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PHYS2012 EMP10_04 DIELECTRICS – MACROSCOPIC VIEW Reference: Young & Freedman Chapter 24 Capacitance & Dielectrics REVIEW Dielectrics are insulators – charges tend not to move easily in non-metallic solids Gauss’s Law E dA qenclosed 0 q f qb 0 qf r 0 Electric Field and Potential V V V E V xˆ yˆ y z x V E.dl Parallel plate capacitor V E dl E d zˆ V E d Capacitors Two conducting plates separated by an dielectric Uses (basic component of most electronic circuits): timing circuits, filtering, smoothing fluctuating voltages, transmission of ac signals, resonance circuits, flash lights in cameras, pulsed lasers, air bag sensors, ac circuits, etc Stores charge on conducting plates, stores electric potential energy due to the work done is separating the charges. Energy stored in the electric field. Capacitance –“ability” to store charge Q C Q=CV V For a parallel plate capacitor C r 0 A d A capacitance only depends upon the d geometry and dielectric Capacitors in series |Q| on each plate V = V1 + V2 +... 1 Ctotal 1 1 ... C1 C2 Capacitors in parallel Q = Q1 + Q2 + ... V across each capacitor Ctotal C1 C2 ... Energy U emp10_04.doc 1 Q2 1 1 QV CV 2 2 C 2 2 1 u r 0 E2 2 14 sep 10 4. 1 Polarization P e 0 E Pn p b P nˆ e r 1 (special case) Px Py Pz x y z (more general) b P Electric displacement 1 DP E 0 E890 FIELDS INSIDE DIELECTRIC MATERIALS E D P Dielectric materials consist effectively of a large number of electric dipoles. An electric dipole consists of two equal and opposite charges +q and –q separated by a vector distance d dipole moment p pe = q d d pe qd points from negative to positive -q +q pe We can consider the polarization of the dielectric in terms of the induced electric dipoles or we can simply the description of dielectric behaviour by discussing different kinds of fields – the electric field E , electric displacement field D and the polarization P to account for the macroscopic properties of dielectrics. If we insert a dielectric material between two charged plates, the voltage across the plates decreases. When we remove it, the voltage goes back up again. The charge upon the plates can’t be affected, what the dielectric does is to reduce the electric field E and hence the voltage V ( V E dl ). Why is the electric field reduced? Electric displacement D Historically, to account for behaviour of a dielectric material in an external electric field, the concept of the electric displacement field was introduced. The free charges Qfree which might consist of electrons on a conductor or ions embedded in the dielectric material give rise to the electric displacement field D . Gauss’s Law can be expressed as emp10_04.doc 14 sep 10 4. 2 D dA free d Q free D f f Qf A Q f Q free f free D f where f is the free charge surface density and f is the free charge volume density. The field lines for D connect free charges (positive to negative). Electric polarization P The molecules within the dielectric material experience an electrostatic force due to an electric field. The molecules are said to be polarized – each molecule becomes a tiny electric dipole. A bound charge which means charge that can’t leave its “home” molecule is produced by the polarization. The effect of the dielectric is due entirely to the bound charge. We can smooth over the internal structure of the material and assign it an average dipole moment pe per unit volume d and define this as the electric polarization P dp P e n pe d where n is the number density of the electric dipoles (number of dipoles per unit volume). The lines of P connect bound charges (negative to positive). The polarization describes the extent to which permanent or induced dipoles become aligned. The polarization gives rise to a surface bound charge density b bound and a volume bound charge density b bound P b nˆ where n̂ is the normal outward pointing unit vector (special case). Thus, the polarization equals the magnitude of the bound (induced) charge per unit area on the surface of the dielectric material. Also, the polarization can be obtained through the relationship b P (no proof, more general) where b is the volume density of the bound charges. To develop a simple model of a dielectric material, we need to make a number of assumptions. For an ideal dielectric material: Homogeneous – properties don’t change with position Isotropic – properties don’t depend upon direction Linear – polarization is proportional to the electric field Stationary – all charges are stationary For ideal dielectrics, we can write P e 0 E r e 1 where e is the dimensionless constant of proportionality, known as the electric susceptibility and r is the dielectric constant or the relative permittivity ( r 1 ) of the material. 