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PH504 – Part 7 Dielectrics In an insulator, an electron is attached to a particular atom. When certain non-conducting materials are used to fill the space between the two conductors of a capacitor the capacitance is found to increase. Such materials are known as dielectrics. Relative permittivity: If the capacitance of a capacitor is Co when the region between the conductors is a vacuum and Cm when it is filled with a given dielectric then the ratio Cm/Co= r defines the relative permittivity or dielectric constant of the dielectric. Dielectric strength: Dielectrics normally act as insulators but above a given electric field (known as their dielectric strength) their insulating property breaks down and they start to conduct. This limits the maximum potential allowed between the two conductors and hence from Q=CV the maximum charge and energy that can be stored. Two fundamental properties of insulating materials (that is, rubber, glass, asbestos, or plastic) are insulation resistance and dielectric strength. These are two entirely different and distinct properties. One of the most common types of insulation is rubber. The voltage that may be applied to a rubber-covered conductor is dependent on the thickness and the quality of the rubber covering. Other factors being equal, the thicker the insulation, the higher may be the applied voltage. Air 3 MV/m Paper 16 MV/m Rubber 20 MV/m roughly Water 30 MV/m Teflon 60 MV/m 1. Polarization of Dielectric 1 If a material contains polar molecules, they will generally be in random orientations when no electric field is applied. An applied electric field will polarize the material by orienting the dipole moments of polar molecules: Parallel Plate with Dielectric Dielectrics represent a class of materials which, although insulators, exhibit a number of effects when placed in an E-field. A good example is their effect on capacitors. Result: decreases the effective electric field between the plates and increases the capacitance of the parallel plate structure. The capacitance of a set of charged parallel plates is increased by the insertion of a dielectric material. The dielectric must be a good electric insulator so as to minimize any DC leakage current through a capacitor The dielectric is characterized by a dielectric constant k, and the capacitance is multiplied by that factor. 2 The capacitance is inversely proportional to the electric field between the plates, and the presence of the dielectric reduces the effective electric field. A capacitor has capacitance C0 when the space between its two conductors is a vacuum, filling this space with a dielectric increases the capacitance to a new value Cm. The ratio Cm/C0=r = k, is known as the relative permittivity of the dielectric. Insertion. The plates of a capacitor are charged to some value Q and then isolated. Because of the relationship Q = CV, if C increases when a dielectric is placed between the conductors, as Q must remain constant V must decrease. Because V is related to the E-field by E=-V, the introduction of the dielectric must decrease E by r. 3 The aim of this lecture is to understand why this occurs. LIH dielectrics In order to simplify our treatment we will consider a sub-class of dielectrics known as LIH ones. Such dielectrics exhibit the following properties. Linearity (LIH): All properties are proportional to the magnitude of any applied E-field. This implies that r is a constant and is independent of applied voltage or E-field. Isotropy (LIH): r is independent of the orientation of the material i.e. it is independent of the direction of the E-field. Homogeneity (LIH): r has the same value at all points in the dielectric. Static relative permittivity: For the majority of dielectrics, r is found to vary with temperature and the frequency (if any) at which the applied E-field is modulated. The value of r in a steady E-field is known as the static relative permittivity. It is this value that is relevant to electrostatic situations. 4 2. The effect of an E-field on a dielectric Because dielectrics are insulators they contain no free charges (charges which are able to move around freely within the material when an E-field is applied). However although unable to move far, the positive and negative charges may be able to move slightly (but in opposite directions) when an E-field is applied. Consider an atom which consists of negatively charged electrons orbiting the positively charged nucleus. In general the centroids of the positive and negative charges coincide and the atom produces no external E-field. Neither the electrons or the nucleus is free to move over large distances when an E-field is applied. However the application of an E-field will cause the electrons to be moved, on average, a small distance in one direction and (possibly) the nucleus to move in the opposite direction. The centroids of the positive and negative charges now no longer coincide and we have an electric dipole which we know produces an external E-field. In this case the atoms are said to be polarised. Referring to the above diagram. For no external field the atoms are unpolarised (centroids of q coincide) and hence they produce no external E-field. When an external field (Eext) is applied the atoms are polarised and form small electric dipoles. In the present case the field moves the positive charges upwards and the negative ones downwards. 