Institutt for fysikk, NTNU TFY4155/FY1003: Elektrisitet og magnetisme Spring 2005

Summary, week 8 (February 22 and 23)

Electric polarization. Dielectrics. [FGT 25.5, 25.6; YF 24.4, 24.5; TM 24.5, 24.6; AF 25.6, 25.7; LHL 20.5; DJG 4.1]

Insulator: No mobile (free) charges (but bound charges) Dielectric: Polarizable insulator. (All insulators are polarizable.)

If a dielectric is placed in an external electric field E_{0}, we obtain alignment of (molecular) electric dipoles along E_{0}, cf øving 6, exercise 2. (Alternatively: Polarization internally in atoms and nonpolar molecules that, at the outset, have zero electric dipole moment.)

Net (macroscopic) effect of the external field: Displacement of bound charge.

±Q_{i} = net induced charge on the surface of the insulator.

Polarization = dipole moment pr unit volume:

P = \frac{p}{V}

Electric dipole moment (repetition!):

p = qa Dielectric in external field E_{0}:

Volume: V = Ad Density of induced surface charge: σ_{i} = Q_{i}/A

Hence: Total dipole moment: p = |p| = Q_{i}d = σ_{i}Ad = σ_{i}V Polarization: P = |P | = p/V = σ_{i}

In general: P · n̂ = P_{⊥} = σ_{i} (n̂ = surface normal, P_{⊥} = component of P perpendicular to the surface)

Electric displacement. [FGT 25.6; YF 24.6; TM 24.6; AF 25.8; LHL 20.5; DJG 4.3]

We use the idealized system which we have used before: Oppositely charged metal plates (infinitely large), now with dielectric in between:

Free charge pr unit area on the metal plates: σ_{f} ... which generate electric field between the plates: E_{f} = σ_{f}/ε_{0} (zero field outside the plates) Induced charge pr unit area on the surface of the dielectric: σ_{i} ... which generate electric field between the plates: E_{i} = σ_{i}/ε_{0} Total field between the plates: E = E_{f} + E_{i} ⇒ E = |E| = E_{f} − E_{i} = (σ_{f} − σ_{i})/ε_{0} Net charge on the surfaces: ±σ = ±(σ_{f} − σ_{i}) ... which generate total field between the plates: E = σ/ε_{0} = (σ_{f} − σ_{i})/ε_{0}, OK!

We have: σ_{i} = P = polarization in the dielectric ( = dipole moment pr unit volume)

Hence:

σ_{f} = σ + σ_{i} = ε_{0}E + P

So we see that the density of free charge σ_{f} is determined by the combination ε_{0}E + P. In many situations, e.g. in experiments, it is precisely the free charge that we are able to control. With dielectrics present, it is therefore often convenient to ”refer to” the vector field

D ≡ ε_{0}E + P

D is called electric displacement.

Here:

D = |D| = σ_{f}

In general (as we found for P):

σ_{f} = D · n̂ = D_{⊥}

where D_{⊥} is the normal component of the electric displacement.

Gauss’ lov for D:

∮_{S} D · dA = Q_{f}

where Q_{f} is net free chrage inside the closed surface S. (Net total charge inside S is Q_{in} = Q_{f} − Q_{i}, with −Q_{i} = net bound charge, associated with the polarization P, inside S.)

Electric susceptibility and permittivity. [FGT 25.5; YF 24.4; TM 24.5, 24.6; AF 25.9; LHL 20.5; DJG 4.4]

Linear response: P proportional to E, i.e., we may write

P = χ_{e}ε_{0}E

where we have introduced χ_{e} = electric susceptibility. NB: Such a linear relation between P and E is not always valid, but in this course, we will assume that it is valid. Also note that E is the total field, and not only the external field.

Hence:

D = ε_{0}E + P
= (1 + χ_{e})ε_{0}E
= ε_{r}ε_{0}E
= εE

Here, we have introduced ε_{r} = 1 + χ_{e} = relative permittivity (”the dielectric constant”) ε = ε_{r}ε_{0} = the permittivity of the medium

Units: [χ_{e}] = [ε_{r}] = 1 (dimensionless) [ε] = [ε_{0}] = C^{2}/Nm^{2}

Point charge q in dielectric with permittivity ε:

Electric field: E(r) = (q/4πεr^{2})r̂ Electric potential: V (r) = q/4πεr I.e.: As for point charge in vacuum, but with ε_{0} → ε > ε_{0}; the medium is polarized and screens the point charge so that E and V are reduced by the factor 1/ε_{r}.

Capacitor and capacitance. [FGT 25.1, 25.5; YF 24.1, 24.2; TM 24.4; AF 25.10; LHL 20.1; DJG 2.5.4]

Capacitor = two separated electric conductors with charge ±Q (Or sometimes: One electric conductor with charge Q, with the other conductor moved to infinity.)

Coulomb’s law ⇒ electric field around the conductors is proportional with Q

It then follows that the potential difference ΔV between the two conductors is also proportional with Q:

C = \frac{Q}{ΔV}

C = the capacitance of the conductor

Unit for capacitance: [C] = [Q/ΔV] = C/V ≡ F (farad)

Symbol in electric circuits:

C is a geometric factor, dependent on the shape and size of the two conductors, and their mutual distance. C also depends on what kind of medium we have in the region between the two conductors. Capacitance is, by definition, a positive quantity. Calculation of C for a given system is done by calculating the potential difference between the two conductors, ΔV = V_{+} − V_{−}, for a given charge ±Q.

Parallel plate capacitor, filled with air (vacuum), with surface area A, and plate distance d:

C = ε_{0} \frac{A}{d}

Parallel plate capacitor, filled with dielectric with relative permittivity ε_{r}, surface area A, and plate distance d:

C = ε_{0}ε_{r} \frac{A}{d}

If the region between the two conductors of the capacitor is first air and then partly or fully filled with a dielectric, the capacitance will always increase. (The same happens if parts of the volume are filled with metal.)

Energy associated with electric field [FGT 25.3; YF 24.3; TM 24.3; AF 25.11; LHL 20.4; DJG 2.4.3]

Density of potential energy, i.e., potential energy pr unit volume, is with electric field E equal to

u = \frac{1}{2}ε_{0}E^{2}

We also found that the potential energy could be associated with the electric charge: If a system has electric potential v(q) when the charge is q, we must perform a work dW = v(q) dq in order to increase the charge from q to q + dq. Hence, the total work needed, and therefore also the total stored potential energy in the system, is

W = U = ∫_{0}^{Q} v(q) dq

i.e., in order to charge the system from zero charge to a final charge Q.

Alternatively, we may calculate the stored potential energy by integrating the energy density u over the whole volume V:

U = ∫_{V} u dV = ∫_{V} \frac{1}{2}ε_{0}E^{2} dV

(Note that V is here volume and not potential. Further: Don’t be confused by the notation used above: I used v(q) to denote the potential difference between the two capacitor plates when they had charge q and −q, i.e., at some arbitrary stage during the charging. The reason for the small v was I wanted to reserve V for the potential difference between the plates after they had been fully charged, i.e., with charge Q and −Q. I try to minimize mixup in the notation, but V is an exception, being used both for potential and for volume.)