0 r is the permittivity of the dielectric material [F.m-1 or C2.N.m-2]. For anisotropic dielectric P and E are not in the same direction and e is not a constant but a tensor. emp10_04.doc 14 sep 10 4. 3 Electric field E The average electric field inside the dielectric material is due to contribution of both the free and the bound charges. How can we explain the reduction in the electric field between the capacitor plates? The bound charges on the surface of the dielectric partly cancel the effect of the free charges, hence reducing the resulting electric field. This can be seen by applying Gauss’s Law to find the average electric field inside the dielectric E Edielectric E dA qenclosed 0 0 E f b q f qb conductor 0 E 0 E 0 - dielectric + + 0 E D P 1 + - free bound D P + - E + The field lines for E connect net charges, free & bound (positive to negative) +Qfree on inner surface P + + + + + + + + + + + + - - - - - - -qbound Symmetry – fields must be uniform – field lines perpendicular to plates +qbound + + + + + + - - - - - - - - - - - - D -Qfree on inner surface Interior points electric field must be zero E E 1 0 ( D P) The reduction in the electric field can be expressed in terms of the dielectric constant for the material r E Edielectric air r emp10_04.doc 14 sep 10 4. 4 Since P is parallel to E , the equation 0 E D P implies that D is also parallel to E and this equation can be written as 1 E DP 0 E 1 0 D E e 0 D 0 1 e E r 0 E E E D r 0 r 1 e r 0 D D r 0 E b r 0 ( f b ) 0 1 b f r Hence, the magnitude of the bound surface charge b of the dielectric is less than the magnitude of the free charge density f on the conductor. b f 1 The above figure shows what happens to the electric field inside the dielectric. In this case, 2/3 electric field lines start on the positive plate are cancelled by negative charges in the dielectric and reappear on the other side. The electric field induces a polarization within the dielectric material – the negative charges move a little to the left and the positive to the right under the influence of the applied electric field. For this figure, e 2 r 3 . D is the same inside and outside the dielectric material and the value of E inside is only 1/3 of its value outside. E is reduced by the factor r inside the dielectric and 2/3 of the E field lines are swallowed by the bound surface charge of the dielectric. If the dielectric were replaced by a metal instead, all the electric field lines would disappear and E would be zero inside, the metal behaves like a dielectric of infinite dielectric constant. E095 E567 emp10_04.doc E906 14 sep 10 4. 5 MAXWELL’S DISPLACEMENT CURRENT When an uncharged capacitor is first connected to a battery, a current is established in the conductors to charge the capacitor. Maxwell showed that it is necessary to assume a current of the same value also flowed in the space between the capacitor plates. Electric displacement current density Jd d free dt dD dE dP 0 dt dt dt dP/dt rate of change of polarization – associated with the actual motion of charges in the dielectric: rotation of permanent dipoles or induced dipoles – displacement of charges – posses a current character. 0 dE current associated with change in electric field strength even when a vacuum is dt between the plates. E040 E851 emp10_04.doc 14 sep 10 4. 6 FREQUENCY RESPONSE OF THE DIELECTRIC CONSTANT The capacitance of any capacitor is directly proportional to the dielectric constant of the material between the capacitor plates. Hence, the dielectric constants of two materials can be readily compared by introducing the materials, in turn, into a given capacitor and determining the resulting capacitances. For a given material, the change in dielectric constant as a function of pressure, temperature, or some other variable can be measured with high precision by employing the material-filled capacitor as the capacitive element in a tuned circuit. Resonance frequency LC tuned circuit f0 1 LC If the circuit is sharply resonant, a small change in the capacitance of the capacitor results in a significant change in the resonant frequency of the circuit. By this means, for example, even the small changes in the dielectric constants of gases which occur when the temperature is altered have been accurately studied. When a DC voltage is applied to a capacitor, the polar molecules in the dielectric orient themselves under the action of the electric field. When the applied voltage is an alternating one, the polar molecules again attempt to line up