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```Lecture 13: Advanced
Examples
• Selected
examples taken from Problem Set 4
• Magnetic Potential (as an example of working
with non-conservative fields)
• Plus Maxwell’s Equations (as a concise
example of the power of vector calculus)
Vector Areas (Problem 4.1)
B
z
z
C
OA
y
A
B=(1,1,1)
B
x
O
y
C=(0,2,0)
C
O
x
z
A=(1,0,0)
C
Vector Area of Triangle OAB
B
y
O
A
Vector Area of Triangle OBC
x
Maximum Projected Area
Area Projected in direction of this unit vector
Solid Angle (Problem 4.2)
• Angle (in radians) is dS/r where S
is a circle of radius r centred
on the origin
z
• Solid Angle (in steradians) is
dA/r2 where dA is a sphere of
radius r centred on the origin
y
x
• Applying divergence theorem to red volume
since on sides of closed volume,
field is everywhere parallel to the
sides so surface integrals are zero
•
•Solution to this “paradox”: divergence is non-zero at the origin: applying definition
of divergence to small sphere around origin
and only a fraction
of these field lines emerge into the cone; therefore
divergence theorem works, if origin (source of field lines) is properly treated
Divergence (Problem 4.3)
• Heat up body so it expands by linear factor
• Note fractional increase in volume is
• Divergence is a measure of local expansion/contraction
Centre of mass of body:
doesn’t move as body
expands
Curl (Problem 4.6)
• Consider first, Stokes Theorem applied
to a simple circular loop in the
z=0 plane
z
y
Surface S has cylindrical sides + a circular
cap (taken to be at +ve infinity)
along sides of cylinder
since
is parallel to sides
on cap
x
• Note that double
integral result is the
same however the
loop is wrapped
around the cylinder,
so result is always
Magnetic Potential
• In the curl-free region can define a magnetic potential
such that
where is the cylindrical
polar angle.
as required by
Ampere’s
Theorem
Since In 2D polars
recalling that
• This scalar potential must now be a multi-valued function of
position
• (i) If loop doesn’t enclose wire
• (ii) if loop goes round wire once
• (iii) if loop goes round wire n times
wire carrying current
out of the paper
Maxwell’s Equations
“No magnetic monopoles”
“Gauss’s Theorem”
Stoke’s Theorem
emf induced is rate of change of magnetic flux
through the circuit
But “Ampere’s Law”
because
Recalling “continuity equation”
of Lecture 8
if this were true
capacitors couldn’t
charge up!
Maxwell’s Displacement
Current
surface
Displacement Current
(4)
cutting space
between
capacitor
plates
I
• Now take
• which is a continuity equation allowing
charge to flow
surface cutting
wire
... and, applying Stoke’s Theorem to get
, and hence , we know that
we now get the same answer for
for a surface passing through
the plates of the capacitor (with zero ) as for a surface, with the
same rim, cutting through the current-carrying wire (with non-zero )
```
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