write the work done in rotating a dipole when it is placed in uniform electric field
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The general expression for the work done by a variable force along a path between the initial position (xi,yi,zi) and the final position (xf,yf,zf) is
W=∫(xf,yf,zf)(xi,yi,zi)F⃗ ⋅ds⃗ .
In the case of an electric dipole consisting of two point charges ±q separated by the vector ℓ⃗ having length ℓ, i.e. p=qℓ, in an external uniform electric field E⃗ , which makes an angle θ with the dipole separation vector ℓ⃗ , the net force on the dipole is zero since the forces on each charge are of the same magnitude, but acting in opposite directions. However, there is a net torque acting on the dipole due to the separation of the charges. To calculate the work done by the torque in rotating the dipole let us consider the work done by the torque in rotating each charge in the dipole separately. The total work can be found by adding the work done in rotating each charge together.
The work done by the torque in rotating each charge can be calculated using the expression for work above. In this case F⃗ is the force doing the turning and ds⃗ is an infinitesimal arc length of a circle of radius rdθ, where r=ℓ/2, i.e. the pivot point for the application of the turning force is located at the midpoint of ℓ⃗ . The force F⃗ is tangential to the circle of radius r and has magnitude qEsinθ in each case. Since I assume that the external field E⃗ provides positive torque on the positive charge in the dipole, i.e. a counter-clockwise torque we have, in the case of each charge, that F⃗ and ds⃗ are in the same direction so
F⃗ ⋅ds⃗ =(qEsinθ)(rdθ).
Therefore we have that the work done by the external field in rotating the electric dipole through some angle is
W=∫θfθiqEsinθrdθ+∫θfθiqEsinθrdθ.
Therefore
W=2∫θfθiqrEsinθdθ.
Since r=ℓ/2 we find that the work done on the dipole by the torque provided by E⃗ is
W=∫θfθiqℓEsinθdθ=∫θfθipEsinθdθ
where the electric dipole is given by p⃗ =qℓ⃗ and has magnitude p=qℓ.
We can express the total work done by the torque in rotating the dipole as
W=∫θfθiτzdθ,
where τz=pEsinθ.
If, in the above derivation, I assumed the opposite direction for the field, i.e. E⃗ is replaced with −E⃗ , a clockwise torque would occur and the work done on the electric dipole would be
W=−∫θfθipEsinθdθ=∫θfθiτzdθ,
where τz=−pEsinθ.
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edited Jan 21 at 23:04
answered Dec 17 '17 at 21:24
Physics_Et_Al
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Analogous to the work
W=FΔs
done by a force F over a distance Δs , there is also a work done by a torque T for a rotational movement
W=TΔθ
where Δθ is the angle of rotation. The torque experienced by an electric dipole with dipole moment p⃗ in an electric field E⃗ is
T⃗ =p⃗ ×E⃗ =pEsinθ
were θ is the angle between the dipole moment and the electric field. Thus the work performed during a rotation of a dipole from an angle θ=α to an angle θ=β is given by
W=∫βαT(θ)dθ=∫βαpEdθsinθ
Hope this helps.
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