Physics 2: Electrostatics
Kalamazoo Valley Community College
John Stahl
Electrostatics
Charge (q): A fundamental measurement of unbalanced electrons (-)
or protons (+). Unit Coulomb or C
Objects with different charges attract, while like charges repel. We
describe this behavior with forces.
The strength of the force is based on how far apart the two objects
are, the amount of charge each carries, and how well the charges
sense each other.
F21
1
2
F12
r
𝑞1 ∙ 𝑞2
𝐹𝐸 = 𝑘𝑒
𝑟2
𝑁 ∙ 𝑚2
𝑘𝑒 = 8.99𝐸9
𝐶2
Example
A charged metal ball with a mass of 35.0g is
tethered to a light string. A second charge is placed
nearby. What angle does the string make to the
vertical if q1 is 1.5mC and q2 is 3.25mC ?
Addtionaly, Solve for the string tension.
a
1
40cm
30cm
2
Start with a free-body diagram of the tethered
ball.
T
F21
q21
q
1
W=mg
The ball feels pulled to the earth by gravity, the Tension in the
string T, points at q, and for the electric force F21 we use trig to
solve for the angle q21. Before we can add up force we have a
lot of setup work to do.
F21
Trig
q21
We need to have an angle for F21, which we will
call q21. Using the right triangle we solve for q12.
The interior angle would be useful if we wanted
to know the direction of F12.
q12 = q21
40cm
q12
R
30𝑐𝑚
𝑡𝑎𝑛(𝜃12 ) =
40𝑐𝑚
𝜃12 = 36.9°
We will also need the hypotenuse, which is the distance between the two charges.
𝑅2 = (30𝑐𝑚)2 +(40𝑐𝑚)2
𝑅 = 50𝑐𝑚
30cm
F12
Electric Force
There is enough information to
calculate the magnitude of the electric
force F21.
𝐹21 = 𝑘𝑒
F21
q21
1
𝑞1 ∙ 𝑞2
𝑟2
𝐹21 =
8.99𝑥109
R=50cm
𝑁 ∙ 𝑚2 1.5𝑥10−6 𝐶 ∙ 3.25𝑥10−6 𝐶
∙
𝐶2
(0.5𝑚)2
𝐹21 = 0.1753𝑁
We have enough of the pieces to start summation of the forces.
2
q12
F12
Setup a table of forces written on the FBD breaking into
their x and y components. We do not know any values
for the string tension and will need to write in variables
into the table.
𝐹21𝑥 = −0.1753𝑁 ∙ cos⁡(36.9°)
𝐹21𝑥 = −0.1402𝑁
When we finish the table we can move on to
adding the forces in the x-direction.
𝐹21𝑦 = 0.1753𝑁 ∙ sin⁡(36.9°)
𝐹21𝑦 = 0.1053𝑁
Points left
𝑊 = −0.035𝑘𝑔 ∙ 9.8𝑠𝑚2
Forces
x
y
T
𝑇 ∙ cos⁡(𝜃)
𝑇 ∙ sin⁡(𝜃)
F21
−0.1402𝑁
0.1053𝑁
W
0
−0.3434𝑁
𝐹𝑥 = 0
In electrostatics the summation of force is
always equal to zero.
𝑇 ∙ cos 𝜃 − 0.1402𝑁 + 0 = 0
𝑇 ∙ cos 𝜃 = 0.1402𝑁
𝑇=
0.1402𝑁
cos 𝜃
Start with the x-direction and solve for the
Tension, while we do not get a number we
have an important relationship for the
next step.
𝐹𝑦 = 0
𝑇 ∙ sin 𝜃 + 0.1053𝑁 − 0.3434𝑁 = 0
𝑇 ∙ sin 𝜃 = 0.2377N
0.1402𝑁
cos 𝜃
Add up the forces in the y-direction.
Replace T with the relationship from
above.
sin⁡(𝜃)
Trig Note: cos⁡
=tan⁡(𝜃)
(𝜃)
∙ sin 𝜃 = 0.2377N
0.2377N
tan 𝜃 =
0.1402𝑁
𝜃 = 59.5°
𝜃 = 59.5°
𝛼 = 180° − 90° − 𝜃
Recall we really wanted a, which is the angle
the string makes with the vertical. Again we
use trig to solve for a.
𝛼 = 30.5°
For the sting tension bring back the relationship between T and q.
0.1402𝑁
𝑇=
cos 𝜃
0.1402𝑁
=
cos⁡(59.5°)
𝑇 = 0.276𝑁