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EMP 1 DIELECTRICS – MACROSCOPIC VIEW Reference: Young & Freedman Chapter 24 Capacitance & Dielectrics Dielectric are insulators – charges tend not to move easily in non-metallic solids How does an object acquire a charge? Why does a charged object attract a neutral object? How do we investigate the electrical properties of an insulator or dielectric material? How does a dielectric change the capacitance? What is meant by potential difference (voltage)? What produces an electric field? BACKGROUND CAPACITORS Capacitors – two conducting plates separated by an dielectric Capacitors – 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 Capacitor – stores charge on conducting plates, stores electric potential energy due to the work done is separating the charges. Capacitance –“ability” to store charge Q C V For a parallel plate capacitor C r 0 A d A capacitance only depends upon the d geometry and dielectric Capacitors in series (charge on each plate is the same) 1 Ctotal 1 1 ... C1 C2 ijcooper/physics/p2/em/emp_01.doc Capacitors in parallel (voltage across each capacitor is the same) Ctotal C1 C2 ... 2 May 2017 EMP1.1 CHARGED CAPACITOR NOT CONNECTED TO A BATTERY (Q = Qfree = constant) Parallel plate capacitor – no dielectric (vacuum or air) How is the charge located on the surfaces of the conducting plates? area of plates A + + + + + + + + + + + + +Qfree on inner surface plate separation d -Qfree on inner surface - - - - - - - - - - - - Charge on plates electric field between plates E V Ex E=V/d V x f V V f Vi E dl i V=Ed Gauss’s Law – the total electric flux through any closed surface is equal to the net charge enclosed within the surface. E E dA E A E Qenclosed o V A Q free d 0 free 0 V Q free o C d free 0 Q free V 0 A d q free free A Energy stored and energy density (energy stored by the field) Q Qfree 1 Q2 1 1 1 1 U CV 2 QV 0 E 2 Ad u 0 E 2 2 C 2 2 2 2 Proof Work done to charge capacitor q dq 1 Q 1 Q2 dW v dq W q dq C C 0 2 C The operation of assembling upon a conductor a group of charges that mutually repel one another requires work and therefore results in the production of potential energy - this potential energy is possessed by the charged conductor itself but it may be more correct to picture the energy stored in the field surrounding the conductor. ijcooper/physics/p2/em/emp_01.doc 2 May 2017 EMP1.2 Electric displacement or electric flux density D is a useful quantity, since its value only depends upon the density of free charges and in not changed by the introduction of the dielectric. D 0 E D Gauss’s Law dA free dv Q free D free Parallel plate capacitor – isotropic dielectric inserted that fills space between plates Fixed charge on plates of capacitor (Q = Qfree same value without or with dielectric inserted) Dielectric fills space between the capacitor plates. dielectric constant (relative permeability) r permeability of dielectric = r 0 no dielectric – subscript 0 with dielectric – subscript r 1 d Capacitance only depends upon the geometry and dielectric C0 0 A d Cd r 0 A d A d Cd C0 C=Q/V Insertion of dielectric C increases reduction in V and E since Q = constant redistribution of the charges in the dielectric material (polarization). r K Vd V0 r Cd 1 C0 Ed E0 r D Dd D0 free f ijcooper/physics/p2/em/emp_01.doc voltage & electric field decrease by the factor r electric displacement remains constant 2 May 2017 EMP1.3 Dielectric increases the breakdown voltage (dielectric strength) can use a larger voltage before a disruptive discharge occurs. Dielectric constant ( r K) and dielectric strength Dielectric Dielectric constant strength V.m-1 Air 1.00058 ~ 3106 Water 80 Mica 7-8 Polystyrene 2.5 Titanium dioxide ceramics 15 - 500 ~ 2107 Barium titanate 500 - 6000 ~ 2106 How can we explain the reduction in the electric field between the capacitor plates? Reduced electric field – net charges induced on the surface of the dielectric (dielectric still neutral) – polarization. P bound b E Ed Ed 1 0 1 0 D free f 1 ( D P) 0 ( free bound ) ( free bound ) E0 r 1 bound free 1 r D r 0 Ed Ed Lines of D connect free charges (positive to negative) Lines of P connect bound charges (negative to positive) Lines of E connect net charges = free & bound (positive to negative) In isotropic materials: D , E and P all have the same direction free free r 0 Vd free d r 0 Cd r 0 A d bound free +Qfree on inner surface P + + + + + + + + + + + + - - - - - - -qbound Symmetry – fields must be uniform – field lines perpendicular to plates +qbound + + + + + + - - - - - - - - - - - - D -Qfree on inner surface ijcooper/physics/p2/em/emp_01.doc Interior points electric field must be zero 2 May 2017 E E 1 0 ( D P) EMP1.4 conductor + - dielectric Gauss’s Law + E 0 + - free bound + - E free bound 0 E + Isotropic materials – assume polarization P proportional to the electric field P e 0 Ed r 1 e Maxwell’s displacement current Switched closed – current through conductors to charge capacitor. Maxwell showed that it is necessary to assume a current of the same value also flowed in the space between the capacitor plates. 