Derive expression for the electric field at the point on the
(a) axial line (b)perpendicular of dipole
Answers
Answer:
Electric field due to an electric dipole at a point on its axial line: AB is an electric dipole of two point charges −q and +q separated by small distance 2d. P is a point along the axial line of the dipole at a distance r from the midpoint O of the electric dipole.
The electric fiedl at the point P due to +q placed at B is,
E
1
=
4πε
0
1
(r−d)
2
q
(along BP)
The electric field at the point P due to −q placed at A is,
E
2
=
4πε
0
1
(r+d)
2
q
(along PA)
Therefore, the magnitude of resultant electric field (E) acts in the direction of the vector with a greater, magnitude. The resultant electric field at P is
E=E
1
+(−E
2
)
E=[
4πε
0
1
(r−d)
2
q
−
4πε
0
1
(r+d)
2
q
] along BP
E=
4πε
0
q
[
(r−d)
2
1
−
(r+d)
2
1
] along BP
E=
4πε
0
q
[
(r
2
−d
2
)
2
4rd
] along BP
If the point P is far away from the dipole, then d≪r
∴E=
4πε
0
q
−
r
4
4rd
=
4πε
0
q
r
3
4d
E=
4πε
0
1
r
3
2p
along BP
[∵ Electric dipole moment p=q×2d]
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