PX275 - E2c - example

to determine the electric field outside a charged hollow sphere

using the divergence theorem:

\iint_{S} \vec{E} \cdot d \vec{s} = ∭_{V} (\vec{\nabla} \cdot \vec{E}) d V

defining:

\vec{\nabla} \cdot \vec{E} = \frac{σ}{ε_{0}} δ (\vec{r} - \vec{a})

remember: $δ (\vec{r} - \vec{a}) = 0$ for $r \neq a$

\iint_{S} \vec{E} \cdot d \vec{s} = ∭_{V} \frac{σ}{ε_{0}} δ (\vec{r} - \vec{a}) d V

∭ δ (\vec{r} - \vec{a}) d V = 4 π a^{2} at \vec{r} = \vec{a}

\begin{aligned} \iint_{S} \vec{E} \cdot d \vec{s} & = 4 π r^{2} E = ∭_{V} \frac{σ}{ε_{0}} (δ (\vec{r} - \vec{a})) d V \\ = \frac{σ}{ε_{0}} 4 π a^{2} \\ ∴ E & = \frac{σ}{ε_{0}} \frac{a^{2}}{r^{2}} \end{aligned}

note: $δ (r - a) = 0 \forall r \neq a$
the electric field inside the sphere must be zero because there is no enclosed charge inside the sphere