apacitance and Dielectrics Physics 3 Lecture 4-
apacitors Device for storing electrical energy which can then be released in a controlled manner onsists of two conductors, carrying charges of q and q, that are separated, usually by a nonconducting material - an insulator Symbol in circuits is It takes work, which is then stored as potential energy in the electric field that is set up between the two plates, to place charges on the conducting plates of the capacitor Since there is an electric field between the plates there is also a potential difference between the plates Physics 3 Lecture 4-
apacitors We usually talk about capacitors in terms of parallel conducting plates They in fact can be any two conducting objects Physics 3 Lecture 4-3
apacitance The capacitance is defined to be the ratio of the amount of charge that is on the capacitor to the potential difference between the plates at this point V ab Units are farad oulomb Volt Physics 3 Lecture 4-4
alculating the apacitance We start with the simplest form two parallel conducting plates separated by vacuum Let the conducting plates have area A and be separated by a distance d The magnitude of the electric field between the two plates is given by E σ ε 0 ε 0 A We treat the field as being uniform allowing us to write d V ab Ed ε0a Physics 3 Lecture 4-5
alculating the apacitance Putting this all together, we have for the capacitance V ab ε 0 A d The capacitance is only dependent upon the geometry of the capacitor Physics 3 Lecture 4-6
farad apacitor Given a farad parallel plate capacitor having a plate separation of mm. What is the area of the plates? We start with And rearrange to solve for A, giving ε0 A A d d ε 0. 0 ( )( 3.0F.0 0 m) 8 8.85 0 m F / m This corresponds to a square about 0km on a side! Physics 3 Lecture 4-7
Series or Parallel apacitors Sometimes in order to obtain needed values of capacitance, capacitors are combined in either Series or Parallel Physics 3 Lecture 4-8
apacitors in Series apacitors are often combined in series and the question then becomes what is the equivalent capacitance? Given what is We start by putting a voltage, V ab, across the capacitors Physics 3 Lecture 4-9
apacitors in Series What happens with? apacitors become charged because of V ab If upper plate of gets a charge of +, Then the lower plate of gets a charge of - Since there is no source of charge at point c, and we have effectively put a charge of on the lower plate of, the upper plate of gets a charge of + harge onservation This then means that lower plate of has a charge of - Physics 3 Lecture 4-0
apacitors in Series We also have to have that the potential across plus the potential across should equal the potential drop across the two capacitors Vab Vac + Vcb V + V We have Then V andv V ab + Dividing through by, we have V ab + Physics 3 Lecture 4-
apacitors in Series The equivalent capacitor will also have the same voltage across it V ab + Physics 3 Lecture 4- The left hand side is the inverse of the definition of capacitance So we then have for the equivalent capacitance + eq If there are more than two capacitors in series, the resultant capacitance is given by eq i i V
apacitors in Parallel apacitors can also be connected in parallel Given what is Again we start by putting a voltage across a and b Physics 3 Lecture 4-3
apacitors in Parallel The upper plates of both capacitors are at the same potential Likewise for the bottom plates We have that Now V or V V V andv and V ab V Physics 3 Lecture 4-4
apacitors in Parallel The equivalent capacitor will have the same voltage across it, as do the capacitors in parallel But what about the charge on the equivalent capacitor? The equivalent capacitor will have the same total charge Using this we then have eq + V eq or V + + + V Physics 3 Lecture 4-5
apacitors in Parallel The equivalent capacitance is just the sum of the two capacitors If we have more than two, the resultant capacitance is just the sum of the individual capacitances eq i i Physics 3 Lecture 4-6
Example a b 3 a b Where do we start? Recognize that and are parallel with each other and combine these to get This is then in series with with 3 The resultant capacitance is then given by + + 3 3 ( + ) + + 3 Physics 3 Lecture 4-7
Example onfiguration A onfiguration B onfiguration Three configurations are constructed using identical capacitors Which of these configurations has the lowest overall capacitance? a) onfiguration A b) onfiguration B c) onfiguration The net capacitance for A is just In B, the caps are in series and the resultant is given by + net net In, the caps are in parallel and the resultant is given by net + Physics 3 Lecture 4-8
