Find potential energy of an electric dipole in an external 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)(xi,yi,zi) and the final position (xf,yf,zf)(xf,yf,zf) is
W=∫(xf,yf,zf)(xi,yi,zi)F⃗ ⋅ds⃗ .W=∫(xi,yi,zi)(xf,yf,zf)F→⋅ds→.
In the case of an electric dipole consisting of two point charges ±q±qseparated by the vector ℓ⃗ ℓ→ having length ℓℓ, i.e. p=qℓp=qℓ, in an external uniform electric field E⃗ E→, which makes an angle θθ with the dipole separation vector ℓ⃗ ℓ→, 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.
W=∫(xf,yf,zf)(xi,yi,zi)F⃗ ⋅ds⃗ .W=∫(xi,yi,zi)(xf,yf,zf)F→⋅ds→.
In the case of an electric dipole consisting of two point charges ±q±qseparated by the vector ℓ⃗ ℓ→ having length ℓℓ, i.e. p=qℓp=qℓ, in an external uniform electric field E⃗ E→, which makes an angle θθ with the dipole separation vector ℓ⃗ ℓ→, 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.
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