ELEC 3105 Basic EM and Power Engineering Electric dipole Force / torque / work on electric dipole Z

The Electric Dipole z P(x, z) Consider electric field and potential produced by 2 charges (+q, -q) separated by a distance d. P(x, z) +q d x -q

The Electric Dipole z P(x, z) The dipole is represented by a vector of magnitude qd and pointing from –q to +q. P(x, z) +q d x -q Note: small letter p Units {p} dipole moment; Coulomb meter {Cm}

Potential at P(x, y) due to charge +q. The Electric Dipole z P(x, z) Potential at P(x, y) due to charge +q. +q d x -q Units {p} dipole moment; Coulomb meter {Cm}

Potential at P(x,y) due to charge -q. The Electric Dipole z P(x,z) Potential at P(x,y) due to charge -q. +q d x -q Units {p} dipole moment; Coulomb meter {Cm}

The Electric Dipole Suppose (x,y) >>> d P(x, z) z +q d -q x Can be rewritten and the expression for the potential simplified. Then:

Suppose (x, z) >>> d The Electric Dipole Suppose (x, z) >>> d Binomial expansion

Suppose (x, z) >>> d The Electric Dipole Suppose (x, z) >>> d Binomial expansion

Suppose (x,y) >>> d The Electric Dipole Suppose (x,y) >>> d z P(x, z) +q d -q x

The Electric Dipole Suppose (x, z) >>> d Potential produced by the dipole

Suppose (x, z) >>> d The Electric Dipole Suppose (x, z) >>> d z P(x, z) +q d -q x

The Electric Dipole Suppose (x, z) >>> d P(x, z) z Cartesian coordinates (x, z) +q d -q x

The Electric Dipole Suppose (x, z) >>> d P(r, , ) z Spherical coordinates (r, , ) +q d -q x

The Electric Dipole z x V Drops off as 1/r2 for a dipole P(r, , ) x V Drops off as 1/r2 for a dipole V Drops off as 1/r for a point charge

The Electric Dipole Now to compute the electric field expression P(x, z)) Cartesian coordinates (x, z) z P(r, , ) Spherical coordinates (r, , ) +q d -q x

Now to compute the electric field expression The Electric Dipole Now to compute the electric field expression

Spherical coordinates (r, , ) The Electric Dipole Spherical coordinates (r, , ) No 𝜙 dependence

The Electric Dipole

Here consider dipole as a rigid charge distribution Force on a dipole in a uniform electric field Here consider dipole as a rigid charge distribution +q d No net translation since -q Opposite direction

Here consider dipole as a rigid charge distribution Force on a dipole in a non-uniform electric field Here consider dipole as a rigid charge distribution +q d net translation since -q And / Or

Force on a dipole in a non-uniform electric field y Force on a dipole in a non-uniform electric field +q Manipulate expression to get simple useful form d -q x

Force on a dipole in a non-uniform electric field y Force on a dipole in a non-uniform electric field After the manipulations end we get: +q d -q x We will obtain this expression using a different technique.

Here consider dipole as a rigid charge distribution Torque on a dipole Here consider dipole as a rigid charge distribution +q d/2 d/2 -q The torque components + and  - act in the same rotational direction trying to rotate the dipole in the electric field.

Review of the concept of torque Torque on a dipole Torque: Pivot Moment arm length Force Angle between vectors r and F

For simplicity consider the dipole in a uniform electric field Torque on a dipole Act in same direction +q d -q Also valid for small dipoles in a non-uniform electric field.

Consider work dW required to rotate dipole through an angle d Work on a dipole By definition When you have rotation If we integrate over some angle range then +q d -q

Work on a dipole For  = 90 degrees W = 0. Thus  = 90 degrees is reference orientation for the dipole. It corresponds to the zero of the systems potential energy as well. U=W

Work on a dipole For  = 0 degrees W = -pE. Thus  = 0 degrees is the minimum in energy and corresponds to the having the dipole moment aligned with the electric field.

Work on a dipole For  = 180 degrees W = pE. Thus  = 180 degrees is the maximum in energy and corresponds to having the dipole moment anti-aligned with the electric field.

Force on a dipole +q d -q Work Force After the manipulations end we get: +q d We will obtain this expression using a different technique. Work -q Force

Exam question: once upon a time Stator dipole +q -q +Q 2R 2r Rotor dipole E on +q F on +q  on +q  on rotor -Q (0,0)

Exam question: Once upon a time D>>R 2R e)  on dipole -Q

Electric flux Density extra topic (1) 𝐷 𝐷 =𝜀 𝐸

From other definitions of flux we can obtain other useful expressions for electrostatics Divergence theorem

Divergence theorem Integrands must be the same for all dV Point function Gauss’s law in differential form Medium dependence

Divergence theorem Integrands must be the same for all dV Point function Gauss’s law in differential form No dependence on the dielectric constant