Maxwell’s equations are a set of four fundamental laws that describe how electric and magnetic fields interact. These laws, formulated by James Clerk Maxwell in the mid-19th century, are the foundation of classical electrodynamics.
1 Gauss’s Law for Electric Fields
Gauss’s law for electric fields deals with electrostatic fields. It states that the electric flux out of a closed surface is proportional to the charge enclosed by that surface.
To understand this law, we first have to introduce the divergence theorem.
1.1 Mathematical Background: Divergence Theorem
Let $\vec{F}$ be some vector field that acts on particles inside a volume $V$ which is enclosed by a surface $A$. We want to measure the net amount of particles flowing out of $V$ due to $\vec{F}$.
The divergence theorem, also known as Gauss’s theorem, relates (1) the flux of a vector field across a closed surface to (2) the divergence of the vector field within the volume enclosed by the surface. Mathematically, this is written as:
\begin{equation} \oiint_{A} \vec{F} \cdot \mathrm{d} \vec{A} = \iiint_{V} (\vec{\nabla} \cdot \vec{F}) \mathrm{d} V \tag{1} \end{equation}
Let’s try to understand left-hand-side intgeral first.
- The vector field $\vec{F}$ evaluated at a particular point $\vec{x}$ is a vector that tells us how a particle located at $\vec{x}$ will move.
- For particle located at $\vec{x}$ to leave $A$, there has to be a component of $\vec{F}$ at $\vec{x}$ that is normal to $A$, otherwise there will be no outward flow.
- If we consider a small surface area $\mathrm{d} A$, having a normal vector $\hat{n}$, the outward flow, a quantity usually termed as flux, due to $\vec{F}$ across $\mathrm{d} A$ is $\vec{F} \cdot (\mathrm{d} A \hat{n})$ or $\vec{F} \cdot \mathrm{d} \vec{A}$.
- To measure the total flow out of $V$ due to $\vec{F}$, we need to compute $\vec{F} \cdot \mathrm{d} \vec{A}$ over the entire surface. This gives rise to the surface integral on the left-hand-side of equation (1).
Now, for the integral on the right-hand-side.
- The vector field $\vec{F}$ evaluated at a particular point $\vec{x}$ is a vector that tells us how a particle located at $\vec{x}$ will move.
- When particles move, three things can happen: the density of particles in some regions increases, the density of particles in some regions decreases, or the density remains the same.
- If we consider an arbitrary point $\vec{x}$, then the density of particles
at $\vec{x}$ due to $\vec{F}$ can be measured by the divergence of $\vec{F}$
at $\vec{x}$, written as $\vec{\nabla} \cdot \vec{F}$.
- A region with $\vec{\nabla} \cdot \vec{F} < 0$ implies that that particles are flowing into that region. Such a region is sometimes called a sink.
- A region with $\vec{\nabla} \cdot \vec{F} > 0$ implies that that particles are flowing out of that region. Such a region is sometimes called a source.
- A region with $\vec{\nabla} \cdot \vec{F} = 0$ implies that that the number of particles flowing into the region is equal to the number of particles flowing out of it.
- Going back to the original problem of measuring the flow of particles outside the volume $V$, it is clear that for there to be a net flow of particles out of $V$, the algebraic sum of $\vec{\nabla} \cdot \vec{F}$ at all points inside $V$ must be greater than 0. This gives rise to the integral on the right-hand-side of equation (1).
1.2 Gauss’s Law
1.2.1 Differential Form
In point form, Gauss’s law is expressed as:
\begin{equation} \vec{\nabla} \cdot \vec{D} = \rho _{v} \tag{2} \end{equation}
This basically says that if we have a point with a positive (negative) charge density $\rho _{v}$, then all the electric field lines point away from (towards) that point.
