Download Continuous Charge Distribution - McMaster Physics and Astronomy

Electric Fields II
Electric Fields
Electric fields are produced by point charges
and continuous charge distributions
Definition:
F  qE


F is force exerted on charge q by an external field E .
Note: q produces a field but it is not external
Continuous Charge Distributions
dq
+
+
+ +
+ +
+ + +
++
+

r
E=?
dE
Source
•Cut source into small (“infinitesimal”) charges dq
•Each produces
(dq)
dE  ke 2 rˆ
r
dq
dE  ke 2
or
r
Steps:
•Draw a coordinate system on the diagram
•Choose an integration variable (e.g., x)
•Draw an infinitesimal element dx
•Write r and any other variables in terms of x
•Write dq in terms of dx
•Put limits on the integral
•Do the integral or look it up in tables.
Example: Uniformly-Charged Thin Rod
(length L, total charge Q)

E ?
d
+
+
L
+ + + + +
+
Charge/Length = “Linear Charge Density”
= constant = Q/L
dq
E   dE   k
r
e
rod
rod
2
+

Solution:
In 2D problems, integrate components separately
to obtain the electric field:
dq
E x   dE x   k 2 (cos )
r
x-component of E
E y   dE y  .......
Example: Uniformly-Charged Ring
y
R
(x,0)
x
Total charge Q, uniform charge/unit length,
radius R
Find:
E at any point (x, 0)
Solution:
Quiz
y
+
+
+
R
O
+
x
Given the semi-circular uniform charge distribution,
what would the direction of the electric field be at the
origin, O?
A) up
B) down
C) left
D) right
Example: Uniformly-Charged
Semicircle
y
+
+
+
Charge/unit length,

Find: E at origin
R

+
, is uniform
x
Solution:
Summary
• Field of several point charges qi:

qi
E   ke 2 rˆi
ri
i
• Field of continuous charge distribution:
dq
E  k
cos( )
r
dq
E  k
sin( )
r
x
e
2
y
e
2