CPS question
Three point charges lie at the
vertices of an equilateral triangle as
shown. All three charges have the
same magnitude, but Charges #1
and #2 are positive (+q) and Charge
#3 is negative (–q).
Charge #2
+q
Charge #1
+q
y
The net electric force that Charges
#2 and #3 exert on Charge #1 is in
x
A. the +x-direction.
B. the –x-direction.
C. the +y-direction.
D. the –y-direction.
–q
Charge #3
E. none of the above
CPS question
Three point charges lie at the
vertices of an equilateral triangle as
shown. All three charges have the
same magnitude, but Charge #1 is
positive (+q) and Charges #2 and #3
are negative (–q).
The net electric force that Charges
#2 and #3 exert on Charge #1 is in
Charge #2
–q
Charge #1
+q
y
x
A. the +x-direction.
B. the –x-direction.
C. the +y-direction.
D. the –y-direction.
–q
Charge #3
E. none of the above
1
Electric field
We can define the “electric
field” as the electric force per
unit of charge
E=
F
q0
q0 is called a
“test charge”
The force exerted on an arbitrary point charge
by a field is:
F = q0 E
Remember: the force on a charged body is
exerted by a field created by other charges
Electric field (rigorous definition)
The presence of the “test charge” may affect the distribution of
charge on the body creating the field (e.g. metal). So the field
before and after may be different we probe it with the charge. We
want to avoid this situation by taking the “limit”:
F
q0 →0 q
0
E = lim
In practice, we take the charge distribution to be fixed
2
Electric fields may be mapped by force on a test
charge
If one measured the force on a test charge at
all points relative to another charge or
charges, an electric field may be mapped.
Electric field of a point charge
F=
1 qq0
rˆ
2
4πε 0 r
⇒ E = 4πε1
0
q
rˆ
r2
3
Electric field of a point charge (cont.)
E=
1
q
rˆ
2
4πε 0 r
The field points away from
the positive charge, and
toward the negative charge
Electron trajectory in a uniform
electric field
ay =
Fy − eE
=
m
m
4
Electron trajectory in a uniform
electric field
x = v0 x t
1
1 − eE 2
y = v0 y t + a y t 2 → y =
t
voy = 0
2
2 m
⇒ y = − 12 mveE
x2
ox
Electric field calculations
r
r
r
r
E = E1 + E2 + E3 + ...
=
1
4πε 0
⎛ q1
⎜ 2
⎜r
⎝ 1
rˆ1 +
⎞
q
q2
rˆ + 23 rˆ3 + ... ⎟⎟
2 2
r3
r2
⎠
5
CPS question
Two point charges and a point P lie
at the vertices of an equilateral
triangle as shown. Both point
charges have the same magnitude q
but opposite signs. There is nothing
at point P.
Charge #1
–q
P
y
The net electric field that Charges
#1 and #2 produce at point P is in
x
A. the +x-direction.
B. the –x-direction.
C. the +y-direction.
D. the –y-direction.
+q
Charge #2
E. none of the above
CPS question
Two point charges and a point P lie
at the vertices of an equilateral
triangle as shown. Both point
charges have the same negative
charge (–q). There is nothing at
point P.
The net electric field that Charges
#1 and #2 produce at point P is in
Charge #1
–q
P
y
x
A. the +x-direction.
B. the –x-direction.
C. the +y-direction.
D. the –y-direction.
–q
Charge #2
E. none of the above
6
Field of an electric dipole
Charge density
Linear distribution:
Surface (2D)
distribution:
Volume (3D)
distribution:
λ=
dQ
→ Q = ∫ λdx
dx
σ=
dQ
→ Q = ∫∫ σds
ds
Surf .
ρ=
dQ
→ Q = ∫∫∫ ρdv
dv
Volume
7
Field of a ring of charge
Field of a line of charge
8
Field of a disk of charge
Field of two parallel charged sheets
9
Electric field lines
Imaginary lines or curves drawn through a region of
space such that their tangent at any point are in the
direction of the electric field vector at that point
Electric field lines (cont.)
•Higher density of lines → stronger field
•Arrows define the direction of the field
•Field lines never intersect (the field is uniquely defined at
each point)
10
Electric dipoles
Pair of charges with equal magnitude and opposite
sign (q,-q), separated by a small distance d
Force and torque on electric dipole
11