with the field and are, in fact, equally successful if the frequency of the AC voltage is low. As the polarity of the voltage changes, the polar molecules obligingly change their direction. When the frequency of the applied field is high, however, the polar molecules may not dielectric have time to orient themselves to the same extent Constant before the polarity changes. For this reason, in a (polar material that possesses permanent polar molecules) molecules, the dielectric constant decreases with increasing frequency. If, on the other hand, the polar molecules in the dielectric are induced ones, resulting from a displacement of the planetary electron systems there is no observed frequency decrease with increasing frequency, because this displacement is practically instantaneous. In most materials, both permanent and induced polar molecules contribute to the polarization. The dielectric constant of water falls from its low frequency value of 80 to less than 2 at optical frequencies (~1014 Hz). E588 emp10_04.doc 14 sep 10 4. 7 REFRACTIVE INDEX Maxwell prediction of electromagnetic waves Electromagnetic waves time-varying electric and magnetic fields whose directions are mutually perpendicular. In unbounded dielectric media the waves are transverse. Velocity of propagation of electromagnetic waves depends upon the electric (permittivity ) and magnetic (permeability ) properties of the medium. For an unbounded medium 1 v For non-magnetic materials 1 1 v 0 r 0 0 For a vacuum c 1 0 0 The change in the velocity of the electromagnetic wave as it passes from one medium to another is responsible for refraction. The refractive index n of non-magnetic materials is n c r 0 0 v 0 0 n r Refraction dispersion of light through a glass prism When the frequency is comparable to the orbital frequency of the electrons in the material, absorption and emission can take place – the index of refraction can display appreciable frequency dependence, e.g., dispersion of visible light in passage through a glass prism. The frequency for light is f ~ 1014 Hz. emp10_04.doc 14 sep 10 4. 8 This prediction of Maxwell’s electromagnetic theory originally served as a basis for criticizing the theory, for example DC values for water and air water n = 1.3 r 81 9 n r It was not known at the time that water contained permanent polar molecules and as a result the value of r decreases with increasing frequency. air n = 1.000294 r = 1.000295 n r The polarization of the air molecules is entirely due to the displacement under the action of the applied electric field of the electron clouds of their constituent atoms – since this displacement occurs with great rapidity, r displays no frequency dependence. emp10_04.doc 14 sep 10 4. 9 FORCES and ENERGY Just as conductor is attracted into an electric field, so too is a dielectric. The bound charges tend to accumulate near the free charge of opposite sign. But the calculation of forces on dielectrics can be very tricky. For example, we assume that the electric field inside a parallel plate capacitor is uniform and zero outside and the direction of the electric field is always perpendicular to the plates. But a dielectric slab is drawn into the field region between the plates, because in reality there is a fringe field around the edges and it is this non-uniform electric field that pulls the dielectric into the capacitor. Although it can be difficult, if not impossible, to calculate the forces directly using Newton’s Laws, it is often a simple matter to derive expressions for the forces using the principle of conservation of energy. The first step in this approach is to clearly identify the system and secondly, how the work done by forces produces changes in the kinetic and potential energy of the system. Assume that an applied external force Fme (the subscript me emphasizes the application of the external force acting on the system) acts on the system to change only the potential energy of the system without any change in the kinetic energy of the system. Then, by the law of conservation of energy, the work done Wme on the system by an external force Fme equals the change in potential energy Usystem of the system. For an incremental displacement dy, we can write dWme Fme dy dU system Fme dU system dy The net force on the system must be zero because there is zero change in the kinetic energy of the system. The internal force F that acts in the opposite direction to the external force Fme is Fme F Force between the plates of a parallel plate capacitor The charge on a conductor resides in a thin surface Fme layer. This is due to the mutual repulsion between charges of like sign, so that the charges on the conductor are trying to get as far away as possible + + + + + + + + + from each other. Then, for a charged parallel plate capacitor, there is an attractive force between the + + + + + + + + + plates due to the thin layers of opposite charge on each plate. F dy - - - - - - - - - emp10_04.doc 14 sep 10 4. 