1.2.2 Integral Form
In the integral form, Gauss’s law is expressed as:
\begin{equation} \oiint_{A} \vec{D} \cdot \mathrm{d} \vec{A} = \iiint_{V} \rho _{v} \mathrm{d} V = Q \tag{3} \end{equation}
This basically says that electric fields are due to charges. If you have a real or imaginary closed surface of any size and shape and there is no charge inside the surface, the electric flux through that surface must be zero. If you were to place some positive charge anywhere inside the surface, the electric flux through the surface would be positive. If you then added an equal amount of negative charge inside the surface (making the total enclosed charge zero), the flux would again be zero.
2 Gauss’s Law for Magnetic Fields
Gauss’s law for magnetic fields ($\vec{B}$) is similar in form but different in content from Gauss’s law for electric fields. Let’s first state the law and then point out differences.
2.1 Differential Form
In differential form:
\begin{equation} \vec{\nabla} \cdot \vec{B} = 0 \tag{4} \end{equation}
2.2 Integral Form
In integral form:
\begin{equation} \oiint_{A} \vec{B} \cdot \mathrm{d} \vec{A} = 0 \tag{5} \end{equation}
The key difference in the electric field and magnetic field versions of Gauss’s law arises from the fact that opposite electric charges (called “positive” and “negative”) may be isolated from one another, while opposite magnetic poles (called “north” and “south”) always occur in pairs. Because magnetic poles always occur in pairs, $\vec{\nabla} \cdot \vec{B}$ is always zero.
If you have a real or imaginary closed surface of any size or shape, the total magnetic flux through that surface must be zero. Note that this does not mean that zero magnetic field lines penetrate the surface – it means that for every magnetic field line that enters the volume enclosed by the surface, there must be a magnetic field line leaving that volume. Thus, the inward (negative) magnetic flux must be exactly balanced by the outward (positive) magnetic flux. This is exactly the same as saying that magnetic field lines form a closed loop. In other words, it is not possible to isolate magnetic poles.
3 Faraday’s Law of Electromagnetic Induction
Faraday demonstrated that an electric current may be induced in a circuit by changing the magnetic flux enclosed by the circuit.
3.1 Differential Form
The differential form of Faraday’s law is:
\begin{equation} \vec{\nabla} \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} \tag{6} \end{equation}
In point form, the above equation tells us that a circulating electric field is produced by a magnetic field that changes with time.
Note that the word “circulating” is emphasized to point out the difference between charge-based electric fields and those produced by changing magnetic fields. Since charge-based electric fields diverge away from points of positive charge and converge toward points of negative charge, such fields cannot circulate back on themselves. In other words, $\vec{\nabla} \times \vec{E} = 0$ for charge-based electrostatic fields. It is easy to see why this is true since $\vec{E}$ is always perpendicular to the surface of the sphere that encloses the charges.
3.2 Integral Form
For the macroscopic view of Faraday’s law, we make use of Stoke’s theorem:
\begin{equation} \oint_{C} \vec{E} \cdot \mathrm{d} \vec{r} = \iint_{A} (\vec{\nabla} \times \vec{E}) \cdot \mathrm{d} \vec{A} = - \frac{\partial}{\partial t} \left( \iint_{A} \vec{B} \cdot \mathrm{d} \vec{A} \right) \tag{7} \end{equation}
The LHS represents the work down by the electric field in moving charges along the boundary curve $C$. This circulating electric field is induced by a changing magnetic field through the open (not closed, otherwise the RHS term is 0) surface $A$. The circulation of the electric field around a circuit has come to be known as an “electromotive force” (emf).
So, moving a magnet toward or away from a conducting path (such as a loop), causes the magnetic flux through the surface bounded by the loop to change, resulting in an induced emf around the boundary of the loop.
4 Ampere-Maxwell Law
For thousands of years, the only known sources of magnetic fields were certain iron ores and other materials that had been accidentally or deliberately magnetized.
