Download 21-7 Electric Field Calculations for Continuous Charge Distributions

Day 4: Electric Field Calculations for
Continuous Charge Distributions
• A Uniform Distribution of Surface charge
• A Ring of Continuous Charge
• A Long Line of Charge
• A Uniformly Charged Disk
• Two Parallel Plates
A continuous distribution of charge may be treated as
a succession of infinitesimal (point) charges, ΔQ,
each generating an electric field, ΔE.
In the lim Q0
E dE
where:

Q dQ
is the contribution of the Electric Field due to dQ at a
radial distance of “r” away
Integrating both sides:
or
1
dQ
E
40  r 2
Note: Remember that the electric field is a vector; you will
need a separate integral for each component.
A Ring of Charge
A thin, ring-shaped object of radius a holds a
total charge +Q distributed uniformly around
it. Let λ be the charge per unit length (C/m).
The electric field at a point P on its axis, at a
distance x from the center is given by:
1
Q
E
where x  a
2
40 x
A Continuous Line of Charge
a very long line (ie: a wire)
of uniformly distributed
charge. Assume x is much
smaller than the length of
the wire, and let λ be the
charge per unit length
(C/m). The magnitude of
the Electric Field at any
point P a distance x away
is:
 1

E
where y  , as   
20 x
2
The Electric Field or a Uniformly
Charged Disk
Charge is distributed uniformly over a thin
circular disk of radius R. The charge per unit
area (C/m2) is σ. The electric field at a point P
on the axis of the disk, a distance z above its
center is:

E
2 0
if z << R
Electric Field Between Two Parallel Plates
The electric field between two large parallel plates or, which are
very thin and are separated by a distance d. One plate carries a
uniform surface charge density σ and the other carries a
uniform surface charge density –σ, where σ = Q/A (Coulomb /
m2 )

E
0
The electric field is uniform if we assume the plates are large
compared to the separation distance