Assuming the Earth's magnetic field averages about 0.50×10−4 T near Earth's surface, estimate the total energy stored in this field in the first 15km above Earth's surface. The radius of Earth is 6380 km.
To estimate the total energy stored in Earth's magnetic field, we can consider it as an idealized magnetic dipole. The energy stored in a magnetic dipole is given by the formula:
E = (μ₀/2) * (m² / r³),
where E is the energy, μ₀ is the permeability constant (4π×10⁻⁷ T·m/A), m is the magnetic moment, and r is the distance from the dipole.
First, let's find the magnetic moment, which is the product of the magnetic field strength and the area of the dipole. The area is given by the formula for the surface area of a sphere: 4πr².
Area = 4πr² = 4π(6380 km + 15 km)².
Convert the radius into meters:
Area = 4π(6395000 m)².
Next, calculate the magnetic moment:
magnetic moment = magnetic field strength * area.
m = (0.50×10⁻⁴ T) * Area.
Now, substitute the calculated values into the equation of energy:
E = (μ₀/2) * (m² / r³),
E = (4π×10⁻⁷ T·m/A / 2) * ((0.50×10⁻⁴ T) * Area)² / (15 km + 6380 km)³.
Evaluate the expression to find the total energy stored.