“Ampere’s law” relating a steady electric current to a circulating magnetic field was well-known by the time Maxwell began his work in the 1850s. However, Ampere’s law was known to apply only to static situations involving steady currents. It was Maxwell’s addition of another source term – a changing electric flux – that extended the applicability of Ampere’s law to time-dependent conditions. More importantly, it was the presence of this term in the equation, now called the Ampere–Maxwell law, that allowed Maxwell to discern the electromagnetic nature of light and to develop a comprehensive theory of electromagnetism.
4.1 Differential Form
The differential form of the Ampere-Maxwell law is:
\begin{equation} \vec{\nabla} \times \vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t} \tag{8} \end{equation}
In words: a circulating magnetic field is produced by a steady electric current $\vec{J}$ and by an electric field $\vec{D}$ that changes with time.
The left side of the differential form of the Ampere–Maxwell law represents the curl of the magnetic field. All magnetic fields, whether produced by electric currents or by changing electric fields, circulate back upon themselves and form continuous loops. Hence, the word circulating was redundant.
Since magnetic fields always form closed-loops, the divergence of the curl of $\vec{H}$ is always equal to 0.
\[ \vec{\nabla} \cdot (\vec{\nabla} \times \vec{H}) = \vec{\nabla} \cdot \left( \vec{J} + \frac{\partial \vec{D}}{\partial t} \right) = 0 \]
\[ \vec{\nabla} \cdot \vec{J} = - \vec{\nabla} \cdot \left( \frac{\partial \vec{D}}{\partial t} \right) \]
Since $\vec{\nabla} \cdot$ and $\frac{\partial}{\partial t}$ are linear operators, they can be interchnaged:
\[ \vec{\nabla} \cdot \vec{J} = - \frac{\partial}{\partial t} (\vec{\nabla} \cdot \vec{D}) \]
Recall from Gauss’s law for electrostatic fields that $\vec{\nabla} \cdot \vec{D} = \rho _{v}$. Therefore:
\begin{equation} \vec{\nabla} \cdot \vec{J} = - \frac{\partial}{\partial t} \rho _{v} \tag{9} \end{equation}
The above equation tells us that local sources of current can only exist if the charge density at that local point changes over time. In the absence of $\frac{\partial \vec{D}}{\partial t}$, we would have arrived at $\vec{\nabla} \cdot \vec{J} = 0$, which is wrong since local sources of current can only be given by a local change of charge. The $\frac{\partial \vec{D}}{\partial t}$ term was added to Ampere’s law to account for this.
4.2 Integral Form
For the macroscopic view of Faraday’s law, we make use of Stoke’s theorem:
\begin{equation} \oint_{C} \vec{H} \cdot \mathrm{d} \vec{r} = \iint_{A} (\vec{\nabla} \times \vec{H}) \cdot \mathrm{d} \vec{A} = \iint_{A} \vec{J} \cdot \mathrm{d} \vec{A} + \frac{\partial}{\partial t} \iint_{A} \vec{D} \cdot \mathrm{d} \vec{A} \tag{10} \end{equation}
The left side of this equation is a mathematical description of the circulation of the magnetic field around a closed path $C$. The right side includes two sources for the magnetic field; a steady conduction current and a changing electric flux through any open surface $A$ bounded by path $C$.
5 Kirchoff’s Circuit Laws
Kirchhoff’s laws, which include Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL), are essentially simplifications of Maxwell’s equations applied to circuit analysis. They provide a powerful approach to understanding electrical circuits without needing to consider the full complexity of Maxwell’s equations.
Kirchhoff’s Current Law (KCL): KCL states that the algebraic sum of currents entering and leaving a node in a network is equal to zero.
Kirchhoff’s Voltage Law (KVL): KVL states that the algebraic sum of the potential differences (voltages) in any loop or mesh in a network is equal to zero.
