PX157 - B4b - electric flux from an arbitrary surface

d ϕ_{E, i j} = {\vec{E}}_{i j} \cdot d {\vec{S}}_{i j}

\begin{aligned} ϕ_{E} & = \sum_{i} \sum_{j} d ϕ_{E, i j} \\ = \sum_{i} \sum_{j} {\vec{E}}_{i j} \cdot d {\vec{S}}_{i j} \\ ∴ ϕ_{E} & = \iint_{S} \vec{E} \cdot d \vec{S} \end{aligned}

ϕ_{E} =\subset \supset \iint_{S} \vec{E} \cdot d \vec{S}

$d \vec{S}$ is always chosen pointing outwards
eg: a point charge, $+ q$ , at the origin.
- choosing a spherical surface with radius, $r$ , concentric with the charge:

\begin{aligned} \vec{E} & = \frac{q}{4 π ϵ_{0} r^{2}} \hat{r} \\ d \vec{S} & = d S \hat{r} \end{aligned}

\begin{aligned} ϕ_{E} & =\subset \supset \iint_{S} \frac{q}{4 π ϵ_{0} r^{2}} \hat{r} \cdot \hat{r} d S \\ = \frac{q}{4 π ϵ_{0} r^{2}} \subset \supset \iint_{S} d S \end{aligned}

- the sphere has a surface area, $S = 4 π r^{2}$ :

∴ ϕ_{E} = \frac{q}{ϵ_{0}}

eg: the electric flux though a sphere concentric with an ideal electric dipole with dipole moment, $\vec{p} = p \hat{z}$
- coordinates to describe any point on the sphere, $θ$ and $ϕ$ , gives position vector:

{\vec{r}}^{'} = r^{'} \cos ϕ^{'} \sin θ^{'} \hat{x} + r^{'} \sin ϕ^{'} \sin θ^{'} \hat{y} + \hat{r} \cos θ^{'} \hat{z}

- the surface vector for a small section at $(θ^{'}, ϕ^{'})$ is given by:

d \vec{S} = \frac{\partial {\vec{r}}^{'}}{\partial θ^{'}} \times \frac{\partial {\vec{r}}^{'}}{\partial ϕ^{'}} d θ d ϕ = r^{' 2} \sin θ^{'} \hat{r} d θ d ϕ

\vec{E} \approx \frac{1}{4 π ϵ_{0} r^{3}} [3 (\vec{p} \cdot \hat{r}) \hat{r} - \vec{p}]

d \vec{S} = d S \hat{r} = r^{2} \sin θ d θ d ψ \hat{r}

\vec{E} \cdot d \vec{S} = \frac{1}{4 π ϵ_{0} r^{3}} r^{2} \sin θ d θ d ϕ \hat{r} [3 (\vec{p} \cdot \hat{r}) \hat{r} - \vec{p}] \cdot \hat{r}

- use $\vec{p} \cdot \hat{r} = p (\hat{z} \cdot \hat{r}) = p \cos θ$ :

\vec{E} \cdot d \vec{S} = \frac{1}{4 π ϵ_{0} r^{3}} r^{2} \sin θ 2 p \cos θ d θ d ϕ \hat{r}

ϕ_{E} = \int_{0}^{2 π} \int_{0}^{π} \frac{1}{4 π ϵ_{0} r^{3}} r^{2} \sin θ 2 p \cos θ d θ d ψ \hat{r}

- in this:

\int_{0}^{π} 2 \sin θ \cos θ d θ = \int_{0}^{π} \sin 2 θ = 0 \1 \n \2 \n