Work Done in Rotating a Dipole

Definition of an Electric Dipole

    An electric dipole is a system of two equal and opposite electric charges separated by a very small distance, generating an electric dipole moment.

Electric Dipole Rotation in Physics

    Dipole rotation in physics examines the behavior of electric dipoles in external fields, focusing on torque and energy.

Dipole Moment

    An electric dipole is a system of two equal and opposite electric charges separated by a very small distance and the dipole moment is the product of the magnitude of one charge and the vector representing the separation.

It is represented by `\vec p`

The vector formula for the electric dipole moment is -

`\vec p = q \times \vec d`

where,

q is the magnitude of each charge and 

`\vec q` is the displacement vector pointing from the negative charge to the positive charge.

S.I. unit of electric dipole moment is Coulomb `\times`meter

Work Done in Rotating a Dipole

    Suppose an electric dipole is placed in a uniform electric field, on which a torque acts, which tries to rotate the electric dipole to bring it to a position of stable equilibrium.

    If we try to rotate an electric dipole from `\theta_1` to `\theta_2`, then we have to do work against the restoring force acting on the electric dipole.

Work Done in Rotating a Dipole

    Let dw be the small amount of work done in rotating a dipole throung `d\theta`

Work done = Torque `\times` angular displacement

`dW = \tau \times d\theta `

`W = \int_{\theta_1}^{\theta_2} \tau d\theta`

`W = \int_{\theta_1}^{\theta_2} p E sin\theta d\theta`

`W = p E \int_{\theta_1}^{\theta_2} sin\theta d\theta`

`W = p E[ - cos\theta ]_{\theta_1}^{\theta_2}`

`W = - p E[ cos\theta ]_{\theta_1}^{\theta_2}`

`W = - p E[ cos\theta_2 - cos\theta_1 ]`

`W =  p E ( cos\theta_1 - cos\theta_2 )`

Spcial Case - 1

If the dipole is rotated by `90^\circ` from equilibrium.

`\theta_1 = 0` to `\theta_2 = 90` than

`W =  p E ( cos 0^\circ - cos 90^\circ )`

`W =  p E ( 1 - 0 )`

`W =  p E `

Spcial Case - 2

In rotating the dipole by `180^\circ` from the equilibrium

`\theta_1 = 0` to `\theta_2 = 180` than

`W =  p E ( cos 0^\circ - cos 180^\circ )`

`W =  p E {1 - ( - 1)}`

`W =  p E {1 + 1}`

`W =  2 p E `

Related Questions

1.    What is an electric dipole?
2.    Define electric dipole.
3.    What is the definition of an electric dipole moment?
4.    What is the SI unit of electric dipole moment?
5.    What is the formula of electric dipole moment?

Numerical Question on Work Done in Rotating a Dipole

Q.    The distance between equal and opposite charges of 2 micro coulombs is 3.0 cm. Calculate the electric dipole moment.     (Ans. 6`\times10^{-8} C m`)

Q.    Two electric charges x and y are separated by distance p . If the distance of any point P is 30 cm from the electric dipole then find the electric field on (i) axil point P (ii) equatorial point P. (Answer 4.0 `\times 10^3  N/C` and `2.0 \times 10^3  N/C`)

Q.    An electric dipole is placed in a uniform electric field `10^4`N/C at an angle `30^\circ`. In this condition electric dipole feels torque of `9 \times 10^{-29}Nm` then find dipole moment of electric dipole. (Answer 1.8 `\times 10^{-29}Cm`)

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