10 Consider a parallel plate capacitor with a medium of dielectric constant r between the plates. Q is kept constant (capacitor charged and disconnected from the battery) and the plate separation y is increased by the application of an external force Charge Q Plate separation y d Capacitance C constant increases decreases Energy storage U increases Potential difference V increases Electric field E constant Force between plates constant C r 0 A y y C Q2 C U 2C Q Q C V C V V C Gauss’s Law Q = constant E = constant F independent of y F is in the direction of decreasing y U The system is the capacitor and the external force Fme acts on one plate to pull the plates apart (increase the separation distance y) with zero change in the kinetic energy of the system. The rate of change of the energy stored and the external force are A Q2 U C r 0 2C y dU dU dC dy dC dy dU Q2 dC 2C2 A dC C r 20 dy y y dU Q 2 C Q 2 y Q2 dy 2 C 2 y 2 r 0 A y 2 r 0 A dU/dy > 0 as expected. The external work done increases the stored potential energy. The force between the plates is dU Q2 Q 2 Fme F Fme dy 2 r 0 A 2 r 0 A The minus sign for F denotes that the force acts in the opposite direction to the movement of one plate, i.e. in the direction of decreasing y. The magnitude of the force is independent of the plate separation y. emp10_04.doc 14 sep 10 4. 11 V is kept constant (capacitor connected to the battery) and the plate separation y is increased by the application of an external force Potential difference V Plate separation y d Capacitance C constant increases decreases Charge Q decreases Electric field E decreases Energy storage U decreases Force between plates decreases C r 0 A y y C Q Q CV C Q V V E y E y 1 U CV2 C U 2 F 1/y2 C The system is the capacitor and the battery. The external force Fme acts on one plate to pull the plates apart (increase the separation distance y) with zero change in the kinetic energy of the system. For the capacitor, Q C V and if the capacitance changes by dC then the charge changes by dQ V dC (V = constant) A A AV dC C r 0 r 20 dQ r 0 2 dy y dy y y dQ is negative indicating that charge is transferred to the battery from the capacitor. Therefore, energy is transferred to the battery to increase its potential energy r 0 AV 2 dU battery V dQ dy y2 and the potential energy of the capacitor is decreased r 0 AV 2 1 1 r 0 AV dU capacitor V dQ V dy dy 2 y2 2 y2 2 The total change in the potential energy of the system is dU system dU battery dU cap r 0 AV 2 r 0 AV 2 r 0 AV 2 dy dy dy y2 2 y2 2 y2 The energy transferred to the battery is twice the energy lost by the capacitor. The application of the external force Fme increases the total energy of the system. The external force is dU system r 0 AV 2 Fme dy 2 y2 and the attractive force between the plates is F Fme r 0 AV 2 2 y2 Historically, the force to separate the plates of the capacitor was used to measure the potential difference V. emp10_04.doc 14 sep 10 4. 12 Removing or inserting a dielectric in a parallel plate capacitor Consider the case of a slab of dielectric material with dielectric constant r inserted between the square plates of a parallel plate capacitor with an area A = L2. An external force Fme acts to remove the dielectric without increasing its kinetic energy. Q is kept constant (capacitor charged and disconnected from the battery) and the dielectric is pulled out of the capacitor by the application of an external force Charge Q Capacitance C constant decreases Energy storage U increases Potential difference V increases C r 0 A y r C Q2 U C U 2C Q Q C V C V V C Dielectric is attracted to the charged plates of the capacitor Force between plates & dielectric The system is the capacitor. When the dielectric has been displaced by a distance x, the capacitance C is equivalent to two capacitors in parallel L L x Lx L C1 0 C2 r 0 C C1 C2 0 x r L x d d d The change in capacitance for an incremental distance dx is dC 0 L L dC 0 1 r dx 0 r 1 1 r 1 dx d d The capacitance decreases as the slab is withdrawn. The change in the potential energy of the system (capacitor) is 1 Q2 U system 2 C dU system dx dU system dx 2 0L Q 2 dC Q 2 d 1 r 2 2 C dx 2 0 L x r L x d Q 2 d 1 r 2 0 L x r L x 2 0 The potential energy of the capacitor increases as the dielectric is removed. The work done by the external force in withdrawing the dielectric increases the potential energy of the capacitor and the external force is given by dU system Q2 d 1 r Fme 0 2 dx 2 0 L x r L x The direction of the external force Fme is in the direction of increasing x. The force F on the dielectric due to the charge on the capacitor plates is attractive and opposes the withdrawal of the dielectric (F = - Fme). emp10_04.doc 14 sep 10 4. 