Q.An infinitely long thin wire carrying a uniform linear static charge density $\lambda$ lies along the $z$-axis. The wire is set into motion along its own length with a uniform velocity $\mathbf{v} = v\,\hat{k}$, i.e. directed along the $+z$-axis. Consider a field point at perpendicular (radial) distance $r$ from the wire, with $\hat{r}$ the radially outward unit vector and $\hat{\phi}$ the azimuthal unit vector circling the wire. Calculate the Poynting vector $\mathbf{S} = \dfrac{1}{\mu_0}(\mathbf{E} \times \mathbf{B})$.

Electromagnetic Wave Relation: Why $c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$

Let's build this from first principles — not just memorising the formula, but understanding why light and all EM waves travel at this specific speed.

1. The Starting Point: Maxwell's Equations in Vacuum

In empty space (no charges, no currents), Maxwell's equations simplify to:

• Gauss's law for electricity: $\nabla \cdot \mathbf{E} = 0$
• Gauss's law for magnetism: $\nabla \cdot \mathbf{B} = 0$
• Faraday's law: $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$
• Ampère-Maxwell law: $\nabla \times \mathbf{B} = \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$

The key insight: a changing electric field creates a magnetic field, and a changing magnetic field creates an electric field. This mutual induction is what sustains the wave.

2. Deriving the Wave Equation for $\mathbf{E}$

Take the curl of Faraday's law:

Now use the vector identity: $\nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E}$

Since $\nabla \cdot \mathbf{E} = 0$ in vacuum, this becomes:

Substitute $\nabla \times \mathbf{B}$ from Ampère-Maxwell:

Result: The electric field satisfies the wave equation:

3. Identifying the Wave Speed

Compare with the standard wave equation for any wave travelling at speed $v$:

Matching terms:

This $v$ is the speed of electromagnetic waves in vacuum — denoted $c$.

Why this is profound: The constants $\mu_0$ (permeability of free space) and $\varepsilon_0$ (permittivity of free space) come from static electricity and magnetism. Yet their combination gives the speed of light — showing light is an electromagnetic wave.

4. The Magnetic Field Follows Suit

Exactly the same derivation starting from Ampère-Maxwell law gives:

So both $\mathbf{E}$ and $\mathbf{B}$ propagate at the same speed $c$.

5. The Crucial Relationship Between $\mathbf{E}$ and $\mathbf{B}$

For a plane wave travelling in the $x$-direction:

• $\mathbf{E}$ oscillates along $y$: $E_y = E_0 \sin(kx - \omega t)$
• $\mathbf{B}$ oscillates along $z$: $B_z = B_0 \sin(kx - \omega t)$

From Faraday's law: $\frac{\partial E_y}{\partial x} = -\frac{\partial B_z}{\partial t}$

Differentiating the wave forms:

Since $\omega = ck$, we get:

$\frac{E_0}{B_0} = \frac{\omega}{k} = c$ …