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. Parallel plate capacitor with compound dielectric (Q = Qfree = constant) Portion of the inter-plate region occupied by a dielectric slab of thickness t, the rest of the region being air filled C 0 A d t t r How do you prove this? ijcooper/physics/p2/em/emp_01.doc 2 May 2017 EMP1.5 Forces between parallel plates / energy in electric field (Q = constant) The charges on the plates are all on the inner surfaces unbalanced electrical attractive force F between the plates. Suppose the top plate is raised a distance dy by an external force so that there is no increase in kinetic energy |F| = |external force| = |attractive force between the plates|. The change in |potential energy| is equal to |work done| by the force necessary to raise the plate |dU| = |F dy| + + + + + + + + + dy + + + + + + + + + F - - - - - - - - - |F| = |dU/dy| 1 Q2 1 r 0 AQ 2 y 2 C 2 dU r 0 AQ 2 F dx 2 y2 U ijcooper/physics/p2/em/emp_01.doc 2 May 2017 EMP1.6 DIELECTRIC BEHAVIOR Insulator (dielectric) – will not permit a current (passage of charge) although local microscopic displacements of charge may take place. All atoms or molecules can be temporarily polarized under the action of an electric field. Permanent polar molecules (H+Cl-) may exist in a dielectric. Separation of charge electric dipole A dipole consists of two equal and opposite charges + and –q separated by a vector distance d dipole moment p = pe = q d pe qd d points from negative to positive -q +q Induced dipole moment – helium atom E -e +2e Zero electric field – helium atom symmetric zero dipole moment -e -e +2e A -e d B effectively charge +2e at A and -2e at B dipole moment p = 2ed p ijcooper/physics/p2/em/emp_01.doc 2 May 2017 EMP1.7 Potential and electric field from an electric dipole V ( P) 1 q q q r2 r1 4 r1 r2 4 r 1r2 Er E r d V ( P) P q d cos d2 4 r 2 cos2 4 r2 r + (d/2)cos q d cos p cos p r 2 2 4 r 4 r 4 r 3 r1 r – (d/2)cos r r extends from the centre of the dipole to the point P (d/2)cos The radial and tangential components of the field at point P are V 2 p cos r 4 r 3 1 V p sin E r 4 r 3 -q Er along the axis of the dipole along the right bisector of the dipole d +q = 0 E = 0 = /2 Er = 0 Electric field approaches zero much more quickly than a point charge. ??? Why ? See Matlab plot page 10 ijcooper/physics/p2/em/emp_01.doc 2 May 2017 EMP1.8 Electric polarization The extent to which permanent or induced dipoles become aligned is described by the electric polarization P dpe n pe dv where n is the number density for the electric dipoles (number of dipoles per unit volume) electric dipole moment per unit volume P Consider, polarization of the dielectric between the plates of a charged parallel plate capacitor dA +f + + + + + + + + + d Throughout the body of the dielectric, the charges on adjacent ends of the polar molecules neutralize one another. At both the top and bottom of the dielectric the charges do not neutralize each other bound surface charges b . For a cylinder of the dielectric of cross-sectional area dA extending from one plate to the other -f -b +b - - - - - - - - - electric dipole moment pe = q d dpe ( b dA)d P dpe dpe ( b dA)d dv dAd dAd P b P b nˆ where n̂ is the normal outward pointing unit vector. Polarization equals the magnitude of the bound (induced) charge per unit area on the surface of the dielectric. Also, the polarization can be obtained through the relationship b P where b is the volume density of the bound charges. Number density The number density for a gas is obtained from the ideal gas equation pV N k B T p N kB T n kB T V ijcooper/physics/p2/em/emp_01.doc 2 May 2017 n p kB T EMP1.9 For a gas at atmospheric pressure and 20 oC, the number density n is p = 1 atm = 1.013105 Pa T = 20 oC = 293 K kB = 1.381023 J.K-1 n = 2.51023 molecules.m-3 For a solid or liquid (density , molecular mass m, number of molecules N,mass of sample msample) the number density n is obtained as follows msample n V Nm V M NA m m M NA n N V n M NA NA M Avogadro’s number NA = 6.021023 molecules.mol-1 Molar mass M (in kilograms) For copper = 8.93103 