13 V is kept constant (capacitor charged and connected to the battery) and the dielectric is pulled out of the capacitor by the application of an external force Potential difference V Capacitance C constant decreases Charge Q Energy storage Ucap decreases Energy storage Ubattery Energy storage Usystem Force between plates & dielectric increases increases C r 0 A y r C Q CV C Q 1 U CV 2 C U 2 Charge transferred to battery |dUbattery| > |dUcap| Dielectric is attracted to the charged plates of the capacitor The system is the capacitor and the battery. When the dielectric has been displaced by a distance x, the capacitance C is equivalent to two capacitors in parallel L L x Lx L C1 0 C2 r 0 C C1 C2 0 x r L x d d d The change in capacitance for an incremental distance dx is dC 0 L L dC 0 1 r dx 0 r 1 1 r 1 dx d d The capacitance decreases as the slab is withdrawn. The change in the potential energy of the capacitor is 1 U cap CV 2 2 dU cap V 2 dC 0 LV 2 1 r 0 dx 2 dx 2 d When the dielectric slab is withdrawn, the capacitance and the potential energy of the capacitor decrease. Therefore, charge must be transferred to the battery increasing its potential energy Q LV C Q V C 0 x r L x V d dQ 0 LV 1 r 0 dx d dU battery LV 2 V (dQ) 0 1 r 0 dx d emp10_04.doc 14 sep 10 4. 14 The total change in the potential energy of the system is dU system dU battery dU cap LV 2 0 LV 2 0 1 1 r r dx dx dx d 2d LV 2 0 1 r 0 dx 2d Again, the energy transferred to the battery is twice the energy lost by the capacitor. dU system The application of the external force Fme increases the total energy of the system and the external force is dU system LV 2 Fme 0 1 r 0 dx 2d and the attractive force between the charged plates and dielectric is LV 2 F Fme 0 1 r 2d E232 E664 E382 PARALLEL PLATE CAPACITOR WITH COMPOUND DIELECTRIC When there is more than one dielectric between the capacitor plates (compound dielectric), we often replace this capacitor by a number of capacitors (single dielectric) connected in a series and/or parallel combination. Consider a parallel plate capacitor with a thin dielectric slab of thickness t (t < d) midway between the plates of the charged capacitor that is not connected to a battery. Compare the following parameters for the capacitor without and without the dielectric slab: capacitance, charge, potential difference and energy stored. Derive an expression for the thickness of the dielectric slab. Air filled capacitor (subscript 0 refers to when the dielectric is not inserted) 0 A Capacitance C0 Charge Q0 Potential difference V0 Energy stored Q2 U0 2C emp10_04.doc d Q0 C0 14 sep 10 4. 15 Capacitor with thin dielectric slab Charge Q1 = Q0 = constant (subscript 1 refers to when the dielectric is inserted) Capacitor charged and disconnected from battery The capacitance is Q C V To determine the capacitance we need to find the potential difference between the plates. We can do this by integrating the electric field along a contour from the negative plate to the positive plate V1 E dl E0 d t E1 t E0 d t E1 t The effect of the dielectric is to reduce the electric field inside the dielectric by the factor r E E1 0 r So the potential and capacitance are t d t r t V0 V0 V1 E0 d t r d t d t Q r V1 0 C0 d Q0 d C0 C0 V1 t d t r The energy stored is t d t 2 2 r 1 Q0 1 Q0 U1 2 C1 2 C0 d C1 t d t r U 0 d U 0 The capacitance values C0 and C1 can be measured to estimate the thickness t of the dielectric C t 1 0 r 1 d C1 r Alternatively, we can consider the system to be two capacitors in series (which gives the same result as before): emp10_04.doc 14 sep 10 4. 