5.1 Kirchoff’s Voltage Law (KVL)
Kirchoff’s Voltage Law can be derived from Maxwell’s third equation. From Faraday’s Law, we know that the electric field around a closed loop is equal to the negative rate of change of magnetic flux through the loop.
\[ \oint_{C} \vec{E} \cdot \mathrm{d} \vec{r} = - \frac{\partial}{\partial t} \left( \iint_{A} \vec{B} \cdot \mathrm{d} \vec{A} \right) \]
In the absence of a changing magnetic field (which is common in simple circuits), the integral of the electric field around a closed loop is zero.
\[ \oint_{C} \vec{E} \cdot \mathrm{d} \vec{r} = 0 \]
This is basically Kirchoff’s Voltage Law.
5.2 Kirchoff’s Current Law (KCL)
Kirchoff’s Voltage Law can be derived from Maxwell’s fourth equation. Recall from equation (9) that we can only have local sources of current if the charge density at that local point changes over time.
\[ \vec{\nabla} \cdot \vec{J} = - \frac{\partial \rho _{v}}{\partial t} \]
Integration of the above equation across a finite volume $V$ bounded by the outer surface $A$ and using Gauss’ law yields:
\begin{equation} \oiint_{A} \vec{J} \cdot \mathrm{d} \vec{A} = \frac{\partial}{\partial t} \iiint_{V} \rho _{v} \mathrm{d} V = - \frac{\partial Q}{\partial t} \tag{11} \end{equation}
This means that (due to Gauss’ law) summing up all local sources of the current density $\vec{J}$ (i.e., $\vec{\nabla} \cdot \vec{J}$) in a volume $V$ corresponds to the negative change with respect to time of all the charges $Q$ in that volume.
If the change of charge with respect to time in that volume is zero, then the volume contains no local sources (or sinks) of current. Hence, $\oiint_{A} \vec{J} \cdot \mathrm{d} \vec{A} = 0$, which gives us Kirchhoff’s current law.
5.3 Lumped Circuit Model
In the realm of electrical circuit theory, we often make an assumption known as the lumped element model. This model assumes that the physical dimensions of circuit components are much smaller than the wavelength of the signal passing through them, meaning that the signal’s phase is the same everywhere within a component at any given moment. Under these conditions, the properties of the components (like resistance, capacitance, inductance) can be described by a single value (a ‘‘lumped’’ value), and Kirchhoff’s laws apply perfectly.
However, as we start to deal with higher frequency signals (like those in radio frequency or optical circuits), the wavelength of the signals can become comparable to, or even smaller than, the physical dimensions of the components and the circuit as a whole. When this happens, the phase of the signal can vary across a single component or conductor, invalidating the assumptions of the lumped element model.
For instance, consider the voltage around a loop in a circuit. According to KVL, the sum of the voltage differences (potential differences) around any closed loop in a circuit is equal to zero. This is only valid when we can ignore the effect of the changing magnetic field, which can vary significantly over the scale of the circuit, leading to a non-zero line integral of the electric field around a loop (i.e., a non-zero total voltage around a loop). This is a consequence of Faraday’s Law:
\[ \oint_{C} \vec{E} \cdot \mathrm{d} \vec{r} = - \frac{\partial}{\partial t} \left( \iint_{A} \vec{B} \cdot \mathrm{d} \vec{A} \right) \]
If the magnetic field through the loop is changing quickly, which can happen at high frequencies, then the right-hand side of this equation can be non-zero. This introduces an additional voltage term that wouldn’t exist under the assumptions of the lumped element model, and hence KVL can be violated.
This does not mean that Maxwell’s equations are violated – on the contrary, Maxwell’s equations always hold. It’s just that Kirchhoff’s laws, as simplifications of Maxwell’s equations based on certain assumptions, are no longer valid under these conditions. When we reach these high frequencies, we must go back to Maxwell’s equations to describe the system accurately. This leads into the domain of waveguides, transmission lines, and antenna theory, where the effects of varying electric and magnetic fields over space must be taken into account.