About 5.12 x 104 m above Earth's surface, the atmosphere is sufficiently ionized that it behaves as a conductor. The Earth and the ionosphere form a giant spherical capacitor, with the lower atmosphere acting as a leaky dielectric. Find the capacitance C of the Earth-ionosphere system by treating it as a parallel plate capacitor.

The fair-weather electric field is about 150 V/m, downward. How much energy is stored in this capacitor?

Due to radioactivity and cosmic rays, some air molecules are ionized even in fair weather. The resistivity of air is roughly 3.00 × 1014 ?· m. Find the resistance of the lower atmosphere that flows between the Earth's surface and the ionosphere.

Calculate total current that flows between the Earth's surface and the ionosphere.

To find the capacitance of the Earth-ionosphere system, we can use the formula for the capacitance of a parallel plate capacitor:

C = ε₀ * A / d

where C is the capacitance, ε₀ is the permittivity of free space (8.85 x 10^-12 F/m), A is the area of the plates, and d is the distance between the plates.

In this case, the atmosphere behaves as a conductor at a height of 5.12 x 10^4 m above the Earth's surface. So, we can consider the Earth's surface and the ionosphere as the plates of the capacitor.

The distance between the plates (d) can be calculated as the difference between the distance from the Earth's surface to the ionosphere (5.12 x 10^4 m) and the radius of the Earth (6.37 x 10^6 m):

d = (5.12 x 10^4 m) - (6.37 x 10^6 m) = -6.31 x 10^6 m (negative since it is directed upward)

The area of the plates (A) can be calculated as the surface area of a sphere with a radius equal to the Earth's radius:

A = 4πr² = 4π(6.37 x 10^6 m)²

Now, we can substitute these values into the capacitance formula to find C.

Next, to find the energy stored in the capacitor, we can use the formula:

E = 0.5 * C * V²

where E is the energy, C is the capacitance, and V is the potential difference (electric field * distance).

In this case, the electric field is given as 150 V/m, and the distance is the same as the previous calculation (-6.31 x 10^6 m). We can calculate V as the product of the electric field and the distance.

Now, to find the resistance of the lower atmosphere, we can use Ohm's Law:

R = ρ * L / A

where R is the resistance, ρ is the resistivity of air, L is the length of the conductor (distance between Earth's surface and the ionosphere), and A is the cross-sectional area of the conductor (area of the plates).

Substitute the given resistivity and the length (the same as the previous calculation) to calculate the resistance.

Finally, to calculate the total current flowing between the Earth's surface and the ionosphere, we can use Ohm's Law:

I = V / R

where I is the current, V is the potential difference (electric field * distance), and R is the resistance (calculated in the previous step).

Substitute the values into the equation to find the total current.

To find the capacitance C of the Earth-ionosphere system, we can use the formula for the capacitance of a parallel plate capacitor:

C = ε₀ * (A / d)

Where:
C is the capacitance
ε₀ is the permittivity of free space (8.85 x 10^-12 F/m)
A is the area of the plates
d is the separation distance between the plates

In this case, the Earth and the ionosphere act as the plates of the capacitor, with the lower atmosphere acting as a dielectric. We need to determine the area and separation distance between the plates.

The Earth's radius is approximately 6.37 x 10^6 m. However, the distance above the Earth's surface is given as 5.12 x 10^4 m. Therefore, the separation distance between the plates will be:

d = (Earth's radius + distance above the surface) - Earth's radius
d = (6.37 x 10^6 + 5.12 x 10^4) m - 6.37 x 10^6 m

Now we can calculate the capacitance:

C = ε₀ * (A / d)
C = 8.85 x 10^-12 F/m * (4π(6.37 x 10^6)^2 m²) / ((6.37 x 10^6 + 5.12 x 10^4) m - 6.37 x 10^6 m)

To find the energy stored in the capacitor, we use the formula:

Energy = 1/2 * C * V^2

Where:
Energy is the stored energy
C is the capacitance
V is the voltage (electric field x separation distance)

The fair-weather electric field is given as 150 V/m. We can use this value to find the voltage:

V = Electric field * separation distance
V = 150 V/m * ((6.37 x 10^6 + 5.12 x 10^4) m - 6.37 x 10^6 m)

Now we can calculate the energy:

Energy = 1/2 * C * V^2

To find the resistance of the lower atmosphere, we can use Ohm's Law:

R = ρ * (L / A)

Where:
R is the resistance
ρ is the resistivity of air (3.00 x 10^14 Ω·m)
L is the length of the path (distance between Earth's surface and the ionosphere)
A is the cross-sectional area of the path

The distance between the Earth's surface and the ionosphere is considered the length of the path, and the cross-sectional area is assumed to be constant. Let's calculate the resistance:

R = 3.00 x 10^14 Ω·m * L / A

Finally, to calculate the total current flowing between the Earth's surface and the ionosphere, we can use Ohm's Law again:

I = V / R

Where:
I is the current
V is the voltage (electric field * separation distance)
R is the resistance

Let's calculate the total current.