Final answer:
To find the energy in the Earth's magnetic field, we use the energy density formula for a magnetic field and multiply it by the volume of the Earth, assuming a uniform field of 0.5 gauss and a radius of 6378 km.
Step-by-step explanation:
To calculate the energy stored in the Earth's magnetic field, we can use the formula for the energy density of a magnetic field, which is \( u = \frac{B^2}{2\mu_0} \), where \( B \) is the magnetic flux density and \( \mu_0 \) is the permeability of free space (\(4\pi \times 10^{-7} \) N/A2).
The total energy is then the energy density multiplied by the volume over which the field is present. In this case, assuming a uniform magnetic field of 0.5 gauss (which is \( 5 \times 10^{-5} \) tesla) over the volume of the Earth with a radius of 6378 km, we can calculate it as follows:
The volume \( V \) of the Earth is given by \( V = \frac{4}{3}\pi r^3 \), where \( r \) is the radius of the Earth in meters. Plugging in the values, we get:
\( V = \frac{4}{3}\pi (6.378 \times 10^6 m)^3 \).
Now, let's calculate the energy density:
\( u = \frac{(5 \times 10^{-5} T)^2}{2 \times 4\pi \times 10^{-7} N/A^2} \).
Finally, the magnetic field energy \( U \) of the Earth is:
\( U = u \times V \).
Note that the answer will be stated to only two significant digits, since the Earth's magnetic field is given to two significant digits in the question. Also, real magnetic fields like Earth's are not actually uniform, but for this exercise, the uniformity assumption simplifies the calculation.