Question

Assume from electrostatics the equations \(\nabla \cdot \mathbf{E}=\rho / \epsilon_{0}\) and \(\mathbf{E}=-\nabla \phi(\mathbf{E}=\) electric field, \(\left.\rho=\text { charge density, } \epsilon_{0}=\text { constant, } \phi=\text { electrostatic potential }\right)\) Show that the electrostatic potential satisfies Laplace's equation (1.1) in a charge-free region and satisfies Poisson's equation (1.2) in a region of charge density \(\rho\)

Short answer

Poisson's equation: \(\abla^2 \phi = \frac{\rho}{\epsilon_0}\). In a charge-free region, it simplifies to Laplace's equation: \(\abla^2 \phi = 0\).

Step-by-step solution

Understand the Given Equations

Given equations from electrostatics are:1. \[abla \cdot \mathbf{E} = \frac{\rho}{\epsilon_{0}}\]2. \[abla \phi = - \mathbf{E}\]

Express Electric Field in Terms of Potential

Substitute \(egabla \phi\) for \(\mathbf{E}\) in the first equation:\[\abla \cdot \left(-\abla \phi\right) = \frac{\rho}{\epsilon_{0}}\]

Simplify the Divergence of Gradient

Use the identity for the divergence of the gradient (which is the Laplacian):\[-\abla^2 \phi = \frac{\rho}{\epsilon_{0}}\]

Derive Poisson's Equation

Rewrite the equation to get Poisson's Equation:\[\abla^2 \phi = -\frac{\rho}{\epsilon_{0}}\]

Analyze the Charge-Free Region

In a charge-free region, \(\rho = 0\). Substitute this into Poisson's equation:\[\abla^2 \phi = 0\]

Verify Laplace's Equation

This simplifies to Laplace's equation:\[\abla^2 \phi = 0\]

Key concepts

Divergence and Gradient

In electrostatics, the concepts of divergence and gradient play a crucial role in understanding electric fields and potentials. Divergence measures how much a vector field spreads out from a point. It's given by \(abla \cdot \mathbf{E}\), where \(\mathbf{E}\) is the electric field. For the electric field \(\mathbf{E}\), the divergence is proportional to the charge density \(\rho\), which can be written as \(abla \cdot \mathbf{E} = \frac{\rho}{\epsilon_{0}}\). Here, \(\epsilon_{0}\) is the permittivity of free space. This equation is a statement of Gauss's law in differential form. The gradient of a scalar field, such as the electrostatic potential \(\phi\), gives the direction and rate of fastest increase of the field. For instance, the relationship between the electric field and the electrostatic potential is \(\mathbf{E} = -abla \phi\). This means the electric field points in the direction of decreasing potential.

Poisson's Equation

Poisson's Equation is a fundamental equation in electrostatics that relates the electrostatic potential \(\phi\) to the charge density \(\rho\). Starting from the previously mentioned relation \(abla \cdot \mathbf{E} = \frac{\rho}{\epsilon_{0}}\) and substituting \(\mathbf{E} = -abla \phi\), we derive \(abla \cdot (-abla \phi) = \frac{\rho}{\epsilon_{0}}\). Simplifying using the identity \(abla \cdot abla \phi = abla^2 \phi\), we get: \[-abla^2 \phi = \frac{\rho}{\epsilon_{0}}\]. This equation is known as Poisson's Equation and can be rearranged to: \[abla^2 \phi = -\frac{\rho}{\epsilon_{0}}\]. Poisson's Equation tells us that the Laplacian (second spatial derivative) of the potential is proportional to the negative charge density. This is crucial in studying how potentials vary in the presence of charges.

Laplace's Equation

Laplace's Equation is a special case of Poisson's Equation for regions where there are no charges. In a charge-free region, the charge density \(\rho = 0\). Substituting \(\rho = 0\) into Poisson's Equation \(abla^2 \phi = -\frac{\rho}{\epsilon_{0}}\), we get: \[abla^2 \phi = 0\]. Solving Laplace's Equation helps determine the potential \(\phi\) in regions devoid of charge. This equation is foundational in fields like electrostatics, fluid dynamics, and gravitational potential. The solutions to Laplace's Equation, known as harmonic functions, exhibit properties that are useful for understanding and predicting physical scenarios free from charge.

Electrostatic Potential

The electrostatic potential \(\phi\) is a scalar quantity representing the electric potential energy per unit charge at a point in space. It is related to the electric field \(\mathbf{E}\) by the equation \(\mathbf{E} = -abla \phi\). This negative gradient indicates that the electric field points in the direction of greatest decrease of potential. The potential allows us to determine the work done in moving a charge within an electric field. It simplifies complex analyses of electric fields by providing a scalar field whose gradient gives the electric field. For example, in spherical coordinates, the potential due to a point charge \(Q\) is given by \(\phi = \frac{Q}{4 \pi \epsilon_{0} r}\), where \(r\) is the distance from the charge.

Charge Density

Charge density \(\rho\) describes how electric charge is distributed in a region of space. It is a key factor in electrostatics equations such as Gauss's law and Poisson’s Equation. There are different types of charge densities:

• Volume charge density (\(\rho\)): Charge per unit volume, typically used in three-dimensional analyses.
• Surface charge density (\(\sigma\)): Charge per unit area on a surface, useful for two-dimensional analyses.
• Linear charge density (\(\lambda\)): Charge per unit length along a line, often used in one-dimensional problems.

Understanding charge density helps in determining the resulting electric fields and potentials in a given spatial configuration.