MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics: 8.02
W01D2_1 Force on a Charged Object: Solution
Three charged objects are located at the positions shown in the figure. Find a vector
expression for the force on the negatively charged object located at the point P.
Getting Started: Let’s label the particles 1,2, and 3 with q1 = q , q2 = q , and q3 = !q .
!
!
!
!
!
Draw the forces F13 and F23 , and the sum of the forces F3 = F13 + F23 on particle 3.
Make a Plan: Coulomb’s Force law between charges is given by the expression
!
qq !
Fij = k i 3 j rij .
rij
!
!
So we need to write down vector expressions for r13 and r23 in terms of units vectors and
then find the scalar distances r13 and r23 in terms of the given quantities in the figure. We
!
!
!
can then add the forces as vectors using the superposition principle F3 = F13 + F23 .
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics: 8.02
Execute the Plan: From the diagram above on the right, we have that
!
r23 = a î + a ĵ, r23 = 2a
!
r13 = a î,
r13 = a
Notice that these vectors point from a charged object (either 1 or 2) to the charged object
(3) that you are calculating the force on.
We can now write down the forces
!
qq !
q2
F23 = k 23 3 r23 = !k
(a î + a ĵ)
r23
( 2a)3
!
q1q3 !
q2
q2
F13 = k 3 r13 = !k 3 a î = !k 2 î
r13
a
a
The vector sum is then
!
!
!
F3 = F13 + F23 = !k
= !k
q2
q2
(a
î
+
a
ĵ)
+
!k
î
a2
( 2a)3
q2 " 1
q2
%
+
1
î
+
!k
ĵ
$
'&
a2 # 2 2
2 2a 2
Evaluate result: Both the x and y components of the force are negative which agrees
with our vector diagram. Our dimensions are also correct for the force.