An electric field and a magnetic field have the same energy density. Find an expression for the ratio \(E / B\) and evaluate this ratio numerically. What are its units? Is your answer close to any of the fundamental constants listed inside the front cover?

Think of an electric field as an invisible force surrounding electric charges, influencing other charged particles within its reach. It's a bit like gravity, but for charged particles—it pulls or pushes them, depending on the charge type.

Mathematically, this field is represented by the variable 'E' and quantified in units of force per charge (newtons per coulomb, N/C). The strength of an electric field determines how much oomph it imparts on a charged particle. The stronger the field, the more significant the force it exerts. So, when we're talking about energy density of an electric field, we're discussing how much energy is stored in a particular volume of space due to this field's presence. The formula for the energy density is \( u_e = \frac{1}{2} \epsilon_0 E^2 \) where \( \epsilon_0 \) is the permittivity of free space.

Much like its electric counterpart, a magnetic field is another kind of field, made evident by the force it exerts on moving charged particles and magnetic materials. Represented by the symbol 'B', a magnetic field's presence can be derived from electric currents or the intrinsic magnetic moments of elementary particles.

The units for magnetic field strength are teslas (T), and it is crucial in many everyday technologies, from hard drives to MRI machines. The energy density of a magnetic field, similar to the electric field, tells us about the energy trapped within a specific space, given by \( u_b = \frac{1}{2} \frac{1}{\mu_0} B^2 \) where \( \mu_0 \) is the permeability of free space.

The permittivity of free space, denoted by \( \epsilon_0 \), is a fundamental constant that features prominently in the equations of electromagnetism. This constant essentially dictates how much electrical resistance is encountered when forming an electric field in a vacuum.

Permittivity affects the capacity to store energy within an electric field; higher permittivity implies more energy can be stored. Permittivity also determines the force of attraction or repulsion between two charges in a vacuum. Its exact value is approximately \( 8.854 \times 10^{-12} \) farads per meter (F/m).

On the flip side, the permeability of free space, symbolized by \( \mu_0 \), is another universal constant, but this time it's all about magnetism. It signifies how well a magnetic field can permeate through a vacuum.

The permeability of free space is integral in determining the force of attraction or repulsion between two magnetic poles. It also contributes to how energy is stored in a magnetic field. With a defined value of approximately \( 4\pi \times 10^{-7} \) henries per meter (H/m), \( \mu_0 \) is fundamental in calculating the speed of light as it relates to the relationship between electric and magnetic fields.

Now, let's sprint over to the speed of light, represented by the letter 'c'. It's not just a breakneck velocity; it's a pivotal element in understanding the relationship between electric and magnetic fields. It's the cosmic speed limit at an approximate \( 3.0 \times 10^8 \) meters per second.

Why is it important here? The speed of light relates to both electric and magnetic fields through Maxwell's equations. Notably, it serves as a bridge between the energy densities of electric and magnetic fields by linking them with \( E / B = c \). This relationship implies that at a point in space where both fields have the same energy density, the field strengths are related by the speed of light. It's not just a speed—it's the underlying rhythm of the electromagnetic universe.