Question

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 1.00 J of energy, assuming the strength of the magnetic field to be \(5.00 \cdot 10^{-5} \mathrm{~T} ?\)

Short answer

Answer: The volume of space near the Earth's surface that contains 1 Joule of energy is approximately 1.005 x 10^9 m^3.

Step-by-step solution

Recall the formula for magnetic field energy density

The energy density of a magnetic field is given by the formula: \(u = \frac{1}{2} \frac{B^{2}}{\mu_0}\) Where \(u\) is the energy density, \(B\) is the magnetic field strength, and \(\mu_0 = 4 \pi \cdot 10^{-7} \mathrm{~ T m/A}\) is the permeability of free space.

Calculate the energy density

Using the given magnetic field strength \(B = 5.00 \cdot 10^{-5} \mathrm{~T}\), we can calculate the energy density \(u\): \( u = \frac{1}{2} \frac{(5.00 \cdot 10^{-5} \text T)^2}{4 \pi \cdot 10^{-7} \text T \cdot \text m/\text A}\) Calculate the energy density: \(u = 9.95 \cdot 10^{-10} \mathrm{ \frac{J}{m^3}}\)

Calculate the required volume

Now we can find the volume \(V\) that contains 1 Joule of magnetic field energy by dividing the total energy by the energy density: \(V = \frac{E}{u}\) Where \(E = 1.00 \text J\) (given) and \(u\) is the energy density calculated in step 2. Plug in the values: \(V = \frac{1.00 \text J}{9.95 \cdot 10^{-10} \frac{\text J}{\text m^{3}}}\) Calculate the volume: \(V = 1.005 \cdot 10^9 \text m^{3}\) So, the volume of space near Earth's surface that contains 1 Joule of energy is approximately \(1.005 \cdot 10^9 \mathrm{~m^3}\).

Key concepts

Energy Density

Energy density is a fundamental concept in physics, especially when dealing with fields such as electromagnetism. It refers to the amount of energy stored in a given system or region of space per unit volume. For magnetic fields, this is crucial to understanding many aspects of their behavior and potential for work. In the context of our exercise, the energy density tells us how much energy is contained within a specific volume of the magnetic field.

The formula to calculate the magnetic field's energy density, \( u = \frac{1}{2} \frac{B^{2}}{\mu_0} \), is derived from the work required to establish the magnetic field within a vacuum. Here, \( B \) represents the magnetic field strength, and \( \mu_0 \) is the permeability of free space. When we substitute the Earth's magnetic field into this formula, we can determine the energy per cubic meter, which is important for applications like the inventor's quest to tap into this energy.

Permeability of Free Space

The permeability of free space, denoted by \( \mu_0 \), is a physical constant that is essential to the study of electromagnetism. It represents the extent to which a magnetic field can permeate a vacuum, which is the 'space' free of any material or substance. In simpler terms, it's a measure of how easy it is for a magnetic field to spread out in empty space.

The value of \( \mu_0 \) is \( 4 \pi \cdot 10^{-7} \text T m/A \), which arises naturally in the equations of electromagnetism, and also relates electric and magnetic units in the International System of Units (SI). When calculating the energy density of a magnetic field, as was demonstrated in our exercise, the permeability of free space has to be factored in to account for the natural 'resistance' that space offers to magnetic forces. Understanding this constant is vital for engineers and scientists as they design systems and tools that interface with magnetic fields.

Magnetic Field Strength

Magnetic field strength, symbolized by \( B \), is one of the most significant characteristics of a magnetic field. It describes the force that a magnetic field exerts on moving electric charges and magnetic materials, commonly measured in teslas (T) in the International System of Units (SI). A higher magnetic field strength means a stronger ability to affect the surrounding environment.

In the given exercise, the Earth's magnetic field strength is quoted as \( 5.00 \cdot 10^{-5} \text T \), which is typical for the Earth's surface. Magnetic field strength is especially relevant in technology and physical sciences, influencing everything from electric motors to the behavior of charged particles in space. The inventor who aims to exploit the Earth's magnetic field will rely on accurately gauging its strength to estimate the potential energy that could be harnessed for practical uses.