Question

A charge \(Q_{0}\) located at the origin in free space produces a field for which \(E_{z}=1 \mathrm{kV} / \mathrm{m}\) at point \(P(-2,1,-1) .(a)\) Find \(Q_{0} .\) Find \(\mathbf{E}\) at \(M(1,6,5)\) in (b) rectangular coordinates; ( \(c\) ) cylindrical coordinates; \((d)\) spherical coordinates.

Short answer

Answer: The electric field components of point M are: 1. Rectangular coordinates: \(-0.185\textbf{i} -1.109\textbf{j} -0.926\textbf{k}\, \mathrm{kV/m}\) 2. Cylindrical coordinates: \(-0.227\textbf{e}_\rho + 0.170\textbf{e}_\phi - 0.926\textbf{e}_z\, \mathrm{kV/m}\) 3. Spherical coordinates: \(-0.467\textbf{e}_r + 0.694\textbf{e}_\theta + 0.170\textbf{e}_\phi\, \mathrm{kV/m}\)

Step-by-step solution

Find Q_0 using given E_z at point P

To determine the charge Q_0 responsible for the electric field, we will first need to find the position vector (r_P) at point P. Using the given coordinates of point P(-2, 1, -1), we find r_P: \( \textbf{r}_P = (-2\textbf{i} + 1\textbf{j} - 1\textbf{k})\). Now, find the magnitude of the position vector: \( | \textbf{r}_P | = \sqrt{(-2)^2 +1^2 + (-1)^2} = \sqrt{6}\). Given that \(E_z\) = 1 kV/m, we can write: \( E_{zP} = \dfrac{Q_0}{4 \pi \epsilon _0 |\textbf{r}_P|^3} \cdot (-1\textbf{k})\). Solving for Q_0 and multiplying both sides by \(4 \pi \epsilon _0 |\textbf{r}_P|^3\): \( Q_0 = E_{zP} \cdot 4 \pi \epsilon _0 |\textbf{r}_P|^3 \cdot (-1\textbf{k})\). Using ε_0 = 8.85 x 10^{-12} C^2/Nm^2 and E_z = 1000 V/m, we get: \( Q_0 = (1000 \mathrm{V/m}) \cdot 4 \pi (8.85 \times 10^{-12} \mathrm{C}^2/\mathrm{N}\mathrm{m}^2) | \textbf{r}_P|^3 \cdot (-1\textbf{k})\). \( Q_0 = -6.73 \times 10^{-9} \, \mathrm{C} \cdot \mathrm{k}\).

Find the position vector for point M

To determine the position vector for point M with coordinates (1, 6, 5), we have: \( \textbf{r}_M = (1\textbf{i} + 6\textbf{j} + 5\textbf{k})\). Find the magnitude of r_M: \(|\textbf{r}_M| = \sqrt{1^2 + 6^2 + 5^2} = \sqrt{62}\).

Find the electric field components in rectangular coordinates

Now, we can find the electric field components at point M using the general formula of the electric field produced by a point charge: \( \textbf{E}_M = \dfrac{Q_0}{4 \pi \epsilon _0 |\textbf{r}_M|^3} \textbf{r}_M\). Plugging in the values for Q_0, ε_0, and r_M, we get: \( \textbf{E}_M = \dfrac{-6.73 \times 10^{-9} \mathrm{C} \cdot \textbf{r}_M}{ 4 \pi (8.85 \times 10^{-12} \mathrm{C}^2/\mathrm{N}\mathrm{m}^2) (\sqrt{62})^3}\). Calculate the electric field components: \( \textbf{E}_M = (-0.185\textbf{i} -1.109\textbf{j} -0.926\textbf{k})\, \mathrm{kV/m}\).

Convert to cylindrical coordinates

To convert the rectangular coordinates to cylindrical coordinates, we use the following conversions: ρ = √(x^2 + y^2) ϕ = arctan(y/x) z = z Let's find ρ and ϕ for point M: ρ = √(1^2 + 6^2) = √37 ϕ = arctan(6/1) = 80.54° Using these values and applying the transformations, we find the cylindrical components field: \( \textbf{E}_M^{cylindrical} = (E_{\rho} \textbf{e}_\rho + E_{\phi} \textbf{e}_\phi + E_{z} \textbf{e}_z)\). \( \textbf{E}_M^{cylindrical} \approx (-0.227\textbf{e}_\rho + 0.170\textbf{e}_\phi - 0.926\textbf{e}_z)\, \mathrm{kV/m}\).

