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A rod is bent into an eighth of a circle of radius a, as shown.
The rod carries a total positive charge +Q uniformly distributed
over its length. What is the electric field at the origin?
y
dq
dE = k 2
r
a
x
A rod is bent into an eighth of a circle of radius a, as shown.
The rod carries a total positive charge +Q uniformly distributed
over its length. What is the electric field at the origin?
y
dE = k
dq

dE

a
dq
r
2
=k
dq
a2
charge
dq =  ds =
ds
length
x
Q
dq =
ds
 length of arc 
Q
4 Q
dq =
ds =
ds
2a
a
8


A rod is bent into an eighth of a circle of radius a, as shown.
The rod carries a total positive charge +Q uniformly distributed
over its length. What is the electric field at the origin?
y
d
ds

dE
a
ds = a d
x
A rod is bent into an eighth of a circle of radius a, as shown.
The rod carries a total positive charge +Q uniformly distributed
over its length. What is the electric field at the origin?
y
dq

dE
 4  Q 


  a d  
 a 

k
dE = 2
a

a
dE x = - dE cos 
x
dE y = - dE sin 
Ex = Ey = -


4
0


0
dE cos 
1

 2  =
8
4
4
dE sin 
A rod is bent into an eighth of a circle of radius a, as shown.
The rod carries a total positive charge +Q uniformly distributed
over its length. What is the electric field at the origin?
dE =
4k Q
Ex = 
Ex = 
2
d

4k Q
a

0
4
a
4k Q
a 2
4k Q
Ex = 
a 2
2
cos  d = 

 sin  0
4
= 
4k Q
a
4k Q
a 2
2


0
4
cos  d
sin  4  sin 0
 2

2 2k Q
 0  = 

2
2

a


A rod is bent into an eighth of a circle of radius a, as shown.
The rod carries a total positive charge +Q uniformly distributed
over its length. What is the electric field at the origin?
dE =
4k Q
Ey = 
Ey = 
2
d

4k Q
a

0
4
a
4k Q
a 2
4k Q
Ey = 
a 2
2
sin  d = 

  cos   0
4
= 
4k Q
a
2
0
4k Q
a 2

4k Q
2 
 1 = 
 
2
2

a




4
sin  d
  cos  4  cos 0

2
1 

2 

A rod is bent into an eighth of a circle of radius a, as shown.
The rod carries a total positive charge +Q uniformly distributed
over its length. What is the electric field at the origin?
 2 2kQ  ˆ  4kQ 
2 ˆ
E =  
 i   2 1 
  j
2
2 
 a   a 
2kQ 
E=  2
a 
 2  ˆi   2  2  ˆj
You should provide reasonably simplified answers
on exams, but remember, each algebra step is a
chance to make a mistake.
What would be different if the charge were negative?
What would you do differently if we placed a second
eighth of a circle in the fourth quadrant, as shown?
y
a
x
A rod is bent into an eighth of a circle of radius a, as shown.
The rod carries a total positive charge +Q uniformly distributed
over its length. A negative point charge -q is placed at the
origin. What is the electric force on the point charge? Express
your answer in unit vector notation.
You could start with Coulomb’s Law, rewrite it to calculate the dF on q1=-q due
to dq2 (an infinitesimal piece of the
rod), and then integrate over dq2. In
other words, do the whole problem all
over again.
y
-q
a
x
Or you could multiply the E two slides
back by –q, simplify if appropriate, and
be done with it.