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Nathan Nguyen
2024-12-06 12:48:11 -06:00
parent ac4e2030c1
commit bb7fc5f6e8
24 changed files with 139 additions and 113 deletions
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14
docs/physics/electrostatics/3-electro-magnetic-potentials.md
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@@ -14,7 +14,7 @@ $$
\vb{E}(\vb{r}) = \frac{1}{q_e} \vb{F_e}(\vb{r}) = - \frac{1}{q_e} \grad{U_e(\vb{r})} = -\grad{V_e(\vb{r})}
$$
The units of electrostatic potential is Joule/Coublomb, also known as a Volt. Thus, the units of the electric field should be expressed in Volts/meter. Similarly,
The units of electrostatic potential is Joule/Coulomb, also known as a Volt. Thus, the units of the electric field should be expressed in Volts/meter. Similarly,
$$
\vb{H}(\vb{r}) = \frac{1}{q_m} \vb{F_m}(\vb{r}) = - \frac{1}{q_m} \grad{U_m(\vb{r})} = -\grad{V_m(\vb{r})}
@@ -80,7 +80,7 @@ $$
W_n = \frac{1}{2} \frac{4 \pi \epsilon_0} \sum_{i = 1}^{N} \sum_{j > i}^{N} \frac{Q_{ei}Q_{ej}}{\abs{\vb{r_i}-\vb{r_j}}}
$$
For the sake of symnetry, sum overall charges and divide by 2.
For the sake of symmetry, sum overall charges and divide by 2.
$$
W_n = \frac{1}{2} \frac{1}{4 \pi \epsilon_0} \sum_{i = 1}^{N} \sum_{j \neq i}^{N} \frac{Q_{ei}Q_{ej}}{\abs{\vb{r_i}-\vb{r_j}}}
@@ -93,7 +93,7 @@ W_n = \frac{1}{2} \sum_{i = 1}^{N}Q_{ei} \sum_{i \neq j}^{N} \frac{1}{4 \pi \eps
= \frac{1}{2}\sum_{i = 1}^{N} Q_{ei} V(\vb{r_i})
$$
We can rewrite this as a Reimann sum and convert to an integral.
We can rewrite this as a Riemann sum and convert to an integral.
$$
W_e = \frac{1}{2} \int_V p_e(\vb{r}) V_e(\vb{r}) \dd V ; \quad
@@ -129,7 +129,7 @@ $$
We know that $\vb{E}(\vb{r}) = -\div{V_e(\vb{r})}$ and $\vb{H}(\vb{r}) = -\div{V_m(\vb{r})}$
Combinind this, as well as the first of the Maxwell equations, we see that
Combined this, as well as the first of the Maxwell equations, we see that
$$
\div{\vb{E}} = -\div{\grad{V_e}} = - \laplacian{V_e} = \frac{\rho_e}{\epsilon_0}
@@ -139,7 +139,7 @@ $$
\div{\vb{H}} = -\div{\grad{V_m}} = - \laplacian{V_m} = \frac{\rho_m}{\mu_0}
$$
The last inequatlity is called the Poisson Equation, or the inhomogenous Laplace equation.
The last inequality is called the Poisson Equation, or the inhomogeneous Laplace equation.
To solve this equation, we define a Green function as follows:
@@ -147,7 +147,7 @@ $$
\laplacian G(\vb{r}, \vb{r'}) = \delta(\vb{r} - \vb{r'})
$$
Now, we can construct a potential function in terms of said green function that satisfies the lapalce equation.
Now, we can construct a potential function in terms of said green function that satisfies the Laplace equation.
$$
V_e(\vb{r}) = - \int_V G(\vb{r}, \vb{r'}) \frac{\rho_e(\vb{r'})}{\epsilon_0} \dd{V'}
@@ -273,4 +273,4 @@ $$
Note that as a quirk of the function, $P_n(1) = 1$ for all $n$.
We can apply these quadrupole and beyond terms to the volate or other equations, however, this becomes very messy.
We can apply these quadrupole and beyond terms to the violate or other equations, however, this becomes very messy.