Problem 7 - Calculus: Early Transcendental Functions

Question

Use Gauss' Law for electricity and the relationship \(q=\iiint \int_{Q} \rho d V\). For \(\mathbf{E}=\left\langle 2 x y, y^{2}, 5 x y\right\rangle,\) find the total charge in the cone \(y=\sqrt{x^{2}+z^{2}}\).

Step-by-Step Solution

Answer

The total charge density within the cone is \(ρ = 2\).

Step 1: Express the Electric Field's Flux in Terms of Triple Integral

According to Gauss' Law, the flux (\(\Phi\)) of the electric field through a closed surface equals the enclosed charge (\(q\)) divided by the Electric constant (\(\varepsilon_{0}\)). In terms of integrals, this can be expressed as \( \Phi = \iint_A \mathbf{E} \cdot d\mathbf{A} = \iiint_V (\nabla \cdot \mathbf{E}) dV = \iiint_{V} {ρ} dV \).

Step 2: Calculate the Divergence of the Electric Field

Before we can put the value of divergence in Gauss' law, we need to calculate the divergence of the given electric field \( \mathbf{E}=\left\langle 2 x y, y^{2}, 5 x y\right\rangle \). The divergence of a vector field is given by the dot product of the del operator and the vector, which in Cartesian coordinates is given by \( \nabla \cdot \mathbf{E} = (\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}) \cdot \mathbf{E}= \frac{\partial}{\partial x}(2xy) + \frac{\partial}{\partial y}(y^2) + \frac{\partial}{\partial z}(5xy) = 2y + 2y = 4y \).

Step 3: Use Divergence in Gauss' Law and Solve for Charge Density \(ρ\)

After representing divergence in Gauss' law, we get \( q = \iiint_{V} ρ dV = \iiint_{V} 4y dV \). The volume element \(dV\) in cylindrical coordinates is given by \( r d r d θ d z \), where \( r = \sqrt{x^{2}+z^{2}} = y \) in the case of our given cone. Substituting this into our integral we get \( q = \int_{0}^{2\pi}\int_{0}^{h}\int_{0}^{r}4r \cdot r dr dθ dz = 2/3 \pi h^3 \). Hence, the charge density \(ρ\) is \(2/3 \pi h^3 / V\), where \(V\) is the volume of the cone.

Step 4: Calculate the Volume of the Cone and Total Charge Density \(ρ\)

The volume (\(V\)) of a cone is given by \(V = 1/3 \pi r^2 h\). Substitute the relationship \(r=y\) into the volume to get \(V = 1/3 \pi y^2 h\). Finally, substituting the volume \(V\) into the charge density, we get the total charge density \(ρ\) to be \(ρ = 2\).

Key Concepts

Electric field

The electric field is a fundamental concept in electromagnetism and is represented as a vector field. This field describes the electric force that a charge would experience at any point in space. The electric field (\(\mathbf{E}\)), given in this problem, is \(\mathbf{E}=\langle 2 x y, y^{2}, 5 x y\rangle\).
It varies with the coordinates, particularly with respect to variables x, y, and z.
Understandably, this field will exert a force, hence influencing rearrangement of charges within the system, reflecting Gauss' Law's primary principle.

Divergence theorem

The divergence theorem, sometimes known as Gauss' Divergence Theorem, is a key piece of vector calculus connecting flux and volume integrals. It states that the total flux of a vector field through a closed surface is equal to the total divergence over the volume inside the surface.
Mathematically, it is expressed as: \[\iint_A \mathbf{E} \cdot d\mathbf{A} = \iiint_V (abla \cdot \mathbf{E}) dV\]

• \(\mathbf{E}\) is the vector field.
• \(d\mathbf{A}\) is the differential area element of the surface.
• \(abla \cdot \mathbf{E}\) is the divergence of the field.

By applying this theorem, we seamlessly translate surface integrals into volume integrals, simplifying the calculation of charge within a region.

Triple integral

A triple integral extends the concept of integration to functions of three variables, often used to compute volumes or masses for a region of space. In our context, it helps calculate the total charge within a volume by integrating charge density:
\(q = \iiint_{V} \rho dV\).
When calculating the charge enclosed within the cone, the triple integral considers radial components, azimuthal angles, and heights, carefully transforming into cylindrical coordinates for ease of integration.
This process aligns with the concept that true space coordinates are a judicious selection depending on the geometric configuration the triple integral spans.

Charge density

Charge density (\(\rho\)) quantifies the amount of electric charge per unit volume in a given region. For this problem, it is derived from the convergence of all charges within the cone-shaped volume. \(\rho:\text{expressed as}\ 4y\)
As charge density interrelates with the Electric Field, it crucially assists in unfolding the electric force impact on charges dispersed within a volume.

• It helps calculate how much charge resides in a specific area of space.
• Understanding this density helps reconstruct how electric fields emerge from charge distributions.

Integrating this density over a volume gives the total charge enclosed, crucial for applying Gauss' Law.