Question

Write a density function for four point charges \(q_{1}=q, q_{2}=-q, q_{3}=q\) and \(q_{4}=-q\), located at the corners of a square of side \(2 a\), lying in the \(x y\) plane, whose center is at the origin and whose first corner is at \((a, a)\).

Short answer

The density function for the given four point charges located at the corners of a square is \(\rho(x,y) = q \delta(x-a)\delta(y-a) - q \delta(x+a)\delta(y-a) + q \delta(x+a)\delta(y+a) - q \delta(x-a)\delta(y+a)\)

Step-by-step solution

Identify Coordinates of Point Charges

Identify the coordinates for the four point charges. Given the first corner (charge \(q_1\)) is at \((a, a)\), the square's layout implies: \n\n- \(q_1\), at coordinates \((a, a)\)\n- \(q_2\), at coordinates \((-a,a)\)\n- \(q_3\), at coordinates \((-a,-a)\)\n- \(q_4\), at coordinates \((a, -a)\)

Associate Each Charge with Its Position

Since each point charge is positioned at a unique point, we use four Dirac delta functions to represent the distribution of the charges. Each Dirac delta function will be at the charge's location. This gives us:- \(\rho_1 = q \delta(x-a)\delta(y-a)\)- \(\rho_2 = -q \delta(x+a)\delta(y-a)\)- \(\rho_3 = q \delta(x+a)\delta(y+a)\)- \(\rho_4 = -q \delta(x-a)\delta(y+a)\)

Derive the Total Charge Density

Finally, we combine all the charge densities to obtain the total charge density function. We add up all \(\rho_1,\) \(\rho_2,\) \(\rho_3\) and \(\rho_4\) to derive:\(\rho(x,y) = \rho_1 + \rho_2 + \rho_3 + \rho_4 = q \delta(x-a)\delta(y-a) - q \delta(x+a)\delta(y-a) + q \delta(x+a)\delta(y+a) - q \delta(x-a)\delta(y+a)\)

Key concepts

Dirac Delta Function

The Dirac delta function plays a crucial role in describing systems where a physical quantity is concentrated at a single point in space. For students tackling electrostatic problems involving point charges, understanding the Dirac delta function is essential.

Imagine the Dirac delta function \( \delta(x) \) as a very sharp peak at \( x = 0 \) where it's infinitely high but also infinitesimally narrow. Despite these properties, its integral over all space is 1, meaning it 'contains' a finite amount of 'stuff' in an infinitely small space. Though it may sound contradictory, this mathematical abstraction is extremely helpful in physics and engineering.

In the context of electrostatics, \( \delta \) functions are used to model point charges. For example, to represent a point charge located at \( x = a \) and \( y = b \) in two dimensions, we would use the product \( q \delta(x-a)\delta(y-b) \), where \( q \) is the magnitude of the charge. This expression tells us that there's a charge \( q \) and it's all concentrated at the point \( (a, b) \) on the \( xy \) plane.

Point Charges

Point charges are one of the fundamental concepts in electrostatics, which represent charges that are so small they can be considered to exist at a single point in space. This model simplifies calculations and is used extensively in theoretical physics to understand the principles governing charged objects.

When dealing with different point charges in a system, like the four charges at the corners of a square in the given exercise, it's important to assign a sign to each charge (positive or negative) to account for their interaction. In electrostatics, opposite charges attract, while like charges repel each other.

By using the concept of superposition, one can determine the net effect caused by multiple point charges on a particular point in space. Superposition states that the total charge density at any point is the algebraic sum of charge densities due to each charge considered independently. In our problem, each charge contributes its own Dirac delta function-modulated density, and they are summed up to find the total charge distribution over the square's plane.

Electrostatics

Electrostatics is a branch of physics that focuses on the study of stationary or slow-moving electric charges. The fundamental law of electrostatics, Coulomb's Law, governs the interaction between electric charges. It states that the electric force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

In the described exercise, we're dealing with the electrostatic scenario of four point charges at fixed positions. To fully grasp this system's behavior, one must consider not only the magnitude and sign of the charges but also their geometric arrangement. In this case, the charges are placed at the corners of a square, introducing symmetry that can simplify calculations.

Understanding the basics of electrostatics, including electric fields, potential, and the principle of superposition can provide deep insights into such systems of point charges. Mastering the use of the Dirac delta function to express charge distributions, in conjunction with knowledge of electrostatic interactions, allows learners to tackle complex charge configurations with confidence.