University Of Zimbabwe HPH102: Electricity and Magnetism miss l g thwala

Capacitance, Polarization and Dielectrics

Objectives f this lecture Discuss how to store energy in a simple configuration: the parallel plate capacitor Discuss the properties of capacitors Have knowledge of the terms: capacitance dielectric polarisation etc. Gain ability to do certain calculations

Capacitance A capacitor is a useful device for storing charge and energy. It consists of two conductors insulated from each other. A typical capacitor, called a parallel-plate capacitor consists of two large metallic plates with area A, separated by a small distance d. When the plates are connected to a charging device, e. g., a battery, charge is transferred from one conductor to the other until the potential difference between the conductors due to their equal and opposite charges equals the potential difference between the battery terminals.

Capacitance The capacitance is given by:   The SI unit of capacitance is the coulomb per volt, called a farad (F). Given any two conductors, we find the potential difference V when there is a charge +Q on one conductor and –Q on the other, and then calculate the capacitance C.

The parallel-plate capacitor Two plates of a capacitor, each of area A, separated by a distance d. When there is a potential difference V between the plates, electric field lines pass from one plate to the other. The field lines are perpendicular to the sheets and go from positive to negative. Except at the edges, the magnitude of the electric field E is V/d. If –d- is small compared with the dimensions of the plates, edge effects are negligible and the field can be assumed to be uniform between the plates, and zero elsewhere. The electric field E between the plates is a function of the charge density, σ (= ± Q/A).

The parallel-plate capacitor The electric field between the plates is given by Since the electric field between the plates is constant, the potential difference between the plates is Ed. Thus The capacitance is Where k = 1 for vacuum which is acting as the dielectric The constant is the permittivity of free space; its numerical value in SI units is 8.85×10 – 12 F/m .

Question The 90-pF capacitor is connected to a 12-V battery and charged to 12 V. How many electrons are transferred from one plate to another? Solution: The charge transferred is given by: Q = CV = (90 x 10-12)(12) = 1.08 x 10-9 C   This is the magnitude of the charge on either plate. The number of electrons in a charge of 1.08 x 10-9 C is: N = = 6.75 x 109

Combinations of Capacitors Two or more capacitors are often used in combination. The Figure shows two capacitors, with capacitance C1 and C2, connected in parallel. The upper plates of the two capacitors are connected together by a conducting wire and are therefore at the same potential. The lower plates are also connected together and are at a common potential. The potential difference between/across each capacitor is V (= Va – Vb). If the capacitances are C1 and C2, then the charges Q1 and Q2 stored on the plates are given by Q1 = C1V and Q2 = C2V  The TOTAL CHARGE STORED on both capacitors is Q = Q1 + Q2 = (C1 + C2) V Two capacitors connected in parallel C1 C2 a b

Two capacitors connected in series. Capacitors in series The Figure shows two capacitors, with capacitance C1 and C2, connected in series. Suppose the potential difference across the capacitor with capacitance C1 is ΔV1 and the potential difference across the capacitor with capacitance C2 is ΔV2. A charge Q on the top plate will induce a charge -Q on the bottom plate of capacitor C1. Since electric charge is conserved, the charge on the top plate of capacitor C2 must be equal to Q. Thus the charge on the bottom plate is equal to -Q. The potential difference across C1 is given by Figure: Two capacitors connected in series.

Two capacitors connected in series. Capacitors in series The potential difference across C2 is given by: The total potential difference across the two capacitors is given by The Equation above shows that the p.d. across the two capacitors, connected in series, is proportional to the charge Q. The system acts like a single capacitor Ceff whose capacitance can be obtained from the following formula: Figure: Two capacitors connected in series.

Two capacitors connected in series. Capacitors in series In general the effective capacitance of three or more capacitors in series is given by Figure: Two capacitors connected in series.

Example: Multi-plate Capacitor A multi-plate capacitor, such as used in radios, consists of four parallel plates arranged one above the other as shown in the diagram. The area of each plate is A, and the distance between adjacent plates is d. Determine the capacitance of this arrangement? The multiple capacitor is equivalent to three identical capacitors connected in parallel The capacitance of each of the three capacitors is given by The total capacitance of the multi-plate capacitor is A Multi-plate Capacitor. Schematic of Multi-plate Capacitor

Electrostatic energy in a capacitor Suppose the electrostatic potential of plate carrying charge +Q is V1 and the potential of plate with charge -Q is V2. The electrostatic energy of the capacitor is then given by The energy stored by the capacitor can be expressed in terms of C and only one of the variables Q or V as: For the parallel plate capacitor, V = Ed, and expressing the energy in terms of the electric field instead of the potential difference, we find Ad is the volume of the region between the plates. equation shows that electrostatic energy can be stored in a capacitor. The quantity ε0E2/2 is called the energy density (potential energy per unit volume).

