Sierzega: Electric Charge, Forces, and Fields
Electrical Force Problem Solving
1. Represent and Reason
The metal balls on the cart in the illustration have equal magnitude charge and are very light. The rods
supporting and connecting them are made of an insulating material and are also light. The cart rests on a
smooth table.
q1
q2
q3
–
+
+
(a) Fill in the table that follows. In this instance we consider only electric forces—not other types of
force.
Draw labeled arrows
representing electric
forces exerted on the
left metal ball.
Represent the ball
with a dot.
Draw labeled arrows
representing electric
forces exerted on the
center metal ball.
Draw labeled
arrows representing
electric forces
exerted on the right
metal ball.
Draw labeled arrows
representing electric
forces exerted on the
whole cart (a system
with three charged
balls).
(b) Will the cart tend to accelerate either to the left or to the right? Explain your answer.
Sierzega: Electric Charge, Forces, and Fields
2. Represent and reason Examine the different representations of the same situation that are shown in
the illustrations and decide whether they are consistent with each other.
Represent verbally
A positively charged ball of mass m hangs at the end of a string, as shown in the figure.
Simplify
Represent Pictorially
Charged objects can be considered
particles
String outside exerts
a force on string
inside system

L
Represent physically
Free-body diagram
y
S on q2

Charge q1 exerts repulsive
electric force on charge q2
inside the system
System
+
+
q1
q2
Earth pulls down on
the ball, a
gravitational force.
q1 on q2
x
E on q2
Mass of earth
Represent mathematically
The x-component form of Newton’s second law is
∑Fx = m ax
TS on q2(x) + Fq1 on q2(x) + FE on q2(x) = m ax
–T sin θ + k q1 q2 / r2 + 0 = 0
The y-component form of Newton’s second law is:
∑Fy = m ay
T S on q2(y) + F q1 on q2 (y) + FE on q2(y) = m ay
T cos θ + 0 – mg = 0
Evaluate
Are the components in the equations consistent with the force arrows in the free-body diagram? Explain.
Sierzega: Electric Charge, Forces, and Fields
3. Represent and reason A positively charged ball of mass m hangs at the end of a
string. Another positively charged ball is secured at the top end of the string to a
wooden support. Fill in the table that follows.
+
+
Draw a free-body
diagram for the hanging
ball if both balls are
positively charged.
Represent the diagram
mathematically using
Newton’s second law.
Draw a freeRepresent the diagram
body diagram for mathematically using
the hanging ball Newton’s second law.
if the top ball is
negatively
charged.
4. Represent and reason Two equal-mass stationary balls hang at the end of
strings, as shown at the right. The ball on the left has electric charge +5Q and
the ball on the right has electric charge +Q. The strings make angles less than
45o with respect to the vertical. Fill in the table that follows.
Draw a freebody diagram
for the left
ball.
Draw a freebody diagram
for the right
ball.
Decide which string
makes a bigger angle
with the vertical or if
they make the same
angle.
Apply Newton’s
second law in
component form
for the right ball.
Based on your analysis,
rank order the forces TS on
Q, F5Q on Q, and F E on Q,
listing the largest force
first.
Horizontal x-axis:
Explain the ranking:
Vertical y-axis:
Sierzega: Electric Charge, Forces, and Fields
5. Evaluate the solution The problem: A 2.0-kg cart with a +2.0 x 10–5 C charge on it sits at rest 1.0 m
to the right of a fixed dome with charge +1.0 x 10–4 C. The cart is released. Determine how fast it is
moving when it is 3.0 m from the fixed-charged dome.
Proposed solution: The situation is shown at the right
(view from above).
Simplify: We assume that the dome and cart are point
particles.
Top View
q1 =1x10–4 C
C
q2 = 2x10–5
+
+
+
vi = 0
Represent physically: See the free-body diagram to the
right.
vf = ?
1.0 m
3.0 m
Represent mathematically and solve:
∑Fx = kq1q2/r = m ax
x
ax = kq1q2/rm
= (9 x 109 N•m2/C2)(1.0 x 10–4 C) )(2.0 x 10–5 C)/(1.0 m)(2.0 kg) = 9.0 m/s2.
v2 = 02 + 2(9.0 m/s2)[(3.0 m) – (1.0 m)]
or
v = 18 m/s.
(a) Identify any missing elements or errors in the solution.
(b) Provide a corrected solution or missing elements if there are errors.
6. Regular Problem Determine the ratio of the electric force to the gravitational force between a proton
and an electron.
qe = -1.60 x 10-19 C
qp = +1.60 x 10-19 C
me = 9.11 x 10-31 kg
mp = 1.67 x 10-27 kg
k = 9.00 x 109 Nm2/C2
G = 6.67 x 10-11 Nm2/kg2