Problem 57 - Calculus: Early Transcendental Functions

Question

Derive the electrostatic field for positive charges \(q\) at (-1,0) and (1,0) and negative charge \(-q\) at (0,0)

Step-by-Step Solution

Answer

The total electric field at point (x,y) due to the three charges is given by \( E_{total} = k \frac{q}{(x+1)^{2} + y^{2}} \hat{i} + k \frac{q}{(x-1)^{2}+y^{2}} \hat{i} + k \frac{-q}{x^{2}+y^{2}} \hat{i} \).

Step 1: State Coulomb's Law

Coulomb's law gives the electric field \( E \) due to a point charge as \( E = k \frac{q}{r^{2}} \), where \( k \) is Coulomb's constant, \( q \) is the charge and \( r \) is the distance from the charge.

Step 2: Calculate the Electric Field due to each Charge

The electric field due to the positive charge at (-1,0) is \( E_{1} = k \frac{q}{(x+1)^{2} + y^{2}} \hat{i} \). Similarly, the electric field due to positive charge at (1,0) will be \( E_{2} = k \frac{q}{(x-1)^{2}+y^{2}} \hat{i} \), and for the negative charge at origin will be \(E_{3} = k \frac{-q}{x^{2}+y^{2}} \hat{i} \). Here, \( \hat{i} \) and \( \hat{j} \) are the unit vectors in the x and y directions respectively, and (x,y) is the point where we want to calculate the electric field.

Step 3: Add Up the Electric Fields

The total electric field at point (x,y) due to the three charges is given by \( E_{total} = E_{1} + E_{2} + E_{3} \).

Step 4: Simplify the Result

By adding up the individual electric fields we obtained in Step 2, we simplify the result to get the expression for the total electric field due to the three charges.

Key Concepts

Coulomb's Law

Coulomb's Law is fundamental to understanding electrostatics. It describes how the electric force between two point charges is related to their magnitudes, the distance between them, and a constant. The law can be defined as:\[ F = k \frac{|q_1 \times q_2|}{r^2} \]where:

• \( F \) is the magnitude of the force between the charges,
• \( k \) is Coulomb's constant, approximately equal to \( 8.9875 \times 10^9 \) N m²/C²,
• \( q_1 \) and \( q_2 \) are the charges,
• \( r \) is the distance between charges.

This inverse square law means the force diminishes as the distance between charges increases. In vector form, the direction of the force on a charge is along the line joining the charges, with the force being attractive if charges are of opposite signs and repulsive if the same.

Electric Field

The electric field is a crucial concept in electrostatics, representing the force per unit charge at a point in space. It is a vector field, meaning it has both magnitude and direction. The electric field due to a point charge can be calculated using the formula:\[ E = k \frac{q}{r^2} \]Here, the direction of \( E \) depends on the sign of the charge:

• If the charge \( q \) is positive, \( E \) points away from the charge.
• If \( q \) is negative, \( E \) points toward the charge.

The electric field vectors radiate outwards from positive charges and inwards toward negative charges. This field helps to predict how other charges will move or experience force when placed within it. It is important to remember that the electric field at a point is the sum of electric fields due to all present charges.

Point Charge

A point charge is an idealized concept in physics representing a charged object whose size is small enough to be considered as a single point. In calculations and theoretical scenarios, treating a charge as a point allows us to simplify the problem and focus on distance effects rather than the dimensions of the objects themselves.

A real-world example might be an atom or a small charged particle where the charge is essentially concentrated at a point. Despite the abstraction, the concept of a point charge is fundamental for analyzing the electric field and forces in electrostatics.

When calculating fields and forces, we consider the point charge as a source of the field, allowing us to use formulas like those derived from Coulomb's Law to find the impacts at various distances. It's crucial to handle point charge calculations with care since they can sometimes lead to singularities at \( r = 0 \), where values become undefined. Thus, while practical, they serve primarily for theoretical work or approximations.

Electrostatics

Electrostatics is the branch of physics that studies electric charges at rest. It is a foundational subject that helps us understand the behavior of charges that are not in motion, unlike electrodynamics, which deals with moving charges and currents.

In electrostatics, we analyze interactions and distributions of charges, using concepts such as electric field, potential, and force. This field of study involves examining how charged objects influence each other while stationary, using tools like Coulomb's Law. It helps in describing phenomena such as:

• The attraction or repulsion between charged bodies,
• The behavior of charges in an electric field,
• The distribution of charges on conducting surfaces.

Understanding electrostatics is critical for applications in electronics, materials science, and various engineering fields. Key principles from electrostatics are used in designing circuits, sensors, and understanding natural events like lightning.