PX157 - B5a - gauss's law

introduction

for $S$ :

ϕ_{E} = \frac{q}{ϵ_{0}}

for any arbitrary surface around $q$ , the same electric field lines are passing. so, for $S^{'}$ :

ϕ_{E} = \frac{q}{ϵ_{0}}

for a set of charges:

\vec{E} = \sum_{i = 1}^{N} {\vec{E}}_{i}

\subset \supset \iint_{S} \vec{E} \cdot d \vec{S} =\subset \supset \iint \sum_{i = 1}^{N} {\vec{E}}_{i} \cdot d \vec{S} = \sum_{i = 1}^{N} \subset \supset \iint {\vec{E}}_{i} \cdot d \vec{S} = \sum_{i = 1}^{N} \frac{q_{i}}{ϵ_{0}}

gauss's law:

\subset \supset \iint_{S} \vec{E} \cdot d \vec{S} = \frac{Q_{e n c l .}}{ϵ_{0}}

eg: find the flux in each region
$\begin{align*}$ \phi_{B} &= -\frac{q}{\epsilon_{0}}\
\phi_{C} &= 0 &
\phi_{D} &= 0
\end{align*}$$