16 0 A Cair C1 E105 E761 d t 1 Cair 1 1 E202 E797 Cdielectric Cdilectric E288 E900 r 0 A t r 0 A d t t r E456 E600 E617 E696 E704 INTERESTING DIELECTRICS Ferroic Materials cooperative solid state phenomena Ferroics - is the generic name given to the study of ferromagnets, ferroelectrics, and ferroelastics. Ferromagnet - a material able to retain a permanent magnetic moment. Ferroelectric - a material able to retain a permanent dipole moment. Ferroelastic - a material able to retain a permanent shape after deformation (e.g. shape memory alloys) Ferroelectrics Crystalline dielectric materials – permanent electric polarization Rochelle salt, potassium dihydrogen phosphate and barium titanate + + - + - Ferroelectric material - + + - + - - + + - + + + - + + - + + + - + + - + + + - + - + + Antif erroelectric material Electrets Isotropic dielectric material - susceptibility is not constant – e.g. electrets microphones. An electret is a stable dielectric material with a permanently-embedded static electric charge (which, due to the high resistance and chemical stability of the material, will not decay for hundreds of years). Electrets are commonly made by first melting a suitable dielectric material such as a plastic or wax that contains polar molecules, and then allowing it to resolidify in a powerful electrostatic field. The polar molecules of the dielectric align themselves to the direction of the electrostatic field, producing a permanent electrostatic "bias". Modern electret microphones use PTFE plastic, either in film or solute form, to form the electret. An electret microphone is a type of condenser microphone, which eliminates the need for a polarizing power supply by using a permanently-charged material. emp10_04.doc 14 sep 10 4. 17 Piezoelectricity generate an electric charge in a material when subjecting it to a mechanical stress, conversely, generate a mechanical strain in response to an applied electric field. Crystals that lack a centre of symmetry of ion distribution are electrically polarized (i.e. they develop surface charges) when they are mechanically stressed. Crystalline dielectric materials – mechanical pressure exerted upon the crystal results in appearance of electric charge along its surface (polarization effect) mechanical stress emf emf mechanical stress – change in dimensions of crystal + Zero applied stress Compressive stress Induces a voltage Applied voltage produces An expansion Quartz crystal – mechanically vibrating ac voltage across its face – frequency of mechanical vibration depends upon the dimensions and other parameters of the crystal ac voltage generated possesses a very constant frequency. Quartz is piezoelectric but not ferroelectric. It has a piezoelectric constant of 2.2×10-12 C.N-1 or V.m-1. If you apply a force of 1 N perpendicular to the axis of the quartz crystal, you get a charge of 2.2 pC; if you apply a volt, the length changes by 2.2×10 -12 m. The piezoelectric constants for barium titanate and potassium sodium tartrate (Rochelle salt) (ferroelectric) may be hundreds of times greater but are quite variable. These materials are widely used in transducers. If you apply a voltage to a quartz crystal at a frequency corresponding to a mode of mechanical oscillation, the resulting resonance is stable and precise. ac voltage applied to the faces of crystal – if frequency of the voltage is identical to natural frequency of crystal’s mechanical vibrations crystal vibrate “violently” ultrasonic waves Transducer: electrical energy mechanical energy. emp10_04.doc 14 sep 10 4. 18 Pyroelectricity - migration of positive and negative charge (and therefore establishment of electric polarization) to opposite ends of a crystal's polar axis as a result of a change in temperature. Some of crystals that lack a centre of symmetry of ion distribution, (i.e. piezoelectrics) can also spontaneously develop electric dipoles (polarize), with the degree of polarization dependent on temperature. Crystal heated or cooled changes in charge at the surface – these are piezoelectric changes from the strain associated with thermal expansion or contraction. Reference http://en.wikipedia.org/wiki/Ferroelectricity Polarization P e 0 E r 1 0 E Most materials are polarized linearly with an external electric field; nonlinearities are insignificant. This is called dielectric polarization. Some materials, known as paraelectric materials, demonstrate a more pronounced nonlinear polarization. The electric permittivity, corresponding to the slope of the polarization curve, is thereby a function of the external electric field. In addition to being nonlinear, ferroelectric materials demonstrate a spontaneous (zero field) polarization. Such materials are generally called pyroelectrics. The distinguishing feature of ferroelectrics is that the direction of the spontaneous polarization can be reversed by an applied electric field, yielding a hysteresis loop. Typically, materials demonstrate ferroelectricity only below a certain phase transition temperature, called the Curie temperature TC, and are paraelectric above this temperature. E893 E040 E600 E900 E095 E617 E906 emp10_04.doc E105 E664 E202 E696 E232 E704 E288 E761 14 sep 10 E382 E797 E456 E851 E567 E890 E588 E893 4. 19