electric field due to dipole moment P on the on its axial line at a distance r from the centre
Answers
Explanation:
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]
E acts in the direction of dipole moment.
Answer:
The electric field due to dipole moment P on its axial line at a distance r from the centre is E = 1/4πε₀×2p/r³
Explanation:
- At a point on its axial line, an electric dipole produces an electric field: Two point charges, q and +q, separated by a short distance 2d form the electric dipole known as AB.
- P is a point on the dipole's axial line located r units away from the electric dipole's midpoint O.
- The electric field at point P caused by the placement of +q at B is,
- E₁ = 1/4πε₀×q/(r-d)² along BP
- When -q is positioned at point A, the electric field at point P results in E₂ = 1/4πε₀×q/(r+d)² (along PA)
- As a result, the resultant electric field (E) has a larger magnitude and acts in the direction of the vector. The electric field at P as a result is, E = E₁ + (-E₂)
- E = 1/4πε₀×q/(r-d)² - 1/4πε₀×q/(r+d)² along BP
- E = q/4πε₀ [1/(r-d)² - 1/(r+d)²] along BP
- E = q/4πε₀ [4rd/ (r²-d²)²] along BP
- Point P will be far from the dipole if d<<r.
- Therefore, E = q/4πε₀ - 4rd/r⁴ = q/4πε₀ - 4d/r³
- Since electric dipole moment p = q×2d
- Therefore, the equation of electric field becomes-
- E = 1/4πε₀×2p/r³
- Here, E acts towards the dipole moment direction.
Thus, the electric field due to dipole moment P on its axial line at a distance r from the centre is E = 1/4πε₀×2p/r³
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