kg.m-3 M = 63.5 g = 63.510-3 kg n = 8.51028 atoms.m-3 Note: the number density of solids is much greater than that of gases. Electric susceptibility e For isotropic dielectrics, (electrical properties identical in all directions) the polarization that occurs due to the applied electric field has the same direction as the field. Also, the magnitude of the polarization is proportional to the field P e E where e is the constant of proportionality, known as the electric susceptibility. For anisotropic dielectric P and E are not in the same direction and e is not a constant but a tensor. Electrets Isotropic dielectric material - susceptibility is not constant – e.g. electrets microphones ijcooper/physics/p2/em/emp_01.doc 2 May 2017 EMP1.10 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. 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 have time to orient themselves to the same extent before the polarity changes. For this reason, in a material that possesses permanent polar 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 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). dielectric constant frequency ijcooper/physics/p2/em/emp_01.doc 2 May 2017 EMP1.11 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 em waves depends upon the electric and magnetic 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 its velocity as it passes from one medium to another is responsible to refraction. Refractive index n (non-magnetic materials) n c r 0 0 v 0 0 n r This prediction of Maxwell’s electromagnetic theory originally served as a basis for criticizing the theory, for example DC values for air and water air n = 1.000294 water n = 1.3 r = 1.000295 n r r = 8.9 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. 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. 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. ijcooper/physics/p2/em/emp_01.doc 2 May 2017 EMP1.12 INTERESTING DIELECTRICS Ferroelectricity crystalline dielectric materials – permanent electric polarization (analogy with ferro-magnetic materials – it has nothing to do with iron) Rochelle salt, potassium dihydrogen phosphate and barium titanate Piezoelectricity 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 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. 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. Pyroelectricity crystal heated or cooled changes in charge at the surface – these are piezoelectric changes from the strain associated with thermal expansion or contraction. ijcooper/physics/p2/em/emp_01.doc 2 May 2017 EMP1.13 ELECTRIC DIPOLE PLOT– MATLAB Why is difficult to plot the potential in a plane passing through the axis of the dipole? Potential: Electric Dipole 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 % electric_dipole.m % Ian Cooper School of Physics,University of Sydney close all clear all clc % emconstants ------------------------------------------------------c = 3.00e-8; % speed of light e = 1.602e-19; % elementary charge eps0 = 8.85e-12; % permittivity of free space NA = 6.02e23; % Avogadro constant me = 9.11e-31; % electron rest mass mp = 1.673e-27; % proton rest mass mn = 1.675e-27; % neutron rest mass h = 6.626e-34; % Planck's constant kB = 1.38e-23; % Boltzmann's constant kC = 8.988e9; % Coulomb constant mu0 = 4*pi*1e-7; % permeability of free space amu = 1.66e-27; % atomic mass unit % Setup ------------------------------------------------------------q = e; % dipole charge d = 1.6795e-018; % dipole separation distance q1 = q; q2 = -q; % separated charges kc = 1/(4*pi*eps0); % constant in Coulomb's Law x1 = d/2; x2 = -d/2; % position of dipole y1 = 0; y2 = 0; scale = 1.25; % plotting region xmax = scale * d; ymax = xmax; xmin = -xmax; ymin = -ymax; % plane above diople num = 100; x = linspace(xmin,xmax,num); y = x; [xx yy] = meshgrid(x,y); r1 = sqrt((xx-x1).^2 + (yy-y1).^2); ijcooper/physics/p2/em/emp_01.doc % distance from charges 2 May 2017 EMP1.14 % to test point to calc. potential r2 = sqrt((xx-x2).^2 + (yy-y2).^2); V1 = kc .* q1 ./ (r1); % potential from each charge V2 = kc .* q2 ./ (r2); Vtot = V1 + V2; Vmax = max(max(Vtot)); sat = 0.5; % saturate the potential Vtot(Vtot > sat*Vmax) = sat * Vmax; % potential near a charge % is extremely large Vtot(Vtot < -0.5*Vmax) = -sat * Vmax; Vtot = Vtot/(max(max(Vtot))); figure(2); % [3D] plot surf(xx/d,yy/d,Vtot,'FaceColor','interp',... 'EdgeColor','none',... 'FaceLighting','phong') daspect([1 1 1]) axis tight; view(-45,20) camlight left; colormap(jet) grid off; axis off colorbar title('Potential: Electric Dipole') ijcooper/physics/p2/em/emp_01.doc 2 May 2017 EMP1.15