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 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 Recall principle of virtual work and 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

Polarization Atom Negative electron cloud Positive nucleus No external electric field With an external electric field Charge polarization occurs in the presence of electric field

Polarization Negative electron cloud Positive nucleus Each atom acquires a small dipole moment . For low intensity electric fields the polarization is expected to be proportional to the field intensity: With an external electric field Atomic polarizability of the atom

Atomic Polarizability And Ionization Potential

Polarization With an external electric field No external electric field If the density of particles per cubic meter is N, the net polarization is: has units of {C/m2}

Some molecules have built in dipole moments due to ionic bonds. Polarization Some molecules have built in dipole moments due to ionic bonds. Negative region O2- H+ 𝑝 𝑝 Positive region Water

Polarization O2- Negative region 𝑝 𝑝 H+ Positive region For low intensity applied fields this polarization of the material is again proportional to the field intensity so a more general expression for polarization is: With the electric susceptibility of the material

Materials where is proportional to are called DIELECTRIC materials. Polarization For low intensity applied fields this polarization of the material is again proportional to the field intensity so a more general expression for polarization is: Materials where is proportional to are called DIELECTRIC materials.

Dielectrics Why is the dielectric constant in the medium different than the dielectric constant in vacuum? The answer is contained in the nature of the material being placed in the electric field.

Dielectrics d Consider the slab of material “dielectric” immersed in an external field . Endface Area A The molecules inside will be polarized due to the presence of the electric field.

Dielectrics + - dipole Consider the slab of material “dielectric” immersed in an external field . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + The molecules inside will be polarized due to the presence of the electric field. + SLAB + + + + + + + + + + + + + + + + Area A Polarization induced surface charge density

Dielectrics - + For the single dipole, the dipole moment is: dipole For the dipoles along one line between the endfaces, the dipole moment is: d

Dielectrics The total dipole moment of the slab is then . + The total dipole moment of the slab is then . Q: Total charge on one face of the slab.

Dielectrics d The polarization or dipole moment per unit volume of the slab is then. + End face Area A

FLUX CAPACITOR MEETS THE FUSOR Electric flux Density 𝐷 =𝜀 𝐸 FLUX CAPACITOR MEETS THE FUSOR

Dielectrics d Electric field is shown normal to the surface. Endface Area A The electric flux density vectors Do = Dd . We are treating only normal components here.

Dielectrics These two charge sheets will produce an electric field directed from the positive sheet towards the negative sheet. Polarization bound charge sheets of each endface. Original slab in external electric field In order to determine the magnitude of the electric field, a parallel plate capacitor analysis can be applied to this charge configuration.

Dielectrics Induced electric field due to the polarization effect in the dielectric. Through Gaussian analysis: Polarization bound charge sheets of each end face. Note: The external field Eo and the induced field Ei are in opposite directions. If electrons where free in the dielectric, then the magnitudes of Eo and Ei would end up the same, their directions would be opposite and the net electric field inside the medium would be zero. Such a material is called a metal due to the free electrons.

Dielectrics d Endface Area A Further manipulations of equations required to obtain desired result. Recall that desired result is: Why mediums have a different dielectric constant from that of vacuum? Endface Area A

Dielectrics Endface Area A d Vector diagram inside dielectric

Dielectrics Endface Area A d Since: and Then Vector diagram

Dielectrics Endface Area A d And: with Vector diagram Then

Dielectrics Endface Area A d The dielectric constant can now be obtain ed using : Vector diagram

Answer: Polarization and orientation of internal dipoles. Dielectrics With the electric susceptibility of the material. Why is the dielectric constant in the medium different than the dielectric constant in vacuum? Answer: Polarization and orientation of internal dipoles.

ELEC 3105 Basic EM and Power Engineering Next slides: forget me not

Dielectric Materials

Polarization Field P = electric flux density induced by E

Electric Breakdown Electric Breakdown

Electric flux Density 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

Boundary conditions Normal Component of ELECTROSTATICS Boundary conditions Normal Component of Gaussian surface on metal interface encloses a real net charge s. Gaussian surface on dielectric interface encloses a bound surface charge sp , but also encloses the other half of the dipole as well. As a result Gaussian surface encloses no net surface charge. Air Dielectric Gaussian Surface

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