Question

Show directly from Maxwell's equations that the charge density \(\rho\) and the electric current density \(j\) obey the conservation law $$ \frac{\partial \rho}{\partial t}+\nabla \cdot j=0 $$

Short answer

By taking the time derivative of Gauss's Law for electricity and the divergence of the Ampère-Maxwell Law, and then combining the results, it can be shown that \(\frac{\partial \rho}{\partial t} + abla \cdot \textbf{j} = 0\), which is the continuity equation indicating the conservation of charge.

Step-by-step solution

State Gauss's Law for Electricity

Recall Gauss's Law for electricity which relates the electric field \textbf(E) to the charge density \textbf(\(\rho\)): \[abla \cdot \textbf{E} = \frac{\rho}{\epsilon_0}\], where \(\epsilon_0\) is the vacuum permittivity.

State the Ampère-Maxwell Law

State the Ampère-Maxwell Law which extends Ampère's Law by including Maxwell's addition of the displacement current density: \[abla \times \textbf{B} = \mu_0 \textbf{j} + \mu_0 \epsilon_0 \frac{\partial \textbf{E}}{\partial t}\], where \(\textbf{B}\) is the magnetic field, \(\mu_0\) is the vacuum permeability, and \(\textbf{j}\) is the current density.

Take the Time Derivative of Gauss's Law

Differentiate Gauss's Law with respect to time to obtain a relationship between the rate of change of charge density and the electric field: \[\frac{\partial}{\partial t}\left(abla \cdot \textbf{E}\right) = \frac{1}{\epsilon_0}\frac{\partial \rho}{\partial t}\].

Apply the Divergence to the Ampère-Maxwell Law

Take the divergence of both sides of the Ampère-Maxwell Law, noting that the divergence of a curl is zero: \[abla \cdot (abla \times \textbf{B}) = abla \cdot \left(\mu_0 \textbf{j} + \mu_0 \epsilon_0 \frac{\partial \textbf{E}}{\partial t}\right)\], which simplifies to \[0 = \mu_0abla \cdot \textbf{j} + \mu_0 \epsilon_0 \frac{\partial}{\partial t}\left(abla \cdot \textbf{E}\right)\].

Combine the Results

Combine the results from steps 3 and 4 to get: \[\mu_0 \epsilon_0 \frac{1}{\epsilon_0}\frac{\partial \rho}{\partial t} = -\mu_0abla \cdot \textbf{j}\]. Simplify this equation using the fact that \(\mu_0 \epsilon_0 = \frac{1}{c^2}\) where c is the speed of light in vacuum, and the fact that these constants can be canceled out due to equality, resulting in the continuity equation: \[\frac{\partial \rho}{\partial t} + abla \cdot \textbf{j} = 0\].

Key concepts

Charge Density Conservation

Charge density conservation is a fundamental principle in electromagnetism, reflecting the law of conservation of electric charge. According to this law, electric charge cannot be created or destroyed; it can only be transferred from one place to another.

This concept is captured mathematically by the continuity equation: \[\frac{\partial \rho}{\partial t} + abla \cdot \textbf{j} = 0\]In this equation, the term \(\frac{\partial \rho}{\partial t}\) represents the time rate of change of charge density \(\rho\) at a given point, and \(abla \cdot \textbf{j}\) denotes the divergence of the current density \(\textbf{j}\), which is a measure of how much electric current is flowing out of a small volume of space. When added together, these two terms must cancel out if charge is conserved. This equation is a local statement, meaning it must hold at every point in space and time.

Gauss's Law for Electricity

Gauss's Law for Electricity is one of Maxwell's four equations that underpin classical electromagnetism. It provides a powerful relationship between electric charge and the resulting electric field. The law is encapsulated by the equation:\[abla \cdot \textbf{E} = \frac{\rho}{\epsilon_0}\]This implies that the electric field (\(\textbf{E}\)) emanating from a volume is proportional to the charge density (\(\rho\)) contained within that volume. The proportionality constant \(\epsilon_0\) is the vacuum permittivity, representing how much resistance is encountered when forming an electric field in the vacuum.

Gauss's Law can be applied using a conceptual surface known as a 'Gaussian surface' to simplify the calculation of electric fields, especially when dealing with symmetric charge distributions.

Ampère-Maxwell Law

The Ampère-Maxwell Law is another of Maxwell's equations that extends the original Ampère's Law by adding a term for the displacement current density. This law relates electric currents and changes in electric fields to magnetic fields, with the equation written as:\[abla \times \textbf{B} = \mu_0 \textbf{j} + \mu_0 \epsilon_0 \frac{\partial \textbf{E}}{\partial t}\]In this formulation, \(\textbf{B}\) represents the magnetic field, while \(\textbf{j}\) is the electric current density. The term \(\frac{\partial \textbf{E}}{\partial t}\) accounts for the changing electric field that contributes to the displacement current. The constants \(\mu_0\) and \(\epsilon_0\) represent vacuum permeability and permittivity respectively. This law demonstrates the deep connection between time-varying electric fields and magnetic fields, and it is crucial for the understanding of electromagnetic waves.

Continuity Equation in Electromagnetism

The Continuity equation in electromagnetism is a mathematical expression of charge density conservation, linking the change in charge density to the current density within a region. Deriving the continuity equation involves taking the time derivative of Gauss's Law and the divergence of the Ampère-Maxwell Law:

By differentiating Gauss's Law with respect to time, you get a relationship for \(\frac{\partial \rho}{\partial t}\) (where \(\rho\) is charge density) and by taking the divergence of the Ampère-Maxwell Law, you show that the divergence of the magnetic field \(\textbf{B}\) is zero.

Ultimately, these steps lead to the equation:\[\frac{\partial \rho}{\partial t} + abla \cdot \textbf{j} = 0\]which states that the rate of change of charge density within a volume is always balanced by the flow of current into or out of the volume. Thus, the total charge is conserved over time. This principle is at the heart of understanding how electric and magnetic fields interact dynamically in space and time.