Paper 3, Section II, 11A

Vector Calculus | Part IA, 2014

(i) Starting with Poisson's equation in R3\mathbb{R}^{3},

2ϕ(x)=f(x)\nabla^{2} \phi(\mathbf{x})=f(\mathbf{x})

derive Gauss' flux theorem

Vf(x)dV=VF(x)dS\int_{V} f(\mathbf{x}) d V=\int_{\partial V} \mathbf{F}(\mathbf{x}) \cdot \mathbf{d} \mathbf{S}

for F(x)=ϕ(x)\mathbf{F}(\mathbf{x})=\nabla \phi(\mathbf{x}) and for any volume VR3V \subseteq \mathbb{R}^{3}.

(ii) Let

I=SxdSx3.I=\int_{S} \frac{\mathbf{x} \cdot \mathbf{d} \mathbf{S}}{|\mathbf{x}|^{3}} .

Show that I=4πI=4 \pi if SS is the sphere x=R|\mathbf{x}|=R, and that I=0I=0 if SS bounds a volume that does not contain the origin.

(iii) Show that the electric field defined by

E(x)=q4πϵ0xaxa3,xa\mathbf{E}(\mathbf{x})=\frac{q}{4 \pi \epsilon_{0}} \frac{\mathbf{x}-\mathbf{a}}{|\mathbf{x}-\mathbf{a}|^{3}}, \quad \mathbf{x} \neq \mathbf{a}


VEdS={0 if aVqϵ0 if aV\int_{\partial V} \mathbf{E} \cdot \mathbf{d} \mathbf{S}= \begin{cases}0 & \text { if } \mathbf{a} \notin V \\ \frac{q}{\epsilon_{0}} & \text { if } \mathbf{a} \in V\end{cases}

where V\partial V is a surface bounding a closed volume VV and aV\mathbf{a} \notin \partial V, and where the electric charge qq and permittivity of free space ϵ0\epsilon_{0} are constants. This is Gauss' law for a point electric charge.

(iv) Assume that f(x)f(\mathbf{x}) is spherically symmetric around the origin, i.e., it is a function only of x|\mathbf{x}|. Assume that F(x)\mathbf{F}(\mathbf{x}) is also spherically symmetric. Show that F(x)\mathbf{F}(\mathbf{x}) depends only on the values of ff inside the sphere with radius x|\mathbf{x}| but not on the values of ff outside this sphere.

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