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Unformatted text preview: PHYSICS 222
Introduction to Classical Physics II
Prof. Ruslan Prozorov
Iowa State University
Fall 2011 LECTURE 4 Electric potential work along the path in a potential field
o The work done raising a basketball against gravity
depends only on the potential energy. It does not
depend on other motions. A point charge moving in a
field exhibits similar behavior. PHYS222  Prof. Ruslan Prozorov  Iowa State University 29 August 2011 2 the work done moving a test charge
As a test
charge, q0,
moves in the
presence of
another
charge q.
F d l Fdl cos Fdr
b WA to B rb qq0
qq0 F d l k 2 dr k
r
r
a
ra rb ra qq0
qq0 k ke U b U a ra rb qq0
U ( r ) ke constant
r PHYS222  Prof. Ruslan Prozorov  Iowa State University 29 August 2011 3 potential energy qq0
U (r ) k
r
o potential energy between
like charges increases as
the charges become close
(repulsion).
o Unlike charges have
potential energy becoming
negative as they become
close (attractive).
PHYS222  Prof. Ruslan Prozorov  Iowa State University 29 August 2011 4 The electrical potential (point charge)
Let’s get rid of a
test charge q0
and define: U r q
V (r ) k
q0
r
Units: Volt
1 V = 1 J/C
PHYS222  Prof. Ruslan Prozorov  Iowa State University 29 August 2011 5 general definition of electric potential
B WA to B F d l U B U A , F q0 E A B WA to B q0 E d l q0 VB VA A B VB VA E d l
A PHYS222  Prof. Ruslan Prozorov  Iowa State University 29 August 2011 6 differential definition
electric potential is only defined up to some constant. It
means that you could add (or subtract) ANY constant and
all physics will remain the same. Why? – Because only
potential DIFFERENCE has physical meaning! Why?, because we actually only need an electric field: F U
F qE in electrostatics: E V
U qV remember from 221: V
V
V
ˆ
ˆ
E gradV V x
y
y
z x
PHYS222  Prof. Ruslan Prozorov  Iowa State University ˆ
z 29 August 2011 7 why? because . . .
for ANY constant C: d f x C dx df x dC df x 0
dx
dx
dx so, we could add ANY constant to V to obtain the
same electric field E V x, y, z V x, y, z C therefore we can write: V r E d l
PHYS222  Prof. Ruslan Prozorov  Iowa State University  indefinite integral defined up to a constant 29 August 2011 8 a charged ring
let’s calculate V(r) along the xaxis. First we need electric field.
2 E x ka Q 2 a V x E d l kQ 0 x x2 a x x
2 2 3/2 a 2 3/2 dx d kQ kQ
a2 x2 x x
2 a 2 3/2 C we can choose convenient constant C. For example,
we can require that V(x) become zero at infinity.
Then C must be zero.
kQ
V x 2
2
a x
PHYS222  Prof. Ruslan Prozorov  Iowa State University 29 August 2011 9 an infinite line of charge
2k we already know electric field: E x
dx
x
V x E d l 2k 2k ln x
C where we placed constant C under
ln, so that ln argument is
dimensionless. We can choose ANY constant C and the
potential diverges for x=0 and x=infinity.
(There is nothing wrong with this as we
only need the potential difference. PHYS222  Prof. Ruslan Prozorov  Iowa State University 29 August 2011 10 an infinite charged plate we already know
E
sign x electric field:
2 0 sign x V x E d l dx 2 0 sign x sign x xC x
2 0
2 0
we can choose any C, so
we used C = 0 PHYS222  Prof. Ruslan Prozorov  Iowa State University 29 August 2011 11 two parallel plates
y what is the potential
difference between two
plates? ++++++++++++
E
 d x E
0 V E d l dy y C
0
0 V y y
0 (E points down – negative y
– direction) Let’s choose C, so that V(y=0)=0 potential difference is: PHYS222  Prof. Ruslan Prozorov  Iowa State University V d
0
29 August 2011 12 an electron in two parallel plates
why is it useful? Because we control V by external source
(a battery) of known voltage, so we do not need to know surface
charge density, ++++++++++++
electron F E  d dV
F eE eV e
dy V
V y y 0
0
d V
Fe e
0
d these are all known parameters
PHYS222  Prof. Ruslan Prozorov  Iowa State University 29 August 2011 13 (old style) CRT computer monitor or TV
Anodes (positive,
accelerate
electrons) Fluorescent screen Hot
Electron
cathode beams
(emits
electrons)
PHYS222  Prof. Ruslan Prozorov  Iowa State University 29 August 2011 14 superposition principle
we know that for an electric
field, if we have several
sources, the resulting field is a
vector sum.
Therefore, V E d l E Ei E d l E d l V
i i i V Vi
PHYS222  Prof. Ruslan Prozorov  Iowa State University 29 August 2011 15 Equipotential surfaces and field lines
o Surfaces of equal potential are perpendicular to electric field.
Their density is proportional to electric field strength PHYS222  Prof. Ruslan Prozorov  Iowa State University 29 August 2011 16 field lines and a conducting surface
charge
induced by
polarization PHYS222  Prof. Ruslan Prozorov  Iowa State University 29 August 2011 17 potential in a conductor
here, it is useful to
use differential
form. Inside a
conductor: E V 0
this means that V CONSTANT
usually, the constant is chosen by the boundary
conditions, so that V is continuous across the boundaries.
PHYS222  Prof. Ruslan Prozorov  Iowa State University 29 August 2011 18 a conducting sphere
inside: E 0, V C1 outside: q
Ek 2
r V r E d l k q
q
dr k C2
r2
r Choose C1 and C2 to have V match at r = R.
Obviously, C2=0 and C1 k
inside:
outside:
PHYS222  Prof. Ruslan Prozorov  Iowa State University q
R q
V k const
R
q
V k
r
29 August 2011 19 ...
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This note was uploaded on 11/14/2011 for the course PHYS 5863005 taught by Professor Meyer during the Fall '09 term at Iowa State.
 Fall '09
 MEYER

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