Dielectrics and Polarization If the space between the plates of a capacitor is filled with an insulator, the capacitance of the capacitor will change compared to the situation in which there is vacuum between the plates. A dielectric is an insulator or non conductor of electricity. Its effects are defined in terms of the ratio of the capacitance of a capacitor with a dielectric between the plates to the capacitance of a capacitor without a dielectric between the plates. when the dielectric is vacuum when there’s a different dielectric that is not vacuum o

The change in the capacitance is caused by a change in the electric field between the plates. The electric field between the capacitor plates will induce dipole moments in the material between the plates. These induced dipole moments will reduce the electric field in the region between the plates. the dielectric constant κ and it is defined as the ra- tio of the capacitance with a dielectric between the plates to the capacitance with a vacuum between the plates, that is,

A parallel plate capacitor with air between the plates has a capacitance of 3.85 μF. If mica is placed between the plates of the capacitor what will the new capacitance be? The value of the dielectric constant for mica is found as κ = 5.40. The capacitance with the dielectric placed between the plates is Cd=kCo = 5.40 x 3.85 x 10−6 F = 2.08 x 10−5 F

Dielectrics and Polarization An isolated neutral atom has Z electrons moving around a nucleus carrying a charge +Ze, (The number Z is called the atomic number of the atom.) Consider an isolated atom where the size of the nucleus is grossly exaggerated and the electrons are represented as being smeared in a cloud around the nucleus. In the absence of an external electric field, the nucleus of an atom is at the centre of the electron cloud, where it experiences no net electrostatic force. It is in a position of stable equilibrium, so that if the nucleus is displaced from the centre of the electron cloud, a restoring force would arise from the mutual attraction of the nucleus and electrons. Non polarized dielectric –a- Polarised dielectric –b-

Dielectrics and Polarization When the atom is placed in an external electric field pointing to the right, the field will push the positively charged nucleus to the right and the negatively charged electrons to the left until a new equilibrium is established. –a- below The net effect is that the centre of the electron distribution is shifted away and no longer coincides with the centre of the nucleus. –b- below In its new equilibrium position the atom is said to be polarized. We now have a dipole Assume a representation of an atom in which the centre of mass has been displaced a distance a. The vector p = Zea which points in the direction of the electric field is the dipole moment of the polarized atom. The dimensions of the dipole moment are [charge][length] and dipole moments are measured in coulomb metres. Atomic description of a dielectric placed in a electric field

Dielectrics and Polarization Since the final electric field E can never exceed the free electric field E0, the relative permittivity ε must be larger than 1. The potential difference across a capacitor is proportional to the electric field between the plates. Since the presence of a dielectric reduces the strength of the electric field, it will also reduce the potential difference between the capacitor plates (if the total charge on the plates is kept constant): The capacitance C of a system with a dielectric is inversely proportional to the potential difference between the plates, and is related to the capacitance C0 of a capacitor with no dielectric in the following manner: Since ε is larger than 1, the capacitance of a capacitor can be significantly increased by filling the space between the plates with a dielectric with a large ε.

Dielectric Strength The electric field E in a dielectric cannot be increased indefinitely. If a certain value is exceeded, sparking occurs and the dielectric is said to break down. The maximum electric field intensity that a dielectric can sustain without breakdown is called the dielectric strength. In the design of capacitors, it is important to know the maximum potential difference that can be applied before breakdown occurs. Many capacitors have air as the dielectric. These types have the advantage that if breakdown occurs, the capacitor is not permanently damaged.

Torque on Electric Dipole The torque produced on an electric dipole by an electric field can be expressed as a vector product with direction given by the right hand rule. The lever arm for each charge with respect to the center is

Energy of an Electric Dipole An electric field produces a torque on a dipolewhich tends to take it to its low energy configuration. To rotate it from the low energy state against the field requires work where the shorter form employs the scalar product

The End… Mini summary A capacitor is a device used to store charge. • The amount of charge Q a capacitor can store depends on two major factors—the voltage applied and the capacitor’s physical characteristics, such as its size. • The capacitance C is the amount of charge stored per volt, or C =QV . • The capacitance of a parallel plate capacitor is C = ε0 Ad , when the plates are separated by air or free space. ε0 is called the permittivity of free space. • A parallel plate capacitor with a dielectric between its plates has a capacitance given by C = κε0 Ad , where κ is the dielectric constant of the material. • The maximum electric field strength above which an insulating material begins to break down and conduct is called dielectric strength The End…