5.
a) Briefly explain how the fundamental theorem of calculus applies to Stoke’s Theorem,
I~
A·d~
ℓ=ZZ ~
∇ × ~
A·d~a
where ~
Ais a vector function, d~
ℓis an infinitesimal step along a closed path that bounds the area of
which d~a is an infinitesimal vector area element.
The fundamental theorem of calculus relates a function summed over a region (the integral) with
another function evaluated at the boundary of the region. In addition, the function summed over the
region is the derivative of the function evaluated at the boundary.
The right-hand side of Stoke’s Theorem is a function evaluated over a region, in this case an area. The
specific function is the curl of the magnetic field. Note that this is the derivative of another function,
the vector magnetic field itself. The left-hand side of Stoke’s Theorem is this function (the magnetic
field) evaluated at the boundary of the region, or along the (closed) path that bounds the area.
A physical argument that doesn’t really apply the fundamental theorem of calculus would be that
the curl of the magnetic field, when summed across the area, cancels everywhere except at the border.
Thus the right-hand side really just sums the curling of the magnetic field that is parallel to the border.
This is then exactly the same as summing the component of the magnetic field that is parallel to the
border, as specified by the left-hand side of the equation.
b) The version of Gauss’s Law used in General Physics involves the charge enclosed by the Gaussian
surface (or, more generally, it depends on the charge density ρ). Why does Gauss’s Law, as it appears
in the set of Maxwell’s Equations listed in Problem (1), not have any dependence on charge density?
Maxwell’s Equations as given in Problem 1 are under the condition of empty space. There are therefore
no charges (or space wouldn’t be empty!). If there are no charges, then ρ= 0 and Gauss’s Law reduces
to ~
∇ · ~
E= 0.
This could also be thought of in terms of the integral form of the equation. If there are no sources or
sinks of electric field lines (i.e. no charges), then the net electric flux through any closed surface must
be zero.
