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Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE

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INTRODUCTION TO DIPOLE IN UNIFORM EXTERNAL FIELD
If a dipole is kept in an external electric field, it experiences a rotating effect. By external electric field, we mean electric field that is not induced by dipole itself. The rotating effect is also called torque on the dipole. How we can calculate the torque on a dipole and what are its applications? This can be done by calculating the net torque on opposite charges of the dipole.

Dipole in Uniform External Field
To find torque on a dipole from an external field, consider there is electric dipole placed in an uniform external field. The uniform external electric field is produced externally and is not induced by dipole.
Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE

Fig: An electric dipole placed in non-uniform external electric field 

The external electric field Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEEwill produce electric force Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE on positive charge in upward direction (same direction as Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEEand on negative charge in downward direction (opposite direction toDipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE).
We can see that the dipole is in transitional equilibrium as net force on the dipole is zero.
What about the rotational equilibrium? Is it also zero? If that was the case, then the dipole would have been stationary in position, but experimentally it is found that the dipole rotates with some angular velocity.
This is because, both the electrostatic force that is,Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEEacts a torque in a clockwise direction, thereby making the dipole to rotate in a uniform external electric field.

Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE

Fig: Uniform and Non-uniform electric field 

Torque always acts in a couple, and its magnitude equals to the product of force and its arm. Arm is the distance between the point where the force acts and the point which rotates the dipole. In the dipole placed in the uniform external electric field, we take origin as the point. Torque is denoted by the symbolDipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE and as it has a direction, it is a vector quantity.

Mathematically,
Magnitude of torque = q E × 2a sin θ
τ = 2 q a E sin θ
Since, the dipole moment (p = 2qa)
τ = p E sin θ
Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE
The vector form of torque is the cross product of dipole moment and electric field.
To understand what cross product is, let’s take an example.

Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEEFig: Torque rotates the dipole in uniform electric field  

The net external force acting on the dipole will be zero, hence it will not translate in space. But the same cannot be said about the net torque on the dipole.    
In short,     
Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE   
Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE
Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE

WhereDipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEErepresents the dipole moment of the electric dipole andDipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEErepresents the external electric field in which the dipole is kept.

Observations in net force and torque
Taking the nature of electric field and position of the dipole, following remarks will come out:

  • If the dipoleDipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEEand external electric field Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE are parallel, that is, angle between them is zero, then the dipole will feel zero torque. That is, no rotational effect.  
  • If the external electric field Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE is non-uniform, then net force on the dipole ≠ 0, and torque will create rotation. Hence, it would be a combined rotational and translational motion.
  • If the dipole external electric field Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE are antiparallel, that is, angle between them is non-zero, then the dipole will feel zero torque.
  • When the electric dipole Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE and electric field Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE are parallel, the direction of net force will be in direction of increasing electric field.
    Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE
Fig: Direction of net force depends on orientation of electric dipole 
  • When the electric dipole Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE and electric field Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE are anti-parallel, then the direction of net force will be in direction of decreasing electric field.
  • Force and Torque on a dipole placed in a uniform external field Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE varies with the orientation of dipole in free space

PHYSICAL SIGNIFICANCE

Dipole in Uniform & Non-Uniform Electric Field Notes | Study Physics For JEE - JEE

Fig: Comb attract dry paper piece

When we comb our dry hair and bring it near to some paper pieces, we find that the comb attracts the paper pieces. The comb gains charge, from our hair by the process of rubbing and induce a charge in the uncharged paper. In another way, the comb polarizes the pieces of paper that is, generate a net dipole moment in the direction of electric field. Also, since the electric field is non-uniform, the paper pieces move in the direction of the comb.

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