A circuit consists of three unequal capacitors,, and 3 which are connected to a battery of emf Ε. The capacitors obtain charges, 3, and have voltages across their plates V, V, and V 3. eq is the equivalent capacitance of the circuit. heck all of the following that apply: Example 3 a) b) 3 c) V V 3 d) E V e) V < V f) eq > A detailed worksheet is available detailing the answers Physics 3 Lecture 4-9
Example 4 What is the equivalent capacitance, eq, of the combination shown? o eq o (a) eq (3/) (b) eq (/3) (c) eq 3 3 + eq + Physics 3 Lecture 4-0
Energy Stored in a apacitor Electrical Potential energy is stored in a capacitor The energy comes from the work that is done in charging the capacitor Let q and v be the intermediate charge and potential on the capacitor The incremental work done in bringing an incremental charge, dq, to the capacitor is then given by dw v dq q dq Physics 3 Lecture 4-
Energy Stored in a apacitor The total work done is just the integral of this equation from 0 to W 0 q dq Using the relationship between capacitance, voltage and charge we also obtain U V where U is the stored potential energy V Physics 3 Lecture 4-
Example 5 Suppose the capacitor shown here is charged to and then the battery is disconnected Now suppose you pull the plates further apart so that the final separation is d Which of the quantities,, V, U, E change? : : V: U: E: harge on the capacitor does not change apacitance Decreases Voltage Increases Potential Energy Increases Electric Field does not change How do these quantities change? Answers: d A + + + + ----- d d V d V d U U d d Physics 3 Lecture 4-3
Example 6 Suppose the battery (V) is kept attached to the capacitor Again pull the plates apart from d to d Now which quantities, if any, change? : : V: U: E: harge Decreases apacitance Decreases Voltage on capacitor does not change Potential Energy Decreases Electric Field Decreases How much do these quantities change? Answers: d Physics 3 Lecture 4-4 d d d d d U U E E d d
Electric Field Energy Density The potential energy that is stored in the capacitor can be thought of as being stored in the electric field that is in the region between the two plates of the capacitor The quantity that is of interest is in fact the energy density Energy Density u V Ad where A and d are the area of the capacitor plates and their separation, respectively Physics 3 Lecture 4-5
Electric Field Energy Density Using A ε0 and V E d d u ε0 we then have E Even though we used the relationship for a parallel capacitor, this result holds for all capacitors regardless of configuration This represents the energy density of the electric field in general Physics 3 Lecture 4-6
Dielectrics Most capacitors have a nonconducting material between their plates This nonconducting material, a dielectric, accomplishes three things ) Solves mechanical problem of keeping the plates separated ) Increases the maximum potential difference allowed between the plates 3) Increases the capacitance of a given capacitor over what it would be without the dielectric Physics 3 Lecture 4-7
Dielectrics Suppose we have a capacitor of value 0 that is charged to a potential difference of V 0 and then removed from the charging source We would then find that it has a charge of 0 V 0 We now insert the dielectric material into the capacitor We find that the potential difference decreases by a factor K V0 V K Or equivalently the capacitance has increased by a factor of K K 0 This constant K is known as the dielectric constant and is dependent upon the material used and is a number greater than Physics 3 Lecture 4-8
Polarization Without the dielectric in the capacitor, we have +++++++++++++++ --------------- The electric field points undiminished from the positive to the negative plate With the dielectric in place we have The electric field between the plates of the capacitor is reduced because some of the material within the dielectric rearranges so that their negative charges are oriented towards the positive plate V 0 V E 0 +++++++++++++++ κ - + - + - + E --------------- - + - + - - + + Physics 3 Lecture 4-9
Polarization These rearranged charges set up an internal electric field that opposes the electric field due to the charges on the plates The net electric field is given by E K E 0 Physics 3 Lecture 4-30
Redefinitions We now redefine several quantities using the dielectric constant We define the permittivity of the dielectric as being apacitance: ε K ε 0 A A K0 Kε 0 ε d d with the last two relationships holding for a parallel plate capacitor Energy Density u 0 Kε E εe Physics 3 Lecture 4-3
Example 7 Two identical parallel plate capacitors are given the same charge, after which they are disconnected from the battery. After has been charged and disconnected it is filled with a dielectric. ompare the voltages of the two capacitors. a) V > V b) V V c) V < V We have that and that K We also have that /V or V / Then V and V K Physics 3 Lecture 4-3 V K