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
Electric Field and Potential
- Parallel plate capacitor
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 V
- For a parallel plate capacitor
capacitance only depends upon the
geometry and dielectric
- Capacitors in series Capacitors in parallel
|Q| on each plate Q = Q1 + Q2 + ...
V = V1 + V2 +... V across each capacitor
- Energy
Polarization
- (special case)
(more general)
Electric displacement
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FIELDS INSIDE DIELECTRIC MATERIALS
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
points from negative to positive
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 , electric displacement field and the polarization 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 and hence the voltage V (). Why is the electric field reduced?
Electric displacement
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 . Gauss’s Law can be expressed as
where f is the free charge surface density and f is the free charge volume density.
The field lines for connect free charges (positive to negative).
Electric polarization
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 per unit volume d and define this as the electric polarization
where n is the number density of the electric dipoles (number of dipoles per unit volume).
The lines of 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 and a volume bound charge density
where 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
(no proof, more general)
where 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
where is the dimensionless constant of proportionality, known as the electric susceptibility and r is the dielectric constant or the relative permittivity () of the material. is the permittivity of the dielectric material [F.m-1 or C2.N.m-2]. For anisotropic dielectric and are not in the same direction and is not a constant but a tensor.
Electric field
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
The field lines for connect net charges, free & bound (positive to negative)
The reduction in the electric field can be expressed in terms of the dielectric constant for the material r
Since is parallel to , the equation implies that is also parallel to and this equation can be written as
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.
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, . 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 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.
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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
dP/dtrate 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.
current associated with change in electric field strength even when a vacuum is between the plates.
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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
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).
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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
For non-magnetic materials
For a vacuum
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
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.
This prediction of Maxwell’s electromagnetic theory originally served as a basis for criticizing the theory, for example DC values for water and air
watern = 1.3
It was not known at the time that water contained permanent polar molecules and as a result the value of decreases with increasing frequency.
airn = 1.000294 = 1.000295
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.
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
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
Force between the plates of a parallel plate capacitor
The charge on a conductor resides in a thin surface 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.
Consider a parallel plate capacitor with a medium of dielectric constant 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 / constantPlate separation y d / increases
Capacitance C / decreases / y C
Energy storage U / increases /
Potential difference V / increases /
Electric field E / constant / Gauss’s Law Q = constant E = constant
Force between plates / constant / F independent of y
F is in the direction of decreasing y
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
dU/dy > 0 as expected. The external work done increases the stored potential energy. The force between the plates is
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.
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 / constantPlate separation y d / increases
Capacitance C / decreases / y C
Charge Q / decreases /
Electric field E / decreases /
Energy storage U / decreases /
Force between plates / decreases / F 1/y2
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, and if the capacitance changes by dC then the charge changes by (V = constant)
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
and the potential energy of the capacitor is decreased
The total change in the potential energy of the system is
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
and the attractive force between the plates is
Historically, the force to separate the plates of the capacitor was used to measure the potential difference V.
Removing or inserting a dielectric in a parallel plate capacitor
Consider the case of a slab of dielectric material with dielectric constant 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 / constantCapacitance C / decreases / r C
Energy storage U / increases /
Potential difference V / increases /
Force between plates & dielectric / Dielectric is attracted to the charged plates of the capacitor
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
The change in capacitance for an incremental distance dx is
The capacitance decreases as the slab is withdrawn. The change in the potential energy of the system (capacitor) is
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
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).
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 / constantCapacitance C / decreases / r C
Charge Q /
Energy storage Ucap / decreases /
Energy storage Ubattery / increases / Charge transferred to battery
Energy storage Usystem / increases / |dUbattery| > |dUcap|
Force between plates & dielectric / 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
The change in capacitance for an incremental distance dx is
The capacitance decreases as the slab is withdrawn.
The change in the potential energy of the capacitor is
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