5 As their centroids no longer coincide they produce an external field E*. Because E-fields point from positive to negative charges, in the present case E* will point downwards and hence will oppose Eext. The total E-field within the dielectric will hence be less than the external field (Eext-E*) explaining our previous observation that the E-field between the conductors of a capacitor decreases when a dielectric is introduced. The effect of an E-field on a dielectric (quantitative treatment) Polarisation. An applied E-field causes the atoms or molecules in a dielectric to become small electric dipoles. The polarisation P at any point in the dielectric is defined as: Polarisation: electric dipole moment per unit volume at that point. If we have a small volume d it will contain a total dipole moment dp of Pd. This is the definition of the polarisation P. Note that polarisation has units C m-2. Why? Consider now a small volume element of length dL and crosssectional area dS. The application of an E-field causes the displacement of positive charge (+dq) out of one end of the element and an equal but negative charge (–dq) out of the other end. This results in a dipole moment (p=sQ) of dp=dq.dL. If instead of dq we use a surface charge density dq=.dS and hence dp=.dS.dL 6 But dS.dL is simply d, the volume of the element and from the definition of P P = dp/d P = So polarisation charge (dq) crossing area dS is PdS. More generally if the area dS is not normal to the polarisation vector P then need a dot product: Polarisation charge crossing dS: is PdS (A) 3. Surface polarisation charges Consider a slab of dielectric in an applied E-field. Along the field direction we can consider the dielectric in terms of a sequence of small elements. If the dielectric is homogeneous (P is the same at all points) then the positive polarisation charge at the end of an element is balanced by an equal but opposite negative charge at the end of the neighbouring element. Only at the surface of the dielectric is there an unbalanced charge. From above if Pn is the normal component of P at the surface then a surface polarisation charge density (P) exists given by P = Pn. 7 Volume Effects If E is not uniform, then P(x,y,z).Or, if the material is not homogeneous then there will also be a volume polarisation charge. The total charge density accumulated per unit volume is: b = - P since the excess charge leaving a small volume is (P)(dx dy dz). Potential The potential produced by a polarized object can be split into the potentials produced by: surface charge density and a volume charge density. Previously we had for a point charge: So we need to replace Q with surface and volume terms, and integrate over the surface and volume. 4. Gauss’s Law in dielectrics, D-field 8 Consider a general situation in which we have both free charges (labelled Qf) and charges due to the polarisation of one or more dielectrics (labelled QP). Gauss’s law can still be applied but we must now take account of both the free and polarisation charges. Applying Gauss’s law to the surface S Note that any dielectric contained entirely within S (e.g. A) does not contribute to the surface flux as it contributes equal amounts of negative and positive flux. Dielectric B however, which is cut by the surface S, will give a contribution to the flux through S as in general there will be a net polarisation charge crossing S. From (A) above the polarisation charge crossing a surface element dS is PdS. Hence the total polarisation charge which leaves the volume enclosed by S is (the integration can be taken over the whole of S as P=0 for all points on S which are outside of the dielectric). The above gives the polarisation charge which has moved outwards through S. Therefore the polarisation charge which remains within S (and is hence relevant for Gauss’s law) is the negative of this value. Hence Gauss ’s law is altered to: / 9 o Rearranging: Because the term 0E + P occurs very frequently it is given a special name, the electric displacement or D-field, symbol D D=0E+P . The units of D are C m-2. Gauss’s law in terms of D is now (integral form) . In words, this states ‘the outward flux of D over any closed surface S equals the algebraic sum of the free charges enclosed by S’. The differential form of Gauss’s law for D is where f is the volume density of free charges. From the form of the above equations we see that while both free and polarisation charges are sources of E, only free charges are sources of D. Because D is given by a modified form of Gauss’s law techniques used to find E for a given charge distribution can also be used to find D. In free space (no dielectric) P=0 and hence D=0E. The D-field a distance r from a point charge Q is then Relationships between E, D and P Electric susceptibility. If the E-field at a point within a dielectric is E and the polarisation at that point is P then the electric susceptibility 10 (e) is defined via the equation P=0eE For an LIH dielectric e is independent of the size and direction of E and position within the dielectric. Because D=0E+P D=0(1+e)E (B) Relationship between e and r Consider a system of free charges which produce an E-field and hence a D-field. If we now introduce a dielectric of relative permittivity r we know that D can not change and hence (B) above tells us that within the dielectric E must decrease to 1/(1+e) of its value in vacuum. However from our definition of r in terms of capacitance we know that introducing a dielectric results in a decrease in E by a factor r. Hence we must have the general result r = 1+e and hence we can also write D=r0E and P=0(r-1)E Also = r0 is used where is the permittivity of the dielectric. Summary of E, D and P A parallel plate capacitor carries charge densities of f. A dielectric partially fills the volume between the plates. 11 D has the same value in the dielectric and in the vacuum. The polarisation P is only non-zero in the dielectric. The polarisation P results in surface polarisation charges P on the top and bottom surfaces of the dielectric. The polarisation surface charges produce an additional Efield, Ep which opposes the vacuum field Evac to give a reduced total field Em within the dielectric. Lines of D pass through the dielectric Some lines of E terminate at the polarisation surface charges. 