Transcript An old rule of thumb

2D case:
W
conservative


 [U (r2 )  U (r1 )]
U ( x, y )
U ( x, y )
Fx  
; Fy  
x
y
2 or 3D cases:
If
or

dU
F  
dr
U ( x, y, z )
U ( x, y, z )
U ( x, y, z )
Fx  
; Fy  
; Fz  
x
y
z
then

U ( r2 )
 


dU 
W   F  dr      dr    dU  U (r2 )  U (r1 )
L

dr
U ( r1 )
W
con


 [U (r2 )  U (r1 )]
Several dimensions: U(x,y,z)
U ( x, y, z )
U ( x, y, z )
U ( x, y, z )
Fx  
; Fy  
; Fz  
x
y
z
Partial derivative is taken assuming all other arguments fixed
Compact notation using vector del, or nabla:

     
F  U ,   i 
j k
x
y
z

dU
Another notation: F   
dr
Geometric meaning of the gradient
U :
Direction of the steepest ascent;
Magnitude
U : the slope in that direction

F  U : Direction of the steepest descent

Magnitude F : the slope in that direction
http://reynolds.asu.edu/topo_gallery/topo_gallery.htm
1)The electric potential V in a region of space is given by
V ( x, y)  A( x  3 y )
2
2
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
c
V (r ) 
3r 3
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 on ring’s axis
z
V(z) =
Q
1
4pe0 (z 2 + R 2 ) 1 2
¶V
Ez = ¶z
From Chapter 2:

E
1
Qz
40 ( z  R )
2
2
3
2

iz
Chapter 3 Summary
In electrostatics, the electric field is conservative:
 
 E  dr  0

r2
 




1
[U (r2 )  U (r1 )]  [V (r2 )  V (r1 )] 
r E  dr   qlim
0 0 q
0
1
1  
 lim
F  dr

q0  0 q 
0 r1
r2
In electrostatics:
U
Fx  
x
U
Fy  
y
V
Ex  
x
V
Ey  
y
If we know V(x,y) we can find the components of electric field E x and E y
Electric potential V is a scalar!
An
old
rule
ofofthumb:
you
have
totostudy
2-3
hours
aaweek
An
old
rule
thumb:
you
have
study
2-3
hours
week
An
old
rule
of
thumb:
you
have
to
study
2-3
hours
a
week
outside
the
class
per
each
credit
hour
An
old
rule
of
thumb:
you
have
to
study
2-3
hours
a
week
outside
the
class
per
each
credit
hour
An
old
rule
of
thumb:
you
have
to
study
2-3
hours
a
week
outside
the
class
per
each
credit
hour
An
old
rule
of
thumb:
you
have
to
study
2-3
hours
a
week
outside
the
class
per
each
credit
hour
An
old
rule
of
thumb:
you
have
to
study
2-3
hours
a
week
outside
the
class
per
each
credit
hour
An
old
rule
of
thumb:
you
have
to
study
2-3
hours
a
week
outside
the
class
per
each
credit
hour
An
old
rule
of
thumb:
you
have
to
study
2-3
hours
a
week
outside
the
class
per
each
credit
hour
An
old
rule
of
thumb:
you
have
to
study
2-3
hours
a
wee
outside
therule
class
per each
credit
hour
An
old
of
thumb:
you
have
to
study
2-3
hours
a
we
outside
the
class
per
each
credit
hour
An
old
rule
of
thumb:
you
have
to
study
2-3
hours
a
w
outside
the
class
per
each
credit
hour
An
old
rule
of
thumb:
you
have
to
study
2-3
hours
a
outside
the
class
per
each
credit
hour
outside
the
class
per
each
credit
outside the class per each credithour
hour
Van de Graaff generator
Have a great day!
Hw: All Chapter 3 problems
and exercises
Reading: Chapter 4