Three points charges +q each kept at the vertices of an equilateral triangle of side of l. Determine the magnitude and sign of the charge to be kept at its centroid so that the charges at the vertices remain in equilibrium.
Answers
Answer:
The charge at any vertex will remain in equilibrium if the net force experienced by this charge due to all other three charges is zero.
Let Q be the required charge to be kept at the centroid G.
Considering the charge at A,
Force
F
1
on charge at A due to charge at B
F
1
=
4πϵ
0
1
a
2
q
2
along
BA
Force
F
2
on charge at A due to charge at C
F
2
=
4πϵ
0
1
a
2
q
2
along
CA
Since angle between
F
1
and
F
2
is 60
∘
.
F
1
+
F
2
=
3
4πϵ
0
1
a
2
q
2
along
GA
Also, the distance of centroid G from any vertex is
3
a
The nature of charge to be kept at G has to be opposite (−ve) so that it exerts a force of attraction on charge (+q) kept at A to balance the force
F
1
+
F
2
Force exerted by (−Q) kept at G on charge (+q) at A=
4πϵ
0
1
(
3
a
)
2
=
4πϵ
0
1
a
2
Q.3q
along
AG
Equating the two forces, being equal and opposite.
3
4πϵ
0
1
a
2
q
2
=−
4πϵ
0
1
a
2
3Qq
⇒Q=−
3
q
.