Question

Let \(\mathbf{F}\) be a radial field \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p},\) where \(p\) is a real number and \(\mathbf{r}=\langle x, y, z\rangle .\) With \(p=3, \mathbf{F}\) is an inverse square field. a. Show that the net flux across a sphere centered at the origin is independent of the radius of the sphere only for \(p=3\) b. Explain the observation in part (a) by finding the flux of \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p}\) across the boundaries of a spherical box \(\left\\{(\rho, \varphi, \theta): a \leq \rho \leq b, \varphi_{1} \leq \varphi \leq \varphi_{2}, \theta_{1} \leq \theta \leq \theta_{2}\right\\}\) for various values of \(p\)

Short answer

Now that we have computed the total flux across the boundaries in part b, we can conclude that: 1. When \(p=3\), the net flux across a sphere centered at the origin is independent of the radius as shown in part a. 2. When \(p=3\), the total flux across the boundaries of the spherical box in part b is always zero, consistent with the conclusion from part a. This exercise demonstrates the principle that for a radial field with a specific power, the net flux across a sphere centered at the origin does not depend on the sphere's radius.

Step-by-step solution

Part a: Find the net flux across a sphere centered at the origin

The flux of a vector field across the surface of a sphere can be found by using the Divergence Theorem: $$\iint_{S}\mathbf{F} \cdot \mathbf{dS}=\iiint_{V}(\nabla\cdot\mathbf{F})dV$$ First, we need to find the divergence of the radial field: $$\nabla\cdot\mathbf{F}=\frac{\partial}{\partial x}\left(\frac{x}{(x^{2}+y^{2}+z^{2})^{p/2}}\right)+\frac{\partial}{\partial y}\left(\frac{y}{(x^{2}+y^{2}+z^{2})^{p/2}}\right)+\frac{\partial}{\partial z}\left(\frac{z}{(x^{2}+y^{2}+z^{2})^{p/2}}\right)$$ To simplify our computations, we will switch to spherical coordinates. Let \(\rho^{2} = x^{2} + y^{2} + z^{2}\). Then, we have: $$\nabla\cdot\mathbf{F}=\frac{1}{\rho^{2}}\frac{\partial}{\partial\rho}\left(\rho^{2}\left(\frac{\rho}{\rho^{p}}\right)\right)=\frac{1}{\rho^{2}}\frac{\partial}{\partial\rho}\left(\rho^{3-p}\right)$$ Now, we will evaluate the integral: $$\iint_{S}\mathbf{F} \cdot \mathbf{dS}=\iiint_{V}(\nabla\cdot\mathbf{F})dV=\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{r}\left(\frac{1}{\rho^{2}}\frac{\partial}{\partial\rho}\left(\rho^{3-p}\right)\right)\rho^{2}\sin{\vartheta}d\rho d\vartheta d\phi$$ Which simplifies to: $$\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{r}(3-p)\rho^{2-p}\sin{\vartheta}d\rho d\vartheta d\phi$$ Notice that this integral does not depend on r when \(p=3\). Therefore, when \(p=3\), the net flux across the sphere is independent of the radius.

Part b: Flux across the boundaries of the spherical box

In this part, we need to find the flux of the radial field, \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p},\) across a spherical box with given polar coordinate bounds. Let's calculate the side faces flux at \(\rho = a\) and \(\rho = b\). Since \(\mathbf{F}\) is a radial field and \(\mathbf{dS}\) is perpendicular to the radius vector on these faces, the side faces do not contribute to the flux. Therefore, the flux across the boundaries comes only from the contribution of the inner and outer surface near the origin. Inner surface at \(\rho = a\): $$\iint_{S_{inner}}\mathbf{F}\cdot\mathbf{dS} = \int_{\varphi_1}^{\varphi_2}\int_{\theta_1}^{\theta_2}\frac{\rho}{|\rho|^{p}}\cdot(-\rho^2\sin\theta)d\theta d\varphi \bigg|_{\rho=a} = -(3-p) a^{3-p}\Delta\varphi\Delta\theta$$ Outer surface at \(\rho = b\): $$\iint_{S_{outer}}\mathbf{F}\cdot\mathbf{dS} = \int_{\varphi_1}^{\varphi_2}\int_{\theta_1}^{\theta_2}\frac{\rho}{|\rho|^{p}}\cdot(\rho^2\sin\theta)d\theta d\varphi\bigg|_{\rho=b} = (3-p) b^{3-p}\Delta\varphi\Delta\theta$$ Total flux across the boundaries: $$\iint_{S}\mathbf{F}\cdot\mathbf{dS} = \iint_{S_{inner}}\mathbf{F}\cdot\mathbf{dS}+\iint_{S_{outer}}\mathbf{F}\cdot\mathbf{dS}= -(3-p) a^{3-p}\Delta\varphi\Delta\theta + (3-p) b^{3-p}\Delta\varphi\Delta\theta$$ Notice that when \(p=3\), the total flux across the boundaries is always zero, which is consistent with our conclusion from part a.

