Methods of Math. PhysicsDr. E.J. Zita, The Evergreen State College
Lab II Rm 2272, [email protected]
Winter wk 3, Thursday 20 Jan. 2011
• Electrostatics & overview• Div, Grad, and Curl
• Dirac Delta
• Modern Physics
• Logistics: PIQs, e/m lab writeup
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
Differential operator “del”
Del differentiates each component of a vector.
Gradient of a scalar function = slope in each direction
Divergence of vector = dot product = outflow
Curl of vector = cross product = circulation =
ˆ ˆx y zx y z
ˆ ˆf f f
f x y zx y z
ˆ ˆyx zVV V
x y zx y z
V
V
Practice: 1.15: Calculate the divergence and
curl of v = x2 x + 3xz2 y - 2xz z2 2(3 ) ( 2 )
ˆ ˆ ...x xz xz
x y zx y z
V
zyx
xzxzxzyx
zyx
ˆˆ
222
V
Ex: If v = E, then div E ≈ charge. If v = B, then curl B ≈ current.
Prob.1.16 p.18
1.22 Gradient
1.23 The operator
1.2.4 Divergence
1.2.5 Curl
1.2.6 Product rules
1.2.7 Second derivatives
2 V V 2 f f Laplacian of scalar Lapacian of vector
Fundamental theorems
For divergence: Gauss’s Theorem (Boas 6.9 Ex.3)
For curl: Stokes’ Theorem (Boas 6.9 Ex.4)
volume surface
d d flux v v a
surface boundary
d d circulation v a v l
Derive Gauss’ Theorem
volume surface
d d flux v v a
Apply Gauss’ thm. to Electrostatics
0
0
0
charge density therefore
volume surface
volume
volume volume
qE d E d
dqd q
d
E d d
E
a
Derive Stokes’ Theorem
surface boundary
d d circulation v a v l
Apply Stokes’ Thm. to Magnetostatics
0
0
0
current density , so
surface boundary
surface
surface surface
B d B d I
dIJ I J d
da
B d J d
B J
a l
a
a a
Separation vector vs. position vector:
Position vector = location of a point with respect to the origin.
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).
222ˆˆˆ zyxrzzyyxx r
2 2 2
ˆ ˆ ˆ' ( ') ( ') ( ')
' ( ') ( ') ( ')
x x x y y y z z z
x x y y z z
r r
r r
r
r
Origin
Source (e.g. a charge or current element)
Point of interest, orField point
See Griffiths Figs. 1.13, 1.14, p.9
(separation vector)rr’
r
Dirac Delta Function
2
ˆ
r
rf
0 0( )
0
if xx
if x
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
and show (p.47) that
( ) ( ) ( )f x x a dx f a
Ch.2: Electrostatics: charges make electric fields
• Charges → E fields and forces
• charges → scalar potential differences
• E can be found from V• Electrodynamics: forces
move charges• Electric fields store
energy (capacitance)
E.dA = q/0=, E = F/q
1 ( ')( ) '
4
rV r d E dl
r
VE
F = q E = m a
W = qV
C = q/V
charges ↔ electric fields ↔ potentials
Gauss’ Law practice:
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).
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 .
Curl
Curl of vector = cross product = circulation
zyx
VVVzyx
zyx
zyx
ˆˆV
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