ELECTRIC FIELDS DUE TO CONTINUOUS CHARGE DISTRIBUTIONS
Approach:
 Break charge distribution into small elements (treat each as a point charge)
 Write vector sum of contributions from elements
 Take limit as elements become infinitesimally small → INTEGRAL
Steps:
 1st: Draw a picture showing contribution from
one charge element
o r̂i is unit vector pointing from qi toward P


o Ei is the contribution to E due to qi

 2 : treat qi as a point charge and find Ei
nd

qi


E
k
rˆ
i
e
2 i
o
ri
 3rd: Do a vector sum over contributions


q
from all
i to get an expression for E

q
Ei  k e  2 i rˆi
ri
i
 4th: go to limit of small elements so sum
becomes exact → integral!

q
E  lim ke  2 i rˆi
qi 0
ri
i
 ke 
dq
rˆ
2
r
VECTOR INTEGRAL! – don’t panic, symmetry helps

dq
E  ke  2 rˆ
r
CHARGE DENSITY (Important concept)
 Use to convert integral over dq into integral over spatial variables (i.e. x, y, z)
o Linear charge density: for a uniformly charged line of length L and total
Q


charge Q, the linear charge density is
L
o Surface charge density: for a uniformly charged plane of area A and
Q


total charge Q, the surface charge density is
A
o Volume charge density: for a uniformly charged space of volume V and
Q


total charge Q, the volume charge density is
V
PROBLEM SOLVING STRATEGY: see page 631
 ASK: Does symmetry simplify the problem?
 Approach:
o For point charges, use superposition
o For continuous charge:
 Identify charge elements
 Build an integral
 Look for symmetry to simplify the integral
We will mostly stick to rods and rings for now.
Example: What is the electric field at a point P a distance x from the left end of
a uniformly-charged (positive) rod of length L and charge Q?
Example: What is the electric field at a point P a distance y above the midpoint
of a uniformly-charged (positive) rod of length L and charge Q?
Example: What is the electric field at a distance x from the centre of a
uniformly charged ring along an axis through the centre of the ring and
perpendicular to the plane of the ring? The ring has radius a and total charge Q.