Question

Flux across concentric spheres Consider the radial fields \(\mathbf{F}=\frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{p / 2}}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}},\) where \(p\) is a real number. Let \(S\) consist of the spheres \(A\) and \(B\) centered at the origin with radii \(0

Short answer

#tag_title# Short Answer When p=0, the total outward flux across the concentric spheres is given by \(36\pi(b^3+a^3)\). In this case, the total outward flux is proportional to the sum of the cubes of the radii of the two spheres. When p=3, the total outward flux across the concentric spheres is always zero, because the divergence of the vector field is zero in this case. This means that the flux across sphere B is equal to the flux across sphere A, and the total flux is independent of the radii a and b.

Step-by-step solution

Part A: Calculate Flux with p = 0

First, calculate the divergence of the given vector field. The divergence of a vector field \(\mathbf{F}\) is given by: $$ \nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} $$ Here, \(\mathbf{F}\) is given as \(\frac{\mathbf{r}}{|\mathbf{r}|^p}\), so when \(p=0\), \(\mathbf{F} = \mathbf{r}\). Thus, $$ \nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(x) + \frac{\partial}{\partial y}(y) + \frac{\partial}{\partial z}(z) = 1 + 1 + 1 = 3 $$ Now, we will use the Divergence Theorem to find the total outward flux across the concentric spheres. The Divergence Theorem states that for a vector field \(\mathbf{F}\) and a closed surface \(S\), $$ \iint_S \mathbf{F} \cdot \mathbf{n} dS = \iiint_E \nabla \cdot \mathbf{F} dV $$ where \(\mathbf{n}\) is the outward-pointing unit normal vector, \(dS\) is the surface area element, \(E\) denotes the region enclosed by \(S\), and \(dV\) is the volume element. For sphere A, the volume of the region enclosed is \(E_A = \frac{4}{3} \pi a^3\). Therefore, $$ \text{Flux across A} = \iiint_{E_A} \nabla \cdot \mathbf{F} dV = \int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{a} 3 r^2 \sin(\theta) dr d\theta d\phi = -36 \pi a^3 $$ For sphere B, the volume of the region enclosed is \(E_B = \frac{4}{3} \pi b^3\). Therefore, $$ \text{Flux across B} = \iiint_{E_B} \nabla \cdot \mathbf{F} dV = \int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{b} 3 r^2 \sin(\theta) dr d\theta d\phi = 36 \pi b^3 $$ Now, find the total flux across concentric spheres: $$ \text{Total Flux} = \text{Flux across B} - \text{Flux across A} = 36 \pi b^3 - (-36 \pi a^3) = 36 \pi (b^3 + a^3) $$ The result tells us that the total outward flux across S is proportional to the sum of the cubes of the radii of the two spheres.

Part B: Calculate Flux with p = 3

We will calculate the flux across the spheres for the case where \(p = 3\), essentially representing an inverse square law. The vector field is given by: $$ \mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|^3} $$ First, let's find the divergence of this vector field: $$ \nabla \cdot \mathbf{F} = \frac{\partial}{\partial x} \left( \frac{x}{(x^2+y^2+z^2)^{\frac{3}{2}}} \right) + \frac{\partial}{\partial y} \left( \frac{y}{(x^2+y^2+z^2)^{\frac{3}{2}}} \right) + \frac{\partial}{\partial z} \left( \frac{z}{(x^2+y^2+z^2)^{\frac{3}{2}}} \right) = 0 $$ Since the divergence of the vector field is zero, the total outward flux across the concentric spheres is always zero. This means that the flux across sphere B is equal to the flux across sphere A. Therefore, in this case, the total flux across the concentric spheres is independent of the radii a and b.

Key concepts

Divergence Theorem

The Divergence Theorem is a powerful mathematical tool used in vector calculus. It relates the flow of a vector field through a surface to the behavior of the vector field within the volume enclosed by the surface.
The theorem is often written as: \[ \iint_S \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_E abla \cdot \mathbf{F} \, dV \] where:

• \(\mathbf{F}\) is a vector field,
• \(S\) is a closed surface,
• \(\mathbf{n}\) is the unit normal to the surface,
• \(E\) is the region inside \(S\).

This means that the integral of the divergence of \(\mathbf{F}\) over the volume \(E\) is equal to the integral of \(\mathbf{F}\) through the boundary surface \(S\).
This theorem is essential for understanding flux in vector fields because it allows us to calculate how much of the vector field flows out of a surface by examining how it diverges within the enclosed volume.

Vector Fields

Vector fields are mathematical constructs that assign a vector to each point in space. Imagine small arrows each pointing in various directions at every point in a space.
These arrows represent quantity and direction, indicating how a particular scalar field, such as temperature or fluid flow, behaves in space.
In our exercise, the vector field is based on the position vector \(\mathbf{r}\), defined as \(\frac{\langle x, y, z\rangle}{(x^2 + y^2 + z^2)^{p/2}}\). This specific kind is particularly useful for modeling physical phenomena such as gravitational and electric fields where magnitude can change with distance.
Understanding vector fields involves visualizing how vectors vary across space, which is crucial in physics and engineering applications for analyzing phenomena like fluid dynamics and electromagnetic fields.

Inverse Square Law

The inverse square law describes how certain physical quantities decrease with distance squared from a source.
Commonly seen in physics, this law applies to forces like gravity and electromagnetism. The mathematical form appears as: \[ F = \frac{k}{r^2} \] where \(F\) is the force, \(k\) is a proportionality constant, and \(r\) is the distance from the source.
In our context, the vector field is defined by \(\mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|^3}\) when \(p = 3\). The inverse square law indicates that the flux of this field through any closed surface is zero when the divergence inside the surface is zero, making it independent of its size or shape.
Recognizing this principle helps in understanding fields that spread out from a point source, like how sunlight wanes the farther one moves from the sun.

Divergence

Divergence is a measure of how much a vector field spreads out from a point. It refers to the way fluid or any field would spread from or compress into a point.
Mathematically, for a vector field \(\mathbf{F} = \langle F_x, F_y, F_z\rangle\), its divergence \(abla \cdot \mathbf{F}\) is calculated as: \[ abla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \] In our problem, divergence helps determine how much of the vector field passes through a surrounding surface. If a field's divergence inside a volume is zero, according to the Divergence Theorem, the net flux through the surface is also zero.
This characteristic of divergence plays a fundamental role in physics, allowing us to analyze how sources and sinks affect a vector field.

Concentric Spheres

Concentric spheres are a set of spheres that share the same center point but have different radii.
Visualize them as layers of an onion, each surrounding the center point. In the exercise, these spheres help analyze how the vector field interacts at different distances from the origin.
By understanding how flux behaves across these spheres, one can determine how physical fields like electromagnetic or gravitational fields behave as they extend outward from a point.
This concept is essential for solving problems where multiple surfaces with different sizes enclose the same field source, allowing for analysis of how fields behave at varying distances from their point of origin.