Having just learned that there is energy associated with magnetic fields, an inventor sets out to tap the energy associated with the Earth's magnetic field. What volume of space near Earth's surface contains \(1 \mathrm{~J}\) of energy, assuming the strength of the magnetic field to be \(5.0 \cdot 10^{-5} \mathrm{~T} ?\)

Understanding the concept of energy density is fundamental when studying magnetic fields. The energy density, represented by the symbol \(u\), is the amount of energy stored in a magnetic field per unit volume. Essentially, it tells us how much magnetic energy is packed into a given space.

The formula for calculating the energy density of a magnetic field is \[u = \frac{1}{2\mu_0} B^2\], where \(B\) represents the magnetic field strength and \(\mu_0\) is the permeability of free space, a constant with the value of \(4\pi \cdot 10^{-7} \mathrm{T\cdot m/A}\). In simpler terms, \(u\) increases with the square of the magnetic field strength, indicating that even small increases in field strength can lead to significant rises in energy density.

When examining the Earth's magnetic field, which is relatively weak, the energy density is accordingly low. This means that to harvest a sizable amount of energy, a large volume of space would be needed. This principle underlies the inventor's quest in the exercise to tap into the energy contained within the Earth's magnetic field.

Magnetic field strength, denoted by \(B\), measures the magnitude of the magnetic field at a specific location. It is expressed in Teslas (T), a unit that quantifies the level of magnetism experienced by a charged particle moving through that field. The magnetic field strength around Earth varies, but near its surface, it is typically around \(5.0 \cdot 10^{-5} \mathrm{T}\).

Despite being invisible to the naked eye, the Earth's magnetic field has real effects, such as the protection it provides from solar winds and the guidance it offers to compass needles. In the context of the exercise, \(B\) represents a key variable in calculating the energy density and, subsequently, the volume of space that would need to be utilized to extract a certain amount of energy from the magnetic field.

Calculating the energy contained in a magnetic field involves using the energy density and the volume of the space in question. Once the energy density \(u\) is known, we can find the volume \(V\) that contains a specific amount of energy \(E\) with the formula \[V = \frac{E}{u}\]. The steps provided demonstrate this calculation, leading to the determination that a colossal volume of approximately \(1.01 \cdot 10^{11} \mathrm{m^3}\) is required to harvest merely 1 Joule of energy from Earth's magnetic field.

This exercise illustrates a practical scenario where understanding the theoretical concepts of magnetic field energy is essential. Although Earth's magnetic field provides various ecosystem services, harnessing a significant amount of energy from it is impractical due to the vast spaces involved, reasserting the value of exploring diverse sources of renewable energy.