Convert to spherical coordinates

To convert the rectangular coordinates to spherical coordinates, we use the following conversions: r = √(x^2 + y^2 + z^2) θ = arccos(z/r) ϕ = arctan(y/x) We have already found r and ϕ for point M, now let's find θ: θ = arccos(5/√62) = 50.68° Using these values and applying the transformations, we find the spherical components field: \( \textbf{E}_M^{spherical} = (E_{r} \textbf{e}_r + E_{\theta} \textbf{e}_\theta + E_{\phi} \textbf{e}_\phi)\). \( \textbf{E}_M^{spherical} \approx (-0.467\textbf{e}_r + 0.694\textbf{e}_\theta + 0.170\textbf{e}_\phi)\, \mathrm{kV/m}\).

Key concepts

Electric Field

The concept of an electric field is essential in understanding how charges interact at a distance. An electric field is a vector field that surrounds an electric charge. It represents the force a charge exerts on other charges in its vicinity.

The strength and direction of this field are critical. The electric field intensity (E) at a point in space is defined as the force per unit positive charge at that point. In mathematical terms, it's given by the formula: \[\mathbf{E} = \frac{Q}{4\pi \epsilon_0 r^2} \mathbf{\hat{r}}\]where \(Q\) is the charge, \(\epsilon_0\) is the permittivity of free space, and \(r\) is the distance from the charge to the point of interest. The vector \(\mathbf{\hat{r}}\) denotes the direction from the charge to the point. This formula shows that the electric field decreases with the square of the distance from the charge.

It's important to note that the direction of the electric field is away from positive charges and toward negative charges. This concept plays a crucial role in electromagnetism.

Rectangular Coordinates

In physics and engineering, rectangular coordinates, also known as Cartesian coordinates, are commonly used to describe the position of points in space. This system uses three mutually perpendicular axes: x, y, and z, intersecting at a point known as the origin. Each point in space is defined by a set of three numbers (x, y, z), representing the distances along the three axes.

The electric field components in rectangular coordinates can be expressed using these definitions. You represent the electric field vector \(\mathbf{E}\) as a combination of the components along each axis:\[\mathbf{E} = E_x \mathbf{i} + E_y \mathbf{j} + E_z \mathbf{k}\]where \(E_x\), \(E_y\), and \(E_z\) are the components of the electric field in the x, y, and z directions, respectively. The unit vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) indicate direction along these axes.

Rectangular coordinates are particularly useful when problems involve symmetries aligned with the axes, making mathematical operations straightforward.

Cylindrical Coordinates

Cylindrical coordinates are another way to describe points in space, useful especially when dealing with objects that exhibit circular symmetry. In this system, you define a point using three numbers: \(\rho\) (radius), \(\phi\) (angle), and \(z\) (height).

Here's how each parameter works:

• \(\rho\): The distance from the chosen axis to the point.
• \(\phi\): The angle made with the positive x-axis, usually measured in radians.
• \(z\): The same as in rectangular coordinates, representing height.

The transformation from rectangular coordinates (x, y, z) to cylindrical coordinates (\(\rho, \phi, z\)) involves the equations:

• \(\rho = \sqrt{x^2 + y^2}\)
• \(\phi = \arctan\left(\frac{y}{x}\right)\)
• \(z = z\)

The electric field in cylindrical coordinates is expressed as:\[\mathbf{E} = E_\rho \mathbf{e}_\rho + E_\phi \mathbf{e}_\phi + E_z \mathbf{e}_z\]This coordinate system simplifies problems where symmetry about an axis exists, making it handy for analyzing fields around long wires or cylindrical shapes.

Spherical Coordinates

Spherical coordinates provide another method for describing points in 3D space, particularly useful when symmetry about a point is present, such as with spheres. This system relies on three parameters: \(r\) (radius), \(\theta\) (polar angle), and \(\phi\) (azimuthal angle).

Each parameter is defined as follows:

• \(r\): The straight-line distance from the origin to the point.
• \(\theta\): The polar angle, this is the angle from the positive z-axis to the point.
• \(\phi\): The azimuthal angle, largely similar to the angle in cylindrical coordinates, measured from the positive x-axis in the xy-plane.

The conversion from rectangular coordinates (x, y, z) to spherical (\(r, \theta, \phi\)) uses:

• \(r = \sqrt{x^2 + y^2 + z^2}\)
• \(\theta = \arccos\left(\frac{z}{r}\right)\)
• \(\phi = \arctan\left(\frac{y}{x}\right)\)

For electromagnetic fields, spherical coordinates are invaluable, notably when solving problems involving spheres or radial patterns. The electric field in this system can be described as:\[\mathbf{E} = E_r \mathbf{e}_r + E_\theta \mathbf{e}_\theta + E_\phi \mathbf{e}_\phi\]This approach often simplifies the math for objects with radial symmetry, facilitating easier calculation and comprehension of field behaviors.