Electric Fields II
I.
Continuous Charge Distribution
We first need to develop a formula for finding the differential electric
field due to an infinitesimal amount of charge, dq.
The electric field can now be found by summing (“integrating”) the
electric field contributions due to all of the differential charges.
NOTE: Both r and
the integral.
r̂ are usually variables and can NOT be removed from
Steps for Solving Integration Problems With Electric Fields:
1.
Break up object into infinitesimal charges dq
2.
Find magnitude of electric field for the infinitesimal charge dq.
a)
find dq in terms of integration variables
b)
find r in terms of integration variables
3.
Find direction of dE by considering force on a positive test charge.
4.
Integration over all charges to get the total electric field.
Uniform Charge Distributions
1.
Linear Charge Distribution:
2.
Surface (Area) Charge Distribution:
3.
Volume Charge Distribution:
4:
Any non-uniform charge distribution can be treated as a uniform
charge distribution if we break it into infinitesimally small pieces:
dq =  dx
or
dq =  dA or
dq =  dV
EXAMPLE:
A 14 cm long bar is charged uniformly with –22 C of charge. What is the electric field
at Point P (36 cm from the middle of the bar) in the figure below.
L2
L2
P
L
x
d
x=0
2
Step 1: Draw a small differential charge element dq as a rectangle of width
dx.
Step 2:
a)
Write dq in terms of the integration variable x.
b)
Write r in terms of the integration variable x.
Step 3:
If a positive test charge is placed at P, it will experience an
attractive force by charge dq.
Step 4: Insert the integrand from Step 3 and evaluate the integral.
Electric Field For A Dipole
When materials like insulators are placed inside an electric field, the material can have an induced
electric field due the shift in the locations of positive and negative charge even though the object
obtains no net charge. This polarization will be considered later in the course. For now we wish to
consider the electric dipole which can be used to model such systems.
Consider two point charges, q and –q lying along the x-axis at d/2 and –d/2. What is the electric field at a
distance z along the z-axis?
Electric Field Due To A Ring Of Charge
We wish to find the electric field at point P which is at a distance z along the z-axis due to a
uniformly charged ring of radius R which lies in the x-y plane with its center at the origin as
shown below.
We always start this type of problem by writing the formula for the electric field due to
infinitesimal piece of charge:
𝑑𝐸⃗ =
𝑘 𝑑𝑞
𝑟̂
𝑟2
We now must figure out how to replace dq, r2 and 𝑟̂ in terms of quantities that will allow us to
sum up (integrate) the contributions for each piece of charge.