derive the expression for electric field at a point on the equatorial line of an electric dipole.
Answers
Answer:
Explanation:
Consider an electric dipole AB consisting charges +q and -q separated by distance 2l and pole strength q. Let, P be the point on the equatorial line at distance d from mid-point of the dipole i.e. O as shown in figure 1.
Now, the electric field, E
1
at point P due to north pole along AP is,
E
1
=
4πϵ
0
1
AP
2
q
E
1
=
4πϵ
0
1
(d
2
+l
2
)
2
q
.........(1) (∵AP
2
=OA
2
+OP
2
)
Similarly, the electric field, E
2
at point P due to south pole along PB is,
E
2
=
4πϵ
0
1
PB
2
q
E
2
=
4πϵ
0
1
(d
2
+l
2
)
2
q
.........(2) (∵PB
2
=OB
2
+OP
2
)
From equations (1) and (2), we get
E
1
=E
2
=E(say)
Now, the net electric field E
P
at P due to the dipole is
E
P
=E
1
+E
2
=2E ........................(3)
Resolving E
1
and E
2
into horizontal and vertical components, as shown in figure 2. The vertical components will be canceled and horizontal will be added together. So, we can write
E
P
=2Ecosθ .............from(3)
E
P
=2(
4πϵ
0
1
(d
2
+l
2
)
2
q
)(
(
d
2
+l
2
)
l
) ............(∵cosθ=
(
d
2
+l
2
)
l
)
For a short dipole, l<<d, therefore
B
P
=
4πϵ
0
1
d
3
2ql
B
P
=
4πϵ
0
1
d
3
Q
........................(∵Q=2ql)