5. Effect of dielectrics on the E-field circuital law in vacuum. Charges make no contribution to the integral of E around a closed path and this is true for both free charges and polarisation charges. Hence this result is still valid in a dielectric: or in a vacuum or dielectric (but not when there is a time varying magnetic field!!). General electrostatic equations in the presence of dielectrics In general all the previously derived equations for V and E can be used if 0 is replaced by 0r. (This is only strictly true for LIH dielectrics). For example if a point charge Q is placed in a dielectric then the electric potential V, at a distance r is given by The only equation which can not be modified in this manner is Coulomb's equation for electrostatic forces. Why? This is because to place a charge in a dielectric we would have to make a small hole in which to place the charge. Hence 12 the charge would not strictly be in the dielectric. The solution of this problem is not trivial but fortunately we are generally interested in E-fields not forces. ． Boundary conditions for D and E These are required for problems where we are concerned with what happens at the interface between two different media (e.g. vacuum and a dielectric, two different dielectrics etc). Consider the interface between two media 1 and 2 (these can be vacuum, conductors or insulators). For generality we assume there is a free charge density f at the interface. E- and D-fields are as shown in the diagram. For a Gaussian surface we take the cylinder shown. If its height is made infinitesimally small then we need only consider the flux through the two ends. Applying Gauss’s law for D: (depends only on free charge) D1dS-D2dS = f dS In the limit dS0 D1n- D2n = Dn = f where Dn is the component of D normal to the surface. Result: the normal component of the D-field is discontinuous by f across any surface. If the free charge is zero, f=0, then the normal component of D is continuous. For the E-field take the rectangular path as the closed loop. If its height is made infinitesimally small then the only contribution to 13 the line integral is along the top and bottom edges where Et is the tangential component of E. Hence E1t = E2t; the tangential component of E is continuous across any interface. Now consider a general case where an E-field (and hence D-field) passes from a first dielectric to a second. If there is no surface charge at the interface then we must have the following (E-tangential) (C) (D-normal) (D) But D1=0rE1 and D2=0rE2 Hence we can write (E) Finally dividing equation (E) by (C) Note: Refraction of lines of D and E at a surface. Note: V is continuous across the interface even in the presence of free charge. 14 6. Stored energy in terms of E and D In the absence of dielectrics, we have shown that electrical potential energy (U) could be written in terms of the E-field (E) as This equation is modified in the presence of dielectrics. For a parallel plate capacitor filled with a dielectric having a relative permittivity r we have the following Hence But Hence This can be shown to be a general result. Hence for situations where E (and D) are not uniform The dot product of D and E is required because for nonLIH dielectrics D and E may not be parallel to each other. Values: r = 1.00059 air = 2–3 ice = 5 -10 glass = 80 water = 4.69 (perp) or 5.06 (parallel) quartz 15 7. Stress on a Conductor (Force on dielectric) Suppose we try to separate two conductors. We have seen that Therefore, at constant Q (open circuit) U = Q2x / ( 2A) where x is the plate spacing. Hence the force is: Fx = - dU/dx = - Q2 / ( 2A) Hence the force per unit area (stress) is: Fx = - Q2 / ( 2A2) = - 2/ 2 - exerted by the conductor. At constant V (closed circuit), energy supplied by battery = VdQ = V2 dC = (Q2/C2) dC. However, change in U = CV2/2 is dU = (Q2/2C2) dC….which is only half the energy! Battery supplies equal amounts of energy to increase internal energy and do external work. Examples: 1. Two long coaxial cylindrical tubes, length L, inner radius a, outer radius b, stand vertically in a tank of water. The inner one is maintained at potential V and the outer one is grounded. To 16 what height h does the water rise in the (narrow) space between the tubes? Take r = 80, V= 300 V, d = a-b = 1mm. For parallel capacitors: C = Cair + C water = (2r (L-h)) o / d + (2r h) / d U = CV2/2 = (2r (L + h(r - 1)) o V2 / 2d The force on the liquid (holding V constant) is: Fh = (U/h )V = 2r (r - 1) o V2 / 2d WHY??? Because of the ‘Fringing Field’: As the dielectric is introduced into the fringe field, the dielectric molecules start to align with the external field. This creates an attractive force as the plus charges of the dielectrics are aligned so that they are closest to the negative plate of the capacitor (and of course the same process occurs for the negative dielectric charges and positive plate of the capacitor). The horizontal components of the fringe field yield a 17 horizontal, attractive force which pulls the dielectric into the space between the capacitor plates. The work done by this force in accelerating the dielectric into the capacitor volume increases the energy stored in the capacitor. Some energy is also converted into electromagnetic radiation and into heat. This force balances the weight of liquid: 2 r d gh Finally, on equating forces,: h = (r - 1) o V2 / 2d2g (independent of L). Values yield 3.2 mm. Conclusions Effects of a dielectric on E, V and C LIH dielectrics Qualitative explanation of dielectrics Quantitative explanation of dielectrics, definitions of P Surface polarisation charge Electric displacement (D-field) Gauss's law for D-fields Relationships between E, D and P (r, e) Electrostatics in the presence of dielectrics Boundary conditions for D and E Electrical potential energy in terms of D and E 18