Transcript EM1 - The Evergreen State College

Methods of Math. Physics
Dr. E.J. Zita, The Evergreen State College, 6 Jan.2011
Lab II Rm 2272, [email protected]
Winter wk 1 Thursday: Electromagnetism
* Overview of E&M
* Review of basic E&M: prep for
charge/mass ratio workshop
* Griffiths Ch.1: Div, Grad, Curl, and d
Introduction to Electromagnetism
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4 realms of physics
4 fundamental forces
4 laws of EM
statics and dynamics
conservation laws
EM waves
potentials
Ch.1: Vector analysis
Ch.2: Electrostatics
4 realms of physics, 4 fundamental forces
Classical Mechanics
Quantum Mechanics
(big and slow:
everyday experience)
(small: particles, waves)
Special relativity
Quantum field theory
(fast: light, fast particles)
(small and fast: quarks)
Four laws of electromagnetism
Electric
Magnetic
Gauss' Law
Gauss' Law
Charges → E fields
No magnetic monopoles
Ampere's Law
Faraday's Law
Currents → B fields
(and changing E→ B fields)
Changing B → E fields
Electrostatics
• Charges → E fields
and forces
• charges → scalar
potential differences
dV
• E can be found from V
• Electric forces move
charges
• Electric fields store
energy (capacitance)
Magnetostatics
• Currents → B fields
• currents make magnetic
vector potential A
• B can be found from A
• Magnetic forces move
charges and currents
• Magnetic fields store
energy (inductance)
Electrodynamics
• Changing E(t) → B(x)
• Changing B(t) → E(x)
• Wave equations for E and B
• Electromagnetic waves
• Motors and generators
• Dynamic Sun
Some advanced topics
• Conservation laws
• Radiation
• waves in plasmas,
magnetohydrodynamics
• Potentials and Fields
• Special relativity
Ch.1: Vector Analysis
A = A x x̂  A y ŷ  A z ẑ,
B = Bx x  B y y  Bz z
Dot product: A.B = Ax Bx + Ay By + Az Bz = A B cos q
Cross product: |AxB| = A B sin q =
x
Ax
y
Ay
z
Az
Bx
By
Bz
Examples of vector products
Dot product: work done
by variable force
W = F dl =  F cos q dl
Cross product:
angular momentum
L = r x mv
Differential operator “del”
Del differentiates each component of a vector.


 
 = x  yˆ  zˆ
x
y
y
Gradient of a scalar function = slope in each direction
f
f
 f
 f = x  yˆ
 zˆ
x
y
y
Divergence of vector = dot product = outflow
Vy
Vz
 Vx
ˆ
ˆ
V = x
y
z
x
y
y
Curl of vector = cross product = circulation
x
V = 
x
Vx
y

y
Vy
z

z
Vz

= x

  yˆ 

  zˆ


Practice: 1.15: Calculate the divergence and
curl of v = x2 x + 3xz2 y - 2xz z
(3xz2 )
(2 xz)
 x 2
ˆ
V = x
y
 zˆ
= ...
x
y
y
x
V = 
x
x2
y

z

y
z
xz2  2 xz

= x

  yˆ 

  zˆ

Ex: If v = E, then div E ≈ charge. If v = B, then curl B ≈ current.
Prob.1.16 p.18

Develop intuition about fields
Look at fields on p.17 and 18.
Which diverge?
Which curl?
Separation vector vs. position vector:
Position vector = location of a point with respect to the origin.
r = x xˆ  y yˆ  z zˆ
r = x2  y2  z 2
Separation vector: from SOURCE (e.g. a charge at position r’)
TO POINT of interest (e.g. the place where you want to find the field, at r).
r = r  r ' = ( x  x ') xˆ  ( y  y ') yˆ  ( z  z ') zˆ
r = r  r ' = ( x  x ')  ( y  y ')  ( z  z ')
2
2
2
Source (e.g. a charge or current
element)
r
(separation vector)
Point of interest, or
Field point
Origin
See Griffiths Figs. 1.13, 1.14, p.9
Fundamental theorems
For divergence: Gauss’s Theorem


v
d

=




volume
v  da = flux
surface
For curl: Stokes’ Theorem
    v  da = 
surface
boundary
v  dl = circulation
Dirac Delta Function
rˆ
f= 2
r
This should diverge. Calculate it using (1.71), or
refer to Prob.1.16. How can div(f)=0?
Apply Stokes: different results on L ≠ R sides!
How to deal with the singularity at r = 0? Consider
 0 if x  0
d ( x) = 
 if x = 0
and show (p.47) that



f ( x) d ( x  a) dx = f (a)
Ch.2: Electrostatics:
charges make electric fields
• Charges make E fields
and forces
• charges make scalar
potential differences
dV
• E can be found from V
• Electric forces move
charges
• Electric fields store
energy (capacitance)
Gauss’ Law practice:
What surface charge density does it
take to make Earth’s field of
100V/m? (RE=6.4 x 106 m)
2.12 (p.75) Find (and sketch) the
electric field E(r) inside a uniformly
charged sphere of charge density r.
2.21 (p.82) Find the potential V(r) inside and outside this
sphere with total radius R and total charge q. Use infinity
as your reference point. Compute the gradient of V in each
region, and check that it yields the correct field. Sketch
V(r).