Question

Write a density function for two point charges \(q_{1}\) and \(q_{2}\) located at \(\mathbf{r}=\mathbf{r}_{1}\) and \(\mathbf{r}=\mathbf{r}_{2}\), respectively.

Short answer

The density function for two point charges \(q_{1}\) and \(q_{2}\) located at \(\mathbf{r_1}\) and \(\mathbf{r_2}\) respectively, can be written as \(\rho(\mathbf{r})=q_{1}\delta(\mathbf{r}-\mathbf{r}_{1})+q_{2}\delta(\mathbf{r}-\mathbf{r}_{2})\).

Step-by-step solution

Establishing the charge density expression for a single point charge

The point charge has the property that all the charge is concentrated in one point in the space. If we denote the location of this charge as \(\mathbf{r}_{1}\), the charge density \(\rho(\mathbf{r})\) for a single point charge \(q\) can be written as \(\rho(\mathbf{r})=q\delta(\mathbf{r}-\mathbf{r}_{1})\), where \(\delta\) represents the Dirac delta function.

Extending the expression for two point charges

For two point charges \(q_{1}\) and \(q_{2}\) located at \(\mathbf{r_1}\) and \(\mathbf{r_2}\) respectively, the charge density should be the sum of the two corresponding Dirac delta function expressions. Therefore, the density function can be written as \(\rho(\mathbf{r})=q_{1}\delta(\mathbf{r}-\mathbf{r}_{1})+q_{2}\delta(\mathbf{r}-\mathbf{r}_{2})\). This equation embodies the density function for two point charges \(q_{1}\) and \(q_{2}\).

Key concepts

Point Charges

Point charges are a foundational concept in electrostatics. They describe the idea of concentrating all the charge of an object at a single point in space, simplifying analysis and calculations.
By treating charges as point charges, it becomes easier to visualize and predict their effects on each other without dealing with complex shapes or sizes.
Mathematically, point charges are used to model phenomena where the size of the charge distribution is negligible compared to the distances involved. Key characteristics of point charges:

• They have a specified amount of charge, either positive or negative.
• They are usually denoted by symbols such as \(q_1\), \(q_2\), etc.
• Their positions are defined by coordinates, like \(\mathbf{r}_1\) and \(\mathbf{r}_2\).

These features make point charges vital in theoretical and practical applications, including electrical field calculations and potential energy considerations.

Dirac Delta Function

The Dirac delta function is a powerful tool in mathematical physics. It is used to represent point charges and other singularities in space.
The function is peculiar because while it is technically not a function in the traditional sense, it serves as an idealized way of modeling such situations.
Conceptually, it can be thought of as an infinitely high, infinitely narrow peak, situated at a specific point in space. Some of the fascinating properties include:

• The function \(\delta(x - a)\) is zero everywhere except at \(x = a\).
• It integrates to one over the entire real line, providing normalization.
• In the context of point charges, it helps localize charge to a specific point with the expression \(\rho(\mathbf{r}) = q \delta(\mathbf{r} - \mathbf{r}_1)\).

This allows scientists and engineers to manipulate equations involving point charges with greater ease and accuracy.

Density Function

Density functions provide a way to describe distributions, such as mass or charge distributions, in space.
In the realm of electrodynamics, a density function can take on a very specific form, particularly in the case of point charges where the Dirac delta function is used.
The density function for point charges allows for the representation of how charge is spread out in space. For instance, consider two point charges, \(q_1\) and \(q_2\), located at \(\mathbf{r}_1\) and \(\mathbf{r}_2\):

• The charge density function is the sum of individual point charge contributions: \(\rho(\mathbf{r}) = q_1 \delta(\mathbf{r} - \mathbf{r}_1) + q_2 \delta(\mathbf{r} - \mathbf{r}_2)\).
• This sum accounts for the entire charge distribution in the space, effectively describing how the charges influence surrounding fields.

Using these functions, one can easily predict electric fields and potentials in a region influenced by point charges.

Electrodynamics

Electrodynamics is the study of electric charges in motion and the fields they generate.
The behavior and interaction of point charges within this framework are central to understanding how electrical systems operate in real-world applications.
This field also bridges the gap between the static cases studied in electrostatics and dynamic scenarios involving changing fields. Key aspects of electrodynamics include:

• Electric and magnetic fields are described using Maxwell's equations, foundational to understanding electrodynamics.
• The movement of charges, such as electrons in conductors, forms the basis of ohmic current.
• Electrodynamics principles enable the understanding of essential technologies, including transformers, motors, and wireless communication.

Through these principles, electrodynamics offers insights into how fields propagate and interact, influenced largely by charge distributions described by concepts like point charges and their density functions.