Question

A \(2-\mu C\) point charge is located at \(A(4,3,5)\) in free space. Find \(E_{\rho}, E_{\phi}\), and \(E_{z}\) at \(P(8,12,2)\).

Short answer

Answer: The electric field components at point P(8, 12, 2) due to a point charge of 2 μC at point A(4, 3, 5) are given by the following expressions: $$ E_{\rho} = \frac{1}{4\pi\epsilon_0} \frac{2\ \mu C}{(106)^2} \cos{\theta} $$ $$ E_{\phi} = \frac{1}{4\pi\epsilon_0} \frac{2\ \mu C}{(106)^2} \sin{\theta} $$ $$ E_z = \frac{1}{4\pi\epsilon_0} \frac{2\ \mu C}{(106)^2} $$ Here, \(\theta\) is the angle between the radial vector (\(\rho_A, \phi_A\)) and (\(\rho_P, \phi_P\)) and can be found using the provided relations in the solution.

Step-by-step solution

Convert cartesian coordinates to spherical coordinates

The cartesian coordinates of point A are (4, 3, 5) and of point P are (8, 12, 2). We will convert them to spherical coordinates using the following relations: $$ \rho = \sqrt{x^2 + y^2 + z^2}, \;\; \phi = \arctan{\left(\frac{y}{x}\right)}, \;\; z = z $$ Converting A(4, 3, 5) to spherical coordinates: $$ \rho_A = \sqrt{4^2 + 3^2 + 5^2} = 7, \;\; \phi_A = \arctan{\left(\frac{3}{4}\right)}, \;\; z_A = 5 $$ Converting P(8, 12, 2) to spherical coordinates: $$ \rho_P = \sqrt{8^2 + 12^2 + 2^2} = 14, \;\; \phi_P = \arctan{\left(\frac{12}{8}\right)}, \;\; z_P = 2 $$

Calculate the distance between A and P

To calculate the electric field components at point P, we need to find the distance (R) between points A and P. We will use the distance formula for 3-dimensional space: $$ R = \sqrt{(x_P - x_A)^2 + (y_P - y_A)^2 + (z_P - z_A)^2} $$ Substituting the values, we get: $$ R = \sqrt{(8 - 4)^2 + (12 - 3)^2 + (2 - 5)^2} = \sqrt{16 + 81 + 9} = \sqrt{106} $$

Calculate the electric field components

We will now use Coulomb's Law in spherical coordinates to calculate the electric field components at point P due to the point charge at point A. The electric field components are given by the following relations: $$ E_{\rho} = \frac{1}{4\pi\epsilon_0} \frac{q}{R^2} \cos{\theta}, \;\; E_{\phi} = \frac{1}{4\pi\epsilon_0} \frac{q}{R^2} \sin{\theta}, \;\; E_z = \frac{1}{4\pi\epsilon_0} \frac{q}{R^2} $$ Here, \(q = 2\ \mu C\), \(\epsilon_0 = 8.85*10^{-12} \ F/m\), and \(\theta\) is the angle between the radial vector (\(\rho_A, \phi_A\)) and (\(\rho_P, \phi_P\)). We can calculate \(\theta\) by using: $$ \cos{\theta} = \frac{(\rho_A * \rho_P * \cos(\phi_P - \phi_A)) + (z_A * z_P)}{R^2} $$ Substituting the values, we get: $$ \cos{\theta} = \frac{(7 * 14 * \cos(\arctan{\left(\frac{12}{8}\right)} - \arctan{\left(\frac{3}{4}\right)})) + (5 * 2)}{106} $$ Now, we will calculate the electric field components \(E_{\rho}\), \(E_{\phi}\), and \(E_{z}\): $$ E_{\rho} = \frac{1}{4\pi\epsilon_0} \frac{2\ \mu C}{(106)^2} \cos{\theta} $$ $$ E_{\phi} = \frac{1}{4\pi\epsilon_0} \frac{2\ \mu C}{(106)^2} \sin{\theta} $$ $$ E_z = \frac{1}{4\pi\epsilon_0} \frac{2\ \mu C}{(106)^2} $$ By plugging in the values and performing the calculations, we can find the electric field components \(E_{\rho}\), \(E_{\phi}\), and \(E_{z}\) at point P(8, 12, 2).

Key concepts

Coulomb's Law

Understanding Coulomb's Law is pivotal when studying the interactions between charged particles. This law states that the force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance separating them. The formula is expressed as:
ewlineewline\[ F = k \frac{{|q_1 q_2|}}{{r^2}} \] where:

• \( F \) is the magnitude of the electric force between the charges,
• \( q_1 \) and \( q_2 \) are the amounts of the charges,
• \( r \) is the distance between the centers of the two charges, and
• given as \( 9 \times 10^9 \ N\cdot m^2/C^2 \), known as Coulomb's constant.

For electric field calculation, Coulomb's Law plays an essential role. The electric field \( E \) at a distance from a point charge can be derived from this law and is expressed as: \[ E = k \frac{{|q|}}{{r^2}} \] where \( q \) represents the charge creating the field and \( r \) is the distance from that charge to the point of interest. When considering the direction, the electric field vector points away from the charge if it is positive and towards the charge if it is negative.
The exercise involves calculating the electric field components at a certain point in space due to a single point charge, using the principles of Coulomb's Law applied in spherical coordinates.

Spherical Coordinates

Moving on to the spherical coordinates system, this three-dimensional coordinate system provides a way to locate points in space using three numbers - the radial distance, the polar angle, and the azimuthal angle. It's particularly useful in problems with spherical symmetry, like those involving point charges as in our exercise. The conversions from Cartesian coordinates \((x, y, z)\) to spherical coordinates \((\rho, \phi, z)\) are given by:
\[\begin{align*}\rho &= \sqrt{x^2 + y^2 + z^2}, \phi &= \arctan\left(\frac{y}{x}\right), \z &= z\end{align*}\]Here, \(\rho\) denotes the radial distance to the point from the origin, \(\phi\) is the azimuthal angle in the plane from the positive x-axis, and \(z\) remains the same as in Cartesian coordinates. In the context of electric fields, using spherical coordinates simplifies the calculation of the field when the sources have spherical symmetry, such as point charges.
In the provided exercise, you're tasked with converting Cartesian coordinates of two points to spherical coordinates to proceed with the calculation of the electric field components. This step is crucial to correctly apply Coulomb's Law in spherical coordinates.

Electromagnetic Theory

Last but not least, we explore the overarching framework of electromagnetic theory, which describes how electric charges create electric fields and how moving charges also generate magnetic fields. This theory is rooted in Maxwell's equations - a set of four fundamental equations that govern all classical electromagnetic phenomena. The key concepts related to electric fields involve understanding how they are generated by charge distributions and how they can exert forces on other charges placed in the field. The theory combines both electric and magnetic fields into the electromagnetic field, with changing electric fields giving rise to magnetic fields, and vice versa.
In our current exercise involving the calculation of the electric field at a given point due to a stationary point charge, we are dealing with a static situation where only the electric aspects of electromagnetic theory are at play. The electric field components \(E_{\rho}, E_{\phi}\), and \(E_z\) represent how a charge at a certain location influences the surrounding space, creating an electric field detectable and quantifiable at any other point in space - a cornerstone concept of electromagnetic theory.