In a particular region of Earth's atmosphere, the electric field above Earth's surface has been measured to be 151 N/C downward at an altitude of 280 m and 167 N/C downward at an altitude of 430 m. Calculate the volume charge density of the atmosphere, assuming it to be uniform between 280 and 430 m. (Hint: You may neglect the curvature of Earth. Why?)
I tried using E=pz/ε, where E was 151+167 and z was (430-280)/2, half the distance between the ends, and got, 3.75E-11 C/m^3, but that was wrong as the online software told me. What should I do?
To calculate the volume charge density of the atmosphere, we need to use the Gauss's Law for electricity. The Gauss's Law states that the electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space.
In this case, we will consider a cylindrical Gaussian surface between the altitudes of 280 m and 430 m. Since the electric field is constant in magnitude but changes in direction, we can neglect the curvature of the Earth as indicated in the hint.
The Gaussian surface has a height equal to the difference in heights, which is (430 m - 280 m) = 150 m.
Now let's calculate the charge enclosed within the Gaussian surface using the given electric field values:
Charge enclosed (Q) = Electric field (E) × Area (A)
The area of the cylindrical Gaussian surface is given by:
Area (A) = π × radius²
Since the Gaussian surface is cylindrical, we only need the radius, which is the average of the altitudes:
Radius (r) = (280 m + 430 m) / 2
Next, we calculate the charges associated with each electric field:
Q1 = E1 × A1
Q2 = E2 × A2
Substituting the values, we have:
Q1 = 151 N/C × π × (280 m + 430 m) / 2)^2
Q2 = 167 N/C × π × (280 m + 430 m) / 2)^2
To find the charge enclosed, we calculate the difference between Q2 and Q1:
Q = Q2 - Q1
Finally, we can calculate the volume charge density (ρ) using the formula:
ρ = Q / Volume
The volume of the cylindrical Gaussian surface is given by:
Volume = π × radius² × height
Substituting the known values, we can find the volume charge density.
To determine the volume charge density of the atmosphere, you will need to use the Gauss's law equation. Gauss's law relates the electric field to the charge enclosed within a surface.
Here's how you can approach the problem step by step:
Step 1: Calculate the electric field difference between the two altitudes:
You correctly calculated the electric field difference by adding the electric field values at both altitudes: E_difference = E_430m - E_280m.
Step 2: Calculate the distance between the altitudes:
You also calculated the distance, z, between the two altitudes as (430 m - 280 m)/2 = 75 m.
Step 3: Determine the charge enclosed within the region:
To find the charge enclosed within the region, you need to use Gauss's law, which states that the flux of the electric field through a closed surface is equal to the total charge enclosed divided by the electric constant (ε₀).
The equation for Gauss's law is:
Φ = Q_enclosed / ε₀
The electric flux (Φ) is related to the electric field (E) and the surface area (A) as:
Φ = E * A
Since the electric field is constant between the altitudes, we can write:
Φ = E_difference * A
Step 4: Calculate the charge enclosed:
Rearrange the formula from Step 3 to solve for Q_enclosed:
Q_enclosed = Φ * ε₀
Step 5: Calculate the volume charge density:
Finally, divide the charge enclosed by the total volume to find the volume charge density:
ρ = Q_enclosed / V
Now, perform the calculations using the given values:
- Electric field difference: E_difference = 167 N/C - 151 N/C = 16 N/C downward.
- Distance: z = 75 m.
- Surface area: A = base area of the region = (430 m - 280 m) * A (where A is the cross-sectional area).
To calculate the cross-sectional area (A), we need to assume that the atmosphere is uniform between 280 m and 430 m. Since the atmospheric volume is uniform in this range, the area can be considered as the product of the height (150 m) and the width (the circumference of Earth, which is approximately 40,000 km or 40,000,000 m).
So the cross-sectional area is: A = 150 m * 40,000,000 m ≈ 6,000,000,000 m².
Now we can calculate the charge enclosed (Q_enclosed):
Q_enclosed = E_difference * A * ε₀.
With the values you provided, ε₀ = 8.854 × 10⁻¹² C²/N*m².
The volume charge density (ρ) can be calculated as:
ρ = Q_enclosed / V,
where V is the volume between the two altitudes (V = A * z).
Perform the calculations using the values obtained, and you should be able to find the correct answer.