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Note 3 Electric Field
Electric Field
The electric field tells you what force a test charge would encounter if it were placed at a
certain location in space near some source charges. The source charges produced the
electric field. The test charge is affected by it. Thus, if I know what the electric field is at a point
in space, I don’t need to know what and where the source charges are when I want to know what
will happen to a test charge.
Here is the definition for the electric field in terms of what it does to the test charge at the
location point called p.
!
!
Felectric on q at point p
Eat point p =
qtest
test
This version is a little more compact.
!
!
Felectric on q at point p = qtest Eat point p
test
If a point charge is the source charge, the electric field it generates at point p is the following. This
is one way to calculate for the electric field in terms of what the source charge does at point p.
!
Edue to q
source
at point p
=k
qsource
r̂from q
rp12
source
to point p
For a positive source charge, the field points away from the source charge. For a negative source
charge, the field points towards the source charge.
The unit of electric field is just force divided by the charge, N/C.
Electric Field and Force Due to a Single Charge
The electric field due a positive source charge, the electric force on a positive test charge and on
a negative test charge.
electric field
force on +charge
force on –charge
+
+
+
page 1
Example: Electric Force Due to a Uniform Electric Field
Somewhere, there is a uniform electric field of 5x104 N/C in the positive x direction. What forces
act on the charges +3 µC and –9 µC immersed in this electric field?
electric field
+3 µC
+
F
F
–
–9 µC
For the +3 µC charge,
!
!
F = q0 E = (+3 ×10−6 C )(5 ×104 N/C )(+x̂) = 0.15 N(+x̂)
For the –9 µC charge,
!
!
F = q 0E = (−9 ×10−6 C )(5 ×104 N/C )(+x̂) = 0.45 N(−x̂)
Positive charges move in the direction of the electric field and negative charges move in the
opposite direction of the electric field.
page 2
Example: Dangling a Charge in a Uniform Electric Field
Somewhere, there is a uniform electric field in the positive x direction. A object with a mass of
0.025 kg and charge of –3 µC is suspended motionlessly from a thread in this electric field making
an angle of 10° from the vertical.
(a) What is the magnitude of the tension in the thread?
(b) What is the magnitude of the electric field?
Suspended means the net force is zero on the object. There are three forces on the mass. Here
they are. They have to be arranged this way so that the net force is zero.
T, tension
Fe, electric force
Fg, gravity
The electric force points to the left because the electric field points to the right and the charge is
negative.
electric field
–
Fe, electric force
According to Newton’s second law, in the x direction,
!
!
!
ΣFx = Fe + Tx = − Fe + Tx = 0
In the y direction,
!
!
!
ΣFy = Fg + Ty = −Fg + Ty = 0
Solve the y direction first since we know the most for that direction.
Ty = Fg
⇒ T cos θ = mg
⇒ T =
mg
(0.025 kg)(9.8 m/s 2 )
=
cos θ
cos(10°)
T = 0.249 N
The first equation now says
Tx = Fe
⇒ T sin θ = q E
⇒ E =
T sin θ (0.249 N )sin(10°)
=
q
3 µC
E = 1.44 ×104 N /C page 3
Electric Field Lines
Electric field lines are lines that follow the electric field vectors without regards to the magnitude.
Here are electric field lines for a point charge.
+
–
Here are electric field lines with two like source charges.
+
+
–
–
Here are electric field lines with two opposite source charges. This is called an electric dipole.
electric field, positive charge
electric field, total
electric field, negative charge
+
–
+
–
page 4