Gauss’s Law in Differential Form

In differential form, Gauss’s Law is

$\vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon_0 \epsilon_r}$

The next part of this post attempts to demystify this law a bit.

Description of Symbols

The $\vec{\nabla}$ on the left is called ‘del’, and it can be written in terms of Cartesian unit vectors and partial derivatives with respect to Cartesian coordinates:

$\vec{\nabla} = \vec{e}_x \frac{\partial}{\partial x} + \vec{e}_y \frac{\partial}{\partial y} + \vec{e}_z \frac{\partial}{\partial z}$.

Here, $\vec{e}_x$, $\vec{e}_y$, and $\vec{e}_z$ are unit vectors for a right-handed Cartesian coordinate system.

$\vec{E} = E_x \vec{e}_x + E_y \vec{e}_y+ E_z \vec{e}_z$ is a vector of an electric field. The variables $E_x$, $E_y$, and $E_z$ are called components, and recall that a vector component can be positive, negative, or zero. Hence the component $E_x$ can be positive, negative, or zero. The same applies to $E_y$ and $E_z$. The electric field vector $\vec{E}$ does not need to be written in terms of Cartesian unit vectors, but this is one option. In particular, the electric field vector $\vec{E}$ could have been written in terms of unit vectors for a spherical coordinate system.

The dot in between $\vec{\nabla}$ and $\vec{E}$ is for a dot product.

$\rho$ is called the volume charge density. Note that $\rho$ is dimensionally equal to charge per unit volume. So, if the unit volume has a value of $1$, then $\rho$ has a value equal to the numerical value of the charge in that unit volume.

$\epsilon_0$ is called the vacuum permittivity. Common units for this quantity are farads per meter.

$\epsilon_r$ is called the relative permittivity, and it is a ratio. Hence $\epsilon_r$ is dimensionless.

Gauss’s Law in a Right-Handed Cartesian Coordinate System

Substituting the Cartesian expression for $\vec{\nabla}$ as well as the Cartesian expression for $\vec{E}$, Gauss’s Law takes the following form.

$\bigg(\frac{\partial}{\partial x} , \frac{\partial}{\partial y} , \frac{\partial}{\partial z} \bigg) \cdot \bigg(E_x, E_y, E_z \bigg)= \frac{\rho}{\epsilon_0 \epsilon_r}$

It is possible to simplify this expression by applying the dot product:

$\frac{\partial}{\partial x} E_x + \frac{\partial}{\partial y} E_y + \frac{\partial}{\partial z} E_z = \frac{\rho}{\epsilon_0 \epsilon_r}$

Partial Derivatives of an Electric Field Components with respect to Cartesian Coordinates

Observe that the left side involves partial derivatives of the Electric Field Variables $E_x$, $E_y$, and $E_z$. So, Gauss’s Law is concerned with how the components of an electric field vector change with respect to spatial coordinates. In this case, those spatial coordinates are Cartesian coordinates.

Since the partial derivative of a constant is zero, a term such as $\frac{\partial}{\partial x} E_x$ is zero if $E_x$ does not depend on the coordinate $x$. Similarly, $\frac{\partial}{\partial y} E_y$ is zero if $E_y$ does not depend on $y$, and $\frac{\partial}{\partial z} E_z$ evaluates to zero if $E_z$ does not depend on $z$.

Note that if $E_z$ were dependent on, say, the coordinate $y$, $\frac{\partial}{\partial z} E_z$ would still evaluate to zero.

Considering a Volume Charge Density of Zero

Suppose the volume charge density $\rho$ is equal to zero:

$\rho = 0$

Recall that volume charge density is the amount of charge in a unit volume. A unit volume cannot be zero, so a volume charge density of zero corresponds to zero charge in a unit volume. So if $\rho=0$, there is no charge in the unit volume.  For this case, one of many solutions to Gauss’s Law is:

$\frac{\partial}{\partial x} E_x = 0$

$\frac{\partial}{\partial y} E_y = 0$

$\frac{\partial}{\partial z} E_z = 0$

Other solutions involve one of the partial derivative terms cancelling one or more of the other partial derivative terms.

If the electric field vector is the zero vector, there must zero charge in the corresponding unit volume, since the vacuum permittivity and the relative permittivity are nonzero by definition.

References

[1] https://en.wikipedia.org/wiki/Del

[2] https://en.wikipedia.org/wiki/Gauss%27s_law

[3] https://www.grc.nasa.gov/WWW/k-12/BGP/vectpart.html

[4] https://en.wikipedia.org/wiki/Relative_permittivity

[5] https://en.wikipedia.org/wiki/Vacuum_permittivity

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