Download 17. Finding Electric Field from Electric Potential


Getting Electric Field E from Electric Potential V (IMPORTANT)
 Useful because V is a scalar and superposition does not require vector sum
 

V


E
  ds
Start by noting:
B
 
implies dV   E  ds
A

E due to a point charge (radial electric field)
 
 only nonzero term in E  ds is Eradial dr
 
dV


E
 ds   Eradial dr
 So:
dV
E


 Implies: radial
dr
ke q
d  ke q 




 Check: dr r
r2


(as it should be)
GENERAL RELATION - ELECTRIC FIELD AND ELECTRIC POTENTIAL:

E
Can get components of
from derivatives of V
Start with:


E  E x iˆ  E y ˆj  E z kˆ and ds  dxiˆ  dy ˆj  dz kˆ
Then:
 
dV   E  ds   E x dx  E y dy  E z dz
Says:
dV
Ex  
dx

Ey  
dV
dy

Ez  
dV
dz

 Components of E are nonzero only for directions along which V changing
o If V is constant over some surface (i.e. an equipotential):

E

has no components along that surface

 Says: E is always perpendicular to equipotential surfaces

E
o Example: for a point charge,
radial, equipotentials are spherical
EQUIPOTENTIAL SURFACES:
 like contour maps
o “valleys” centred on negative charge
o “hills” centred on positive charge
Example: Electric dipole
(contours marked in volts)
 Electric field lines start
on positive charge,
end on negative
charge
 Electric field lines
perpendicular to
equipotentials at
crossing
Example: Move charge  q 0 from A → B
1st: What is the change in potential energy of
the charge  q 0
 U  q0 VB  VA   q0 24V   16V 
 So: U  q0  40 V
2nd: How much work must be done ON  q 0 charge to move it from A → B?
 External force must do Wext  q0  40 V
o External force does positive work. Has to push to move positive
charge from lower electric potential to higher electric potential
3rd: How much work does electric field do when  q 0 is moved from A → B?
 Electric field does Welect  U  q0  40 V
 Along path, electric force is opposite to displacement (work negative)
dV
E


How to understand x
dx
Ey  

 E perpendicular to equipotentials
everywhere.

 Magnitude of E is related to spacing
of equipotentials.
 Electric field magnitude highest
where equipotentials most closely
spaced.
 Same as the relation between slope
and spacing of contours on a map
o Ball would roll along path
perpendicular to contour

o Equivalent to direction of E on
equipotential map
http://atlas.nrcan.gc.ca/site/english/maps/topo/map
dV
dy
Ez  
dV
dz

Example: Calculate E by starting with electric potential V for an electric dipole
Example: In some region of space, the dependence of electric potential on
location is given by:
V  x, y, z   5 x  3 x 2 y  2 yz 2
where the coordinates are in m.

(a) What is E at a point specified by coordinates (x,y,z)?

E
(b) What is the magnitude of
at the point (1, 0, -2) ?
Will now go on to talk about electric potential, electric field, and Gauss’s Law for
Continuous charge distributions
Strategy:
 Break charge distribution into small elements
 Sum over the effects of the carge elements
 Convert the sum into an integral
Will then go on to electric potential and electric field near and in conductors.