)]()([ 12 rUrUW veconservati
y
yxUF
x
yxUF yx
),(
;),(
2D case:
)]()([ 12 rUrUW con
Ifrd
dUF
)()( 12
)(
)(
2
1
rUrUdUrdrd
dUrdFW
rU
rUL
z
zyxUF
y
zyxUF
x
zyxUF zyx
),,(;
),,(;
),,(
or
then
2 or 3D cases:
z
zyxUF
y
zyxUF
x
zyxUF zyx
),,(;
),,(;
),,(
Several dimensions: U(x,y,z)
Compact notation using vector del, or nabla:
kz
jy
ix
UF
,
Another notation:rd
dUF
Partial derivative is taken assuming all other arguments fixed
Geometric meaning of the gradient :U
Direction of the steepest ascent;
Magnitude : the slope in that directionU
:UF
Direction of the steepest descent
Magnitude : the slope in that direction F
http://reynolds.asu.edu/topo_gallery/topo_gallery.htm
1)The electric potential V in a region of space is given by
)3(),( 22 yxAyxV where A is a constant. Derive an expression for the electric field at any point in this region.
2)The electric potential V in a region of space is given by
33)(
r
crV
where c is a constant. The source of the field is at the origin. Derive an expression for the electric field at any point in this region.
Exercise 5 p. 52
An electron moves from one point to another where the second point has a larger value of the electric potential by 5 volts. If the initial velocity was zero, how fast will the electron be going at the second point?
Problem 3 p. 45
Electric potential V is a scalar!
An old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hourAn old rule of thumb: you have to study 2-3 hours a week outside the class per each credit hour
Outline
• Area vector
• Vector flux
• More problems
• Solid angle
• Proof of Gauss’s Law
Electric field lines
These are fictitious lines we sketch which point in the direction of the electric field.
1) The direction of at any point is tangent to the line of force at that point.
2) The density of lines of force in any region is proportional to the magnitude of in that regionE
E
Lines never cross.
1R 2R
Density is the number of lines going through an area (N)
divided by the size of the area
21
14 R
NdensityRAt
22
24 R
NdensityRAt
24 r
NdensityranyAt
204
1
r
qE
For a charge q located at the origin
Edensity
It is important that the force is proportional to 21r
Gauss’s Law
The total flux of electric field out of any closed surface is equal to the charge contained inside the surface divided by .0
S
enclosedQSdE
0
What is or flux of any vector, e.g. velocity of a water flow?
SdE
Consider a flow with a velocity vector . Let S be a small area perpendicular to .
v
v
a) The volume of water flowing through S per unit time is vS
S
a)
v
b) Now S is tilted with respect to . The volume of water flowing through S per unit time is
vcosvS
n
S
b)
vnSS
Area vector
is the angle between velocity vector and unit vector
normal to the surface S.
v
n
SvvS
cosFlux:
S S
Flux of electric field E
The flux of E
SdEd
S
s SdE
SdE
Have a great day!
Hw: All Chapter 4 problems and exercisesReading: Chapter 4
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