there are four point charges placed at the vertices of a square of side a. an known charge is placed at the centre of the cubic square such that system is in equilibrium. also indicate the sign of unknown charge.
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Answer:
O is the center of square and point P is at height h from the center
OA=OB=OC=OD=a
The magnitude of the electric force due to each charge is
4πε
0
1
a
2
+h
2
The components of the the force perpendicular to OP sum up to zero because of the symmetrical distribution of charges about OP. Hence, the resultant force at P is upward along OP. The magnitude of the force is given by
R
up
=4×
4πε
0
1
h
2
+a
2
cosθ
=
πε
0
(h
2
+a
2
)
h
2
+a
2
h
=
πε
0
(h
2
+a
2
)
3/2
Qqh
F
down
=weight=mg
For equilibrium, mg=
πε
0
(h
2
+a
2
)
3/2
Qqh
or Q=πε
0
hq
mg
(h
2
+a
2
)
3/2
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