Flux

Definition

Flux is a measure of vector field strength along a barrier. A barrier can be a curve in 2D, a surface in 3D, a volume in 4D, or an $n$-dimensional shape in $(n+1)$-dimensional space, ${\mathbb{R}}^{n+1}$, where a vector field of $n$-dimensions is present.

The word flux is derived from the Latin word fluxus, meaning “flow”. It is common to use flux to describe physical materials flowing through an imaginary surface, like water in a pipe flowing through an imaginary circular disk. Because of this, many people also visualize flux as vector field lines “flowing” or “moving” through a surface in 3D space. For example, we describe magnetic field lines as flux, as flowing through space, even though there is nothing moving.

2D

$$\Phi ={\int }_C\vec{F}\cdot \hat{n}ds$$

where $\vec{F}$ is the vector field in ${\mathbb{R}}^2$, $C$ is some smooth curve in ${\mathbb{R}}^2$, $\hat{n}$ is the unit vector normal to $C$ at all points, and $ds$ is an infinitesimal arc length of $C$.

3D

$$\Phi ={\iint }_S\vec{F}\cdot \hat{n}dS$$

where $\vec{F}$ is the vector field in ${\mathbb{R}}^3$, $S$ is some smooth surface in ${\mathbb{R}}^3$, $\hat{n}$ is the unit vector normal to $S$ at all points, and $dS$ is an infinitesimal area of $S$.

Derivation

1-Dimension

Suppose we have a one-dimensional space with a one-dimensional vector field, $\vec{f}(x)$, present everywhere. Then to calculate flux here (although the concept of flux may not be the same as in higher dimensions), we must designate a point $x_0$ which will be our mark for measuring the vector field strength.

2-Dimensions

Suppose we have a two-dimensional space, ${\mathbb{R}}^2$, with a two-dimensional vector field, $\vec{f}(x,y)$, present everywhere. Suppose we define a curve $C$ in ${\mathbb{R}}^2$, and we want to calculate the flux passing through $C$. Suppose $C$ is parameterized by a function $\vec{r}(t)$. We can perform a line integral over $C$, comparing the vector at any point along it, $(x_0,y_0)$, to the normal vector to $C$ using a dot product, with the final result of,

$${\int }_C\vec{f}\cdot \hat{n}ds$$

which we will now derive.

The arc length of the curve $C$ is $L$ and an infinitesimally small arc length segment is $ds$, where $s(t)$ is the partial arc length, as here.

$$s(t_0)={\int }_a^{t_0}||\vec{r}{}^{\prime }(t)||dt\to ds=||\vec{r}{}^{\prime }(t)||dt$$

The normal vector to $C$ at any point is found by first finding the tangent, then computing the perpendicular vector to that tangent. The tangent to $C$ at any point is $\vec{r}{}^{\prime }(t)$, and the unit tangent at any point can be called $\vec{T}(t)$, and is $\vec{T}(t)=\frac{\vec{r}{}^{\prime }(t)}{||\vec{r}{}^{\prime }(t)||}$. Then, in 2D, a vector $\vec{b}$ can be perpendicular to a vector $\vec{a}$ if, by decomposing $\vec{a}$ into $x$- and $y$-components, $\vec{b}=⟨-a_y,a_x⟩$ or $=⟨a_y,-a_x⟩$. There are two possible perpendicular vectors to $\vec{T}(t)$, and it doesn’t matter which is used so long as it is consistent along the curve. We will choose to calculate the perpendicular vector, $\vec{N}(t)$, in this way,

$$\vec{T}(t)=⟨T_x(t),T_y(t)⟩\to \vec{N}(t)=⟨-T_y(t),T_x(t)⟩$$

Then, we can normalize this vector into a unit normal vector,

$$\hat{n}(t)=\frac{\vec{N}(t)}{||\vec{N}(t)||}$$

or,

$$\hat{n}(t)=\frac{1}{||\vec{r}{}^{\prime }(t)||}⟨-{\vec{r}}_y{}^{\prime }(t),{\vec{r}}_x{}^{\prime }(t)⟩$$

Now we have everything we need to form an expression that shows a computation of the vector field strength through a curve:

$$\Phi ={\int }_C\vec{f}(x,y)\cdot \hat{n}ds={\int }_a^b\vec{f}(\vec{r}(t))\cdot \hat{n}ds$$

Then, with the earlier expression $ds=|\vec{r}{}^{\prime }(t)|dt$ and the expression for $\hat{n}$,

$$\Phi ={\int }_a^b\vec{f}(\vec{r}(t))\cdot \hat{n}||\vec{r}{}^{\prime }(t)||dt={\int }_a^b\vec{f}(\vec{r}(t))\cdot ⟨-r_y(t),r_x(t)⟩dt$$

where $r_x(t)$ and $r_y(t)$ are the $x$- and $y$- components that make up $\vec{r}(t)$.

3-Dimensions

Suppose we have a three-dimensional space, ${\mathbb{R}}^3$, with a three-dimensional vector field, $\vec{F}(x,y,z)$, present everywhere. Suppose we define a flat surface $S$ in ${\mathbb{R}}^3$, and we want to calculate the flux passing through $S$.

$n$-Dimensions