8.022 - Class 1 - 9/6/2006
October 20, 2007
Coursework
3 hour exams and one nal
Problem sets due in class on Tuesdays
E/M- Electromagnetic Forces
Table 1: Table of Forces
Force Interaction Particles Exchange Particles Strength
Gravity
matter/energy
ESG 8.022 Fall 2006 Exam 1
Instructor: Michael Shaw
Wednesday, October 11th, 2006 @ 1:00PM
1
Spherical Charge Distribution (35 p oints)
Consider the following charge distribution in 3-dimensional Euclidean space.
c1 r
r r1
r1
2
= (r) =
c2 (r r2 ) r
r
8.022 Lecture Notes Class 9 - 09/20/2006
Given line of charge with density x
Line Charge
r
Find E () everywhere
1
a
E d = Qenclosed
0
S
Qenclosed = L
L
r
L2r E () =
0
r
E () =
r
20 r
No because E = 0.
No z because vectors cancel out
E (r) 1
r
E (r)
Like a
ESG 8.022 Fall 2006 Final Exam
Instructor: Michael Shaw
Tuesday, December 19th, 1:30PM to 4:30PM
Instructions
Show work on all problems. Partial credit cannot be granted without adequate progress.
Please explain any nonstandard notation.
Complete 4 out of
8.022 Lecture Notes Class 10 - 09/21/2006
Electric Potential
x
E ( ) =
V
=
V ( ) =
x
1
40
V
( )d3 x
x
| |
xx
Vx0 ( ) is work done to move charge from x0 to x.
x
Find V on xy-plane.
(Hint: cylindrical
coordinates)
1
40
V ( ) =
x
V (r, , z ) =
V (r, , 0) =
8.022 Lecture Notes Class 8 - 09/19/2006
What is the E-Field on the axis of a circular loop of a uniformly
charged thin wire with total charge q ?
dEz = dE cos
= dE a2z+z 2
E (z ) =
=
=
=
dq
r
C r3
2 q (cos +sin +zk)
i
j
1
rd
2 +z 2 )3/2
40 0 2a
(a
qz
8.022 Lecture Notes Class 7 - 09/18/2006
So exactly what is curvilinear? And whats this orthonormal stu?
(=1?)
Gradient in Spherical
Let f (x) = f (r, , )
f d
x
df =
= f (er dr + e rd + e r sin d)
df
dr dr
=
+
df
d d
+
df
d d
So ,
df
1 df
1 df
f = er + e
ESG 8.022 Fall 2006 Exam 3
Instructor: Michael Shaw
December 4, 2006 @ 1:00PM
1
Useful Formulae
You may nd some of the following formulae useful. Then again, you may not.
Maxwells Equations: E = ; E = B ; B = 0; B = 0 J + 0 0 E
0
t
t
v
Lorentz Force Law:
8.022 Lecture Notes Class 5 - 09/13/2006
b
d
vl
d
va
a
T d
S
b
a
V
df
dx = f (b) f (a)
dx
b
(T ) d = T (b) T (a)
l
a
(T ) d =
l
b
a
dT
x
( ) d is path-independent i ( ) = f ( ) .
vx
l
vx
1
40
r
(x ) 3
x
d x = E ( )
3
r
=
xx
1
( ) |( |) d3 x
x
40
x x3
8.022 Lecture Notes Class 4 - 09/12/2006
Dot Operator
dT =
=
T
x dx
+
T
y dy
+
T
z dz
T dl
T = 0 stationary point
T =
( T , T , T )
x y z
Whats T ? A scalar function of a vector (usually 3 , sometimes 2 )
T = T (x, y, z ) (x, y, z ) charge density
V (x,
8.022 - Class 3 - 9/11/2006
October 20, 2007
E (z ) =
=
1
40
L
L (x2 +z 2 )3/2 (xi + z j )dx
L
1
+ .
1/2
40 (x2 +z 2 )
L
Dimensionless Integrals
u = x2 + z 2
du = 2xdx
x = z tan
dx = z sec2 d
8.022 Class Notes - Class 3 - 9/11/06
2
L
L
z
0
(z 2 +l2 )3/2
8.022 - Class 2 - 9/7/2006
October 20, 2007
Figure 1: Six point charges of magnitude q in x-y plane.
1. Find E (x, y )
2. Find E (x, 0)
3. Find E (x, y ) for x > 1
4. Remove charge 6. Find E (0, 0)
Solutions
2
8.022 Class Notes - Class 2 - 9/7/06
1.
q 6
k
ESG 8.022 Fall 2006 Exam 2
Instructor: Michael Shaw
November 7, 2006@ 2:00PM
1
Short Answer Questions
a. Find the magnetic eld whose vector potential is A = A0 (x y )
z
b. How is it possible that an observer on a moving train sees clocks on the ground t