what will happen to torque if field is non uniform
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It experiences a net force given by
[math]\mathbf{F} = \nabla (\mathbf{p} \cdot \mathbf{E})[/math]
where [math]\mathbf{p}[/math] is the dipole moment and [math]\mathbf{E}[/math] is the electric field.
Note that this formula applies to a single dipole, so if you have a polarized material in which the polarization varies spatially, when applying this formula to calculate the force density at a given point in the material, you must keep [math]\mathbf{p}[/math] fixed and vary only [math]\mathbf{E}[/math].
[math]\mathbf{F} = \nabla (\mathbf{p} \cdot \mathbf{E})[/math]
where [math]\mathbf{p}[/math] is the dipole moment and [math]\mathbf{E}[/math] is the electric field.
Note that this formula applies to a single dipole, so if you have a polarized material in which the polarization varies spatially, when applying this formula to calculate the force density at a given point in the material, you must keep [math]\mathbf{p}[/math] fixed and vary only [math]\mathbf{E}[/math].
A1learner:
can U solve it properly
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The torque ττ on an electric dipole with dipole moment p in a uniform electric field E is given by
τ=p×E
τpE
where the "X" refers to the vector cross product.
I will demonstrate that the torque on an ideal (point) dipole on a non-uniform field is given by the same expression.
I use bold to denote vectors.
Let us begin with an electric dipole of finite dimension, calculate the torque and then finally let the charge separation d go to zero with the product of charge q and d being constant.
We take the origin of the coordinate system to be the midpoint of the dipole, equidistant from each charge. The position of the positive charge is denoted by r+r and the associated electric field and force by E+E and F+F, respectively. The notation for these same quantities for the negative charge are similarly denoted with a - sign replacing the + sign.
The torque about the midpoint of the dipole from the positive charge is given by
τ+=r+×F+
τrF
where
F+=qr+×E+(r+)
FqrEr
Similarly for the negative charge contribution
τ−=r−×F−
τrF
where
F−=−qr−×E−(r−)
FqrEr
Note that
r−=−r+
rr
We can now write the total torque as
τtot=τ−+τ+=qr+×(E(r+)+E(r−))
τtotττqrErEr
It is clear that in taking the limit as the charge separation d goes to zero, the sum of electric fields will only contain terms of even order in d.
Noting that
|r+|=d2
rd2
and defining in the usual way
p=qd=q(r+−r−)
pqdqrr
We can write that
τtot=p×E(0)+ second order in d
τtotpE0 second order in d
As we take the limit in which d goes to zero and the product qd is constant, the second order term vanishes.
Thus, for an ideal (point) dipole in a non-uniform electric field, the torque is given by the same formula as that of a uniform field.
Note that it is not correct to start with the expression for a force on an ideal/point dipole in a non-uniform field and then calculate torque from this force. To derive this expression one ends up first taking the limit of a point dipole (on which there is zero force in a uniform field) and then one finds a torque of zero, which is incorrect. One must start with the case of a finite dipole, calculate torque and only then pass to the limit.
When p and E are parallel and anti-parallel, the torque is zero, so yes zero is possible. But the case in which p and E are anti-parallel is one of an unstable equilibrium, and a small angular perturbation will cause the dipole to experience a torque which attempts to align the dipole with the electric field.
τ=p×E
τpE
where the "X" refers to the vector cross product.
I will demonstrate that the torque on an ideal (point) dipole on a non-uniform field is given by the same expression.
I use bold to denote vectors.
Let us begin with an electric dipole of finite dimension, calculate the torque and then finally let the charge separation d go to zero with the product of charge q and d being constant.
We take the origin of the coordinate system to be the midpoint of the dipole, equidistant from each charge. The position of the positive charge is denoted by r+r and the associated electric field and force by E+E and F+F, respectively. The notation for these same quantities for the negative charge are similarly denoted with a - sign replacing the + sign.
The torque about the midpoint of the dipole from the positive charge is given by
τ+=r+×F+
τrF
where
F+=qr+×E+(r+)
FqrEr
Similarly for the negative charge contribution
τ−=r−×F−
τrF
where
F−=−qr−×E−(r−)
FqrEr
Note that
r−=−r+
rr
We can now write the total torque as
τtot=τ−+τ+=qr+×(E(r+)+E(r−))
τtotττqrErEr
It is clear that in taking the limit as the charge separation d goes to zero, the sum of electric fields will only contain terms of even order in d.
Noting that
|r+|=d2
rd2
and defining in the usual way
p=qd=q(r+−r−)
pqdqrr
We can write that
τtot=p×E(0)+ second order in d
τtotpE0 second order in d
As we take the limit in which d goes to zero and the product qd is constant, the second order term vanishes.
Thus, for an ideal (point) dipole in a non-uniform electric field, the torque is given by the same formula as that of a uniform field.
Note that it is not correct to start with the expression for a force on an ideal/point dipole in a non-uniform field and then calculate torque from this force. To derive this expression one ends up first taking the limit of a point dipole (on which there is zero force in a uniform field) and then one finds a torque of zero, which is incorrect. One must start with the case of a finite dipole, calculate torque and only then pass to the limit.
When p and E are parallel and anti-parallel, the torque is zero, so yes zero is possible. But the case in which p and E are anti-parallel is one of an unstable equilibrium, and a small angular perturbation will cause the dipole to experience a torque which attempts to align the dipole with the electric field.
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