Key concepts

Divergence Theorem

The Divergence Theorem, also known as Gauss's theorem, is a fundamental result in vector calculus that relates the flow (flux) of a vector field through a closed surface to the behavior of the vector field inside the volume bounded by the surface. Mathematically, it is expressed as:
\[ \iint_{S} \mathbf{F} \cdot d\mathbf{S} = \iiint_{V} (abla \cdot \mathbf{F}) dV \]
This powerful theorem simplifies the calculation of the flux of a vector field through a closed surface by converting it into a volume integral over the divergence of the field. The divergence, \(abla \cdot \mathbf{F}\), measures the rate at which the vector field 'diverges' from a point, essentially quantifying the 'source' or 'sink' strength at that point. When applied to a radial field, the Divergence Theorem can help to determine how the flux through a spherical surface depends on the radius of the sphere and the nature of the field itself.

Radial Field

A radial field is a type of vector field where the vector at each point is directed radially away from or towards a central point. Mathematically, for a point \(\mathbf{r} = \langle x, y, z \rangle\), a radial field can be expressed as:
\[ \mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|^{p}} \]
Here, the value of \(p\) modifies the magnitude of the field vectors. For instance, when \(p=3\), a significant property emerges: the field is known as an inverse square field because the magnitude of the vectors decreases with the square of the distance from the origin. This property has profound implications, as illustrated in the exercise, where it is shown that the net flux across a sphere is independent of its radius specifically when \(p=3\). This can be interpreted physically, for example, in the context of gravitational and electrostatic fields, which also diminish as the inverse square of the distance from the source.

Spherical Coordinates

Spherical coordinates are an alternative coordinate system for representing points in three-dimensional space, frequently used when dealing with problems having spherical symmetry. A point in this system is described by three coordinates: the radial distance \(\rho\), the polar angle \(\theta\) (angle from the positive z-axis), and the azimuthal angle \(\varphi\) (angle from the positive x-axis in the xy-plane). The relationships between Cartesian and spherical coordinates are given by:
\[ x = \rho \sin\theta \cos\varphi \]
\[ y = \rho \sin\theta \sin\varphi \]
\[ z = \rho \cos\theta \]
Using spherical coordinates allows for the simplification of the vector field and flux calculations through symmetry, as showcased in the exercise's conversion of the divergence of a radial field to spherical coordinates, streamlining the process to solve for the flux across a spherical surface.

Vector Calculus

Vector calculus is an area of mathematics concerned with differentiation and integration of vector fields, primarily in three-dimensional Euclidean space. Central to vector calculus are operators such as gradient, divergence, and curl that measure various aspects of vector fields. The gradient indicates the rate and direction of the steepest increase of a scalar field, the divergence measures the magnitude of a source or sink at a given point in a vector field, and the curl quantifies the rotation of the field around a point. Through the use of these operators, one can describe and analyze physical phenomena like fluid flow, electromagnetic fields, and force distributions. In our exercise, the computation of the divergence of the vector field is an essential application of vector calculus, allowing us to apply the Divergence Theorem and evaluate the flux.

Partial Derivatives

Partial derivatives play a significant role in vector calculus, particularly in the computation of operators such as divergence. A partial derivative represents the rate at which a function changes as only one variable changes while keeping the other variables constant. In the context of a three-dimensional vector field \(\mathbf{F} = (F_x, F_y, F_z)\), the divergence of \(\mathbf{F}\) is given by the sum of the partial derivatives of its components with respect to their respective variables:
\[ abla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \]
In the original exercise, calculating the divergence of the radial field required finding the partial derivatives of the field's components with respect to Cartesian coordinates, which was then greatly simplified by the conversion to spherical coordinates. Understanding how to compute partial derivatives is crucial for determining the divergence and thereby using the Divergence Theorem to find the flux through surfaces.