atmosphere. It can also help us better predict and prepare for weather patterns, as well as study the effects of human activities on the Earth's climate.

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