Electromagnetic field theory and Maxwell's equations notes — Unit 2

To study electromagnetism, we first master the mathematical language used to describe it: vector calculus.

• Scalar Field: A region of space where every point has a scalar value assigned to it (e.g., Temperature, Electric Potential $V$).
• Vector Field: A region of space where every point has a vector assigned to it, possessing both magnitude and direction (e.g., Electric Field $\vec{E}$, Magnetic Field $\vec{B}$).

The Del Operator ($\nabla$) is a differential vector operator:

When the $\nabla$ operator is applied to a scalar function $\phi(x,y,z)$, it produces a vector field known as the Gradient.

⇒Physical Significance

The gradient points in the direction of the maximum rate of increase of the function. Its magnitude represents that maximum rate of change. For example, the electric field $\vec{E}$ is the negative gradient of the electric potential $V$:

Divergence is the dot product of the $\nabla$ operator with a vector field $\vec{A}$. The result is a scalar quantity.

⇒Physical Significance

Divergence measures the net outward flow (flux) of a vector field per unit volume from a point.

• Positive Divergence: The point acts as a "source" (e.g., a positive charge).
• Negative Divergence: The point acts as a "sink" (e.g., a negative charge).
• Zero Divergence: The field is "solenoidal" (e.g., magnetic field, since there are no monopoles).

Curl is the cross product of the $\nabla$ operator with a vector field $\vec{A}$. The result is another vector field.

⇒Physical Significance

Curl measures the rotation or "circulation" of a vector field around a point. If $\nabla \times \vec{A} = 0$, the field is called irrotational (or conservative, like the electrostatic field).

These two theorems act as mathematical bridges, allowing us to convert between integrals. They are strictly required to convert Maxwell's equations from Integral form to Differential form.

⇒Gauss's Divergence Theorem

Converts a closed Surface integral into a Volume integral. It states that the total outward flux of a vector field through a closed surface equals the volume integral of its divergence.

⇒Stokes's Theorem

Converts a closed Line integral into a Surface integral. The line integral of a vector field around a closed boundary equals the surface integral of its curl.

Electric flux ($\Phi_E$) is the total number of electric field lines passing normally through a given area.

⇒Gauss's Law

The total electric flux emerging from any closed surface is $1/\epsilon_0$ times the total charge $q$ enclosed by it.

If volume charge density is $\rho$, then $q = \iiint \rho dV$.

To get the differential form, we start with Gauss's Law: $\iint \vec{E} \cdot d\vec{S} = \frac{1}{\epsilon_0} \iiint \rho dV$.

Apply Gauss's Divergence Theorem to the left side:

Equating the two volume integrals:

Since the volume is arbitrary, the integrands must be equal:

In magnetism, isolated magnetic poles (monopoles) do not exist. Magnetic field lines are always continuous, closed loops.

Because whatever flux enters a closed surface must exit it, the net magnetic flux through any closed surface is identically zero.

Applying Divergence Theorem to the surface integral gives the differential form:

Michael Faraday observed that a changing magnetic flux induces an Electromotive Force (EMF) in a circuit. Lenz's law dictates the negative sign (opposition to change).

Since EMF is the work done in moving a unit charge around a closed loop, it equals the line integral of the electric field ($\oint \vec{E} \cdot d\vec{l}$).

Start with Integral form: $\oint \vec{E} \cdot d\vec{l} = -\iint \frac{\partial \vec{B}}{\partial t} \cdot d\vec{S}$.

Apply Stokes's Theorem to the left side to convert line integral to surface integral:

Equate the two surface integrals:

This shows that a time-varying magnetic field produces a circulating (rotational) electric field.

Ampere's Circuital Law states that the line integral of the magnetic field $\vec{B}$ around any closed path is $\mu_0$ times the current enclosed by that path.

⇒The Inconsistency

Applying Stokes's theorem gives $\nabla \times \vec{B} = \mu_0 \vec{J}$. If we take the divergence of both sides, $\nabla \cdot (\nabla \times \vec{B}) = \mu_0 (\nabla \cdot \vec{J})$. The divergence of a curl is always exactly zero. But the continuity equation says $\nabla \cdot \vec{J} = -\partial\rho / \partial t$.

This means Ampere's law implies that charge density $\rho$ can never change (steady currents only). It completely fails for time-varying currents, like charging a capacitor.

To fix the inconsistency, Maxwell hypothesized that a changing electric field also acts as a current, creating a magnetic field. He introduced the Displacement Current ($I_d$).

The total current density is now the sum of conduction current ($\vec{J}_c$) and displacement current ($\vec{J}_d$).

Substituting the total current into Ampere's Law gives the final, generalized equation.

Applying Stokes's Theorem yields the differential form:

In free space or vacuum, there are no charges and no conduction currents. Therefore:

• Volume charge density $\rho = 0$
• Conduction current density $\mathbf{J}_c = 0$

Maxwell's Equations simplify to:

How do we prove light is an electromagnetic wave? We take the curl of Maxwell's 3rd Equation.

Using 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 free space, the left side is just $-\nabla^2 \mathbf{E}$. Substitute the 4th equation into the right side:

The equation $\nabla^2 \mathbf{E} = \mu_0\epsilon_0\frac{\partial^2 \mathbf{E}}{\partial t^2}$ is a standard 3D wave equation of the form $\nabla^2 u = \frac{1}{v^2} \frac{\partial^2 u}{\partial t^2}$.

Comparing the two, the velocity of the wave $v$ (or $c$) in vacuum is:

Plugging in the constants $\mu_0 = 4\pi \times 10^{-7}$ and $\epsilon_0 = 8.85 \times 10^{-12}$, we get exactly $c \approx 3 \times 10^8$ m/s, proving that light is an electromagnetic wave.

Let an EM wave propagate in the $x$-direction. This means spatial variations exist only in $x$ (so $\partial/\partial y = 0$ and $\partial/\partial z = 0$).

From $\nabla \cdot \mathbf{E} = 0$, we get $\partial E_x / \partial x = 0$. This means the electric field has no oscillating component in the direction of propagation.

Similarly from $\nabla \cdot \mathbf{B} = 0$, we find $\partial B_x / \partial x = 0$. Therefore, both $\mathbf{E}$ and $\mathbf{B}$ oscillate strictly perpendicular to the direction of wave propagation, making it a Transverse wave.

⇒Poynting Vector ($\mathbf{S}$)

Represents the rate of flow of electromagnetic energy per unit area per unit time, pointing in the direction of wave propagation.

⇒Poynting Theorem

A statement of the conservation of energy for electromagnetism. It states that the rate of energy transfer into a volume equals the rate of work done on the charges inside plus the rate of increase of electromagnetic energy stored in that volume.

⇒Unit 2 Mastery

• State the physical significance of Gradient, Divergence, and Curl.
• Write the statements of Gauss's Divergence and Stokes's Theorems.
• Explain the concept of Displacement Current and why it is necessary.
• Derive all four Maxwell's Equations in differential form.
• Derive the EM Wave equation in free space and calculate its velocity.
• Prove that electromagnetic waves are transverse in nature.
• State and mathematically define the Poynting Theorem.