Advertisements
Advertisements
Question
An electric dipole of dipole moment`vecp` consists of point charges +q and −q separated by a distance 2a apart. Deduce the expression for the electric field `vecE` due to the dipole at a distance x from the centre of the dipole on its axial line in terms of the dipole moment `vecp`. Hence show that in the limit x>> a, `vecE->2vecp"/"(4piepsilon_0x^3)`
Solution
Electric Field on Axial Line of an Electric Dipole
Let P be at distance r from the centre of the dipole on the side of charge −q. Then, the electric field at point P due to charge −q of the dipole is given by
`vecE_-q=-q/(4piepsilon_0(r+a)^2)hatp`
Where `hatp `is the unit vector along the dipole axis (from − q to q).
Also, the electric field at point P due to charge +q of the dipole is given by
`vecE_+q=q/(4piepsilon_0(r-a)^2)hatp`
The total field at P is
`vecE=vecE_+q+vecE_-q=q/(4piepsilon_0)[1/(r-a)^2-1/(r+a)^2]hatp`
`=>vecE=q/(4piepsilon_0)(4ar)/(r^2-a^2)^2hatp `
Given:
r = x
`vecE=q/(4piepsilon_0)(4ax)/(x^2-a^2)^2hatp`
For x >> a,
`vecE=(4qa)/(4piepsilon_0x^3)hatp`
`vecE=(2vecp)/(4piepsilon_0x^3)" "[.:vecp=(qxx2a)hatp]`
APPEARS IN
RELATED QUESTIONS
Drive the expression for electric field at a point on the equatorial line of an electric dipole.
Define dipole moment of an electric dipole. Is it a scalar or a vector?
Three charges are arranged on the vertices of an equilateral triangle, as shown in the figure. Find the dipole moment of the combination.
An electric dipole is placed at an angle of 30° with an electric field intensity of 2 × 105 N/C. It experiences a torque equal to 4 Nm. The charge on the dipole, if the dipole length is 2 cm, is ______.
Electric charges q, q, - 2q are placed at the comers of an equilateral triangle ABC of side l. The magnitude of electric dipole moment of the system is ____________.
An electric dipole is placed at an angle of 30° to a non-uniform electric field. The dipole will experience ________.
An electric dipole of moment p is placed parallel to the uniform electric field. The amount of work done in rotating the dipole by 90° is ____________.
Two charges –q each are fixed separated by distance 2d. A third charge q of mass m placed at the mid-point is displaced slightly by x(x << d) perpendicular to the line joining the two fixed charged as shown in figure. Show that q will perform simple harmonic oscillation of time period.
`T = [(8pi^3 ε_0 md^3)/q^2]^(1/2)`
A square surface of side l (m) in the plane of the paper. A uniform electric field E(V/m) also in the plane of the paper is limited only to the lower half of the square surface, the electric flux (in SI units) associated with the surface is ______.
In an electric dipole, what is the locus of a point having zero potential?
