Doc: Electric Dipole Class 12 Notes | EduRev

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ELECTRIC DIPOLE

  • Definition: When two charges of equal magnitude and opposite sign are separated by a very small distance, then the arrangement is called electric dipole.
  • Total charge of the dipole is zero but electric field of the dipole is not zero as charges q and -q are separated by some distance and electric field due TO them when added is not zero.
    Doc: Electric Dipole Class 12 Notes | EduRev

We define a quantity called Dipole Moment Doc: Electric Dipole Class 12 Notes | EduRevfor such a system such that:
Doc: Electric Dipole Class 12 Notes | EduRev
where Doc: Electric Dipole Class 12 Notes | EduRev conventionally, represents the direction from –q to +q.
Axis of a dipole is the line joining –q to +q

  • Midpoint of the axis of the dipole is called the centre of the dipole.
  • All the distances in the space are measured from the centre of the dipole.
  • Perpendicular bisector of the axis of the dipole is called the equatorial line of the dipole.

Q. Why are we defining a dipole? Why can’t we treat it as simply two charge system? Why are we giving it special treatment?
Ans. Dipole is commonly occurring system in nature. We need to generalize our results in context with the dipole system to avoid repeated single point charge calculations using Coulomb’s Law. After these results we would be able to directly apply simplified results derived here to dipole systems.

We approach the calculation of electric field due to an electric dipole in following manner:
(a) Field of an electric dipole at points in equatorial plane:
We now find the magnitude and direction of electric field due to dipole.

Doc: Electric Dipole Class 12 Notes | EduRev

  • P point in the equatorial plane of the dipole at a distance r from the centre of the dipole. Then electric field due to -q and +q are

Doc: Electric Dipole Class 12 Notes | EduRev
Doc: Electric Dipole Class 12 Notes | EduRev
and they are equal in magnitude. Note that Doc: Electric Dipole Class 12 Notes | EduRev is the unit vector along the dipole axis (from -q to + q)

  • From figure we can see the direction of
    Doc: Electric Dipole Class 12 Notes | EduRev
    Their components normal (perpendicular) to dipole cancel away and components along the dipole add up.
  • Dipole moment vector points from negative charge to positive charge so in vector form.

Doc: Electric Dipole Class 12 Notes | EduRev

Substituting the values ofDoc: Electric Dipole Class 12 Notes | EduRev calculated above also, by geometry,
Doc: Electric Dipole Class 12 Notes | EduRev 
Doc: Electric Dipole Class 12 Notes | EduRev
Doc: Electric Dipole Class 12 Notes | EduRev
Now, very frequently we measure electric field at large distances from the dipole ie. r>>a
Therefore, by approximation,
Doc: Electric Dipole Class 12 Notes | EduRev
We know that, by definition,
Doc: Electric Dipole Class 12 Notes | EduRev
Hence,
Doc: Electric Dipole Class 12 Notes | EduRev
Observe, the - sign, it represents that electric field at the equator is in the opposite direction to the dipole moment of the electric dipole i.e. +q to –q.
Doc: Electric Dipole Class 12 Notes | EduRev(b) Field of an electric dipole for points on the axis:Doc: Electric Dipole Class 12 Notes | EduRev

  • Let P be the point at a distance r from the centre of the dipole on side of charge +q as shown in the figure

Doc: Electric Dipole Class 12 Notes | EduRev
Doc: Electric Dipole Class 12 Notes | EduRev
Where Doc: Electric Dipole Class 12 Notes | EduRevis the unit vector along the dipole axis (from -q to + q)
Doc: Electric Dipole Class 12 Notes | EduRev
Thus,
Doc: Electric Dipole Class 12 Notes | EduRev
or
Doc: Electric Dipole Class 12 Notes | EduRev
for r >> a
Doc: Electric Dipole Class 12 Notes | EduRev
As we know that, by definition of dipole moment,
Doc: Electric Dipole Class 12 Notes | EduRev

  • Unit of dipole moment is Coulomb meter (Cm).
  • Thus, in a nutshell, in terms of electric dipole moment, electric field due to a dipole at large distances (r >> a)

(i) At point on equatorial plane ( r >> a)
Doc: Electric Dipole Class 12 Notes | EduRev
(ii) At point on dipole axis (r>>a)
Doc: Electric Dipole Class 12 Notes | EduRev

Note: Dipole field at large distances falls off as 1/r3.

Now, we can generalize the calculation of electric field at any general point in space due to the dipole using the above results.
Doc: Electric Dipole Class 12 Notes | EduRev

Any general point in space, can be located using the polar coordinates r and 𝜃, where the origin can be placed at the center of the dipole, as shown in the above figure.

Now, for any general point P in space located at distance r from centre and inclined at an angle 𝜃 with the axis of the dipole, we can imagine components of the original dipole with dipole moment Doc: Electric Dipole Class 12 Notes | EduRev such that the P lies on the equator of one component and on the axis of the other component.

Now, Lets express our dipole moment Doc: Electric Dipole Class 12 Notes | EduRev as,
Doc: Electric Dipole Class 12 Notes | EduRev

Doc: Electric Dipole Class 12 Notes | EduRev

WhereDoc: Electric Dipole Class 12 Notes | EduRevis the component of the original dipole moment, such that point P is located on the axis of this dipole, i.e.
Doc: Electric Dipole Class 12 Notes | EduRev
Now, at P,
Doc: Electric Dipole Class 12 Notes | EduRev
We know that,
Doc: Electric Dipole Class 12 Notes | EduRev
Thus,
Doc: Electric Dipole Class 12 Notes | EduRev
Therefore,
Doc: Electric Dipole Class 12 Notes | EduRev
One of the component will be along the axial component of electric dipole i.e.Doc: Electric Dipole Class 12 Notes | EduRev and the other component will be along the equatorial component of electric dipole i.e. Doc: Electric Dipole Class 12 Notes | EduRev

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