EM1_L2

Dirac delta function

is what is called a “generalized function”. Used to model point charges in space. The Dirac delta function is defined like so:

$$\begin{array}{r}\delta (x)=\{\begin{array}{ll}0& x\ne 0\\ \infty & x=0\end{array}\end{array}$$

such that

$${\int }_{-\infty }^{\infty }\delta (x)dx=1.$$

In three dimensions, it is used like so:

$${\delta }^3(\mathbf{r})=\delta (x_1)\delta (x_2)\delta (x_3)$$

The integral of the 3d delta function over all of 3-space yields 1:

$$\begin{array}{rl}& {\int }_{x_1=-\infty }^{x_1=\infty }{\int }_{x_2=-\infty }^{x_2=\infty }{\int }_{x_3=-\infty }^{x_3=\infty }\delta (x_1)\delta (x_2)\delta (x_3)d{x}_{3}d{x}_{2}d{x}_1\\ & ={\int }_{x_1}{\int }_{x_2}\delta (x_1)\delta (x_2)\left({\int }_{x_3}\delta (x_3)d{x}_3\right)d{x}_{2}d{x}_1\\ & =\left({\int }_{x_3}\delta (x_3)d{x}_3\right){\int }_{x_1}{\int }_{x_2}\delta (x_1)\delta (x_2)d{x}_{2}d{x}_1\\ & =\left({\int }_{x_1}\delta (x_1)d{x}_1\right)\left({\int }_{x_2}\delta (x_2)d{x}_2\right)\left({\int }_{x_3}\delta (x_3)d{x}_3\right)\\ & =1\end{array}$$

Thus, $Q{\delta }^3(\mathbf{r}-\mathbf{a})$ models the charge density due to a point charge located at $\mathbf{a}$.

The triple integral over all 3-space will be denoted by a vanilla integral. Also, $d{x}_{1}d{x}_{2}d{x}_3$ will be denoted by $d^{3}r$.

Properties of the Dirac delta

$\int \delta (x)dx=1$

$\int \delta (x)f(x)dx=f(0)$

$\int \delta (\lambda x)f(x)dx=\frac{1}{|\lambda |}f(0)$

$\int \delta (x^2-a^2)f(x)dx$

$\delta$ contributes only when $x=a$ or $x=-a$.
Thus,

$$\int \delta ((x+a)(x-a))f(x)dx=\frac{1}{2|a|}(f(a)+f(-a))$$

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A more general version of the previous property.

$$\int \delta (g(x))f(x)dx=\sum _{\lambda \in \{\lambda |g(\lambda )=0\}}\frac{1}{|g^{\prime }(\lambda )|}f(\lambda )$$

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Electric fields

The electric field can be modeled like so

$$\begin{array}{r}\mathbf{E}(\mathbf{r})=k\int \frac{\rho ({\mathbf{r}}^{\prime })(\mathbf{r}-{\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^3}{d}^3{r}^{\prime }\end{array}$$

The electric field due to a point charge located at $\mathbf{a}$ is

$$\begin{array}{rl}& \mathbf{E}(\mathbf{r})=k\int \frac{Q{\delta }^3({\mathbf{r}}^{\prime }-\mathbf{a})(\mathbf{r}-{\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^3}{d}^3{r}^{\prime }\\ =& kQ\frac{\mathbf{r}-\mathbf{a}}{|\mathbf{r}-\mathbf{a}{|}^3}\end{array}$$

Dipole moment

Let charges $Q$ and $-Q$ be located at $\mathbf{r}=\mathbf{a}$ and $\mathbf{r}=\mathbf{a}+\mathbf{d}$ respectively.

Then,

$$\begin{array}{r}\rho ({\mathbf{r}}^{\prime })=Q[{\delta }^3({\mathbf{r}}^{\prime }-\mathbf{a})+{\delta }^3({\mathbf{r}}^{\prime }-\mathbf{a}-\mathbf{d})]\end{array}$$

expand the second term using Taylor’s theorem:

$$\begin{array}{r}{\delta }^3({\mathbf{r}}^{\prime }-\mathbf{a}-\mathbf{d})={\delta }^3({\mathbf{r}}^{\prime }-\mathbf{a})+(-d_i)\frac{\partial }{\partial x_i^{\prime }}\end{array}$$