CapacitanceParallel Plate CapacitorCylindrical Capacitor (Cable)Spherical CapacitorCapacitors in ParallelCapacitors in SeriesEnergy Stored in a CapacitorEnergy Stored in an Electric FieldDielectricsPHY2061 Enriched Physics 2 Lecture Notes CapacitanceCapacitanceDisclaimer: These lecture notes are not meant to replace the course textbook. The content may be incomplete. Some topics may be unclear. These notes are only meant to be a study aid and a supplement to your own notes. Please report any inaccuracies to the professor.A set of conductors can store electric charge. The net charge Q=0=qq, but the magnitudeof charge on each conductor is |q|.This charge q is proportional to the potential difference between the conductors: q C V V V V Vq CV+ -= D D = - �=The constant of proportionality between charge and potential difference is Ccapacitance. Unit is Farad (F) Coulomb/Volt.6121 F = 10 F1pF = 10 Fm--To set up a potential difference between 2 conductors requires an electric “pump”, such as a battery (see next chapter).A larger capacitance implies that a large charge q is stored for the same potential difference V.Capacitance depends only on the geometry of the conductors, not the charge q or voltage V. We can see this through examples.D. Acosta Page 1 1/13/2019+qqCV++qqPHY2061 Enriched Physics 2 Lecture Notes CapacitanceParallel Plate CapacitorConsider the top view of the 2 plates:Create a Gaussian surface (box) that extends inside and outside one of the conductor surfaces.Gauss’ Law enc0Sqde� F = � =�E A�0=E inside a conductor0d� =E A on left/right edges0�E on front outside face onlyenc00Sqd EAq EAee� F = � = =� =�E A�The electric potential difference between the 2 plates is given by: opposite directionsV V V d d dsV dVd++ --D = - =- � � =-D =D� =�E s E s EEESo for parallel plates:D. Acosta Page 2 1/13/2019V+dAE+EdsPHY2061 Enriched Physics 2 Lecture Notes Capacitance0 0 02120 02C 8.85 10 8.85 pF/mNmV Aq EA A Vd dq C VACde e ee e-D� �= = = D� �� �= D� = = � =Cylindrical Capacitor (Cable)Let inner conductor have radius a, and outer radius b.Take Gaussian surface as cylinder between conductors (E=0 inside conductors).enc0002 0 on cylinder ends12SqdE rL q dqEL rep epeF = � =� = � =� =�E AE A�0000 opposite directions, but opposite again2ln22ln /2ln /b ba aV V V d d ds ds drq drV dr drL rq bVL aLq Vb aLCb apepepepe++ --+-D = - =- � � =- =-D =- = =D =� �� = D� �� �� =�� � �E s E s EE ESpherical CapacitorLet inner sphere have radius a, and outer radius b.Take Gaussian surface as sphere between conductors (E=0 inside conductors).Gauss’ Law 2 qK a r br� = < <ED. Acosta Page 3 1/13/2019PHY2061 Enriched Physics 2 Lecture Notes Capacitance2 opposite directions, but opposite again1 1 111b ba abaV V V d d ds ds drdrV dr dr KqrV Kq Kqr a babq VK b aabCK b a++ --+-D = - =- � � =- =-D =- = =� � � �D = - = -� � � �� � � �� �� = D� �-� �� =-�� � �E s E s EE ECapacitors in ParallelConsider N capacitors all connected in parallel to the same source of potential difference V. Across each capacitor i the charge on one of the plates is: i iq C V=The total charge on all the plates with the same electric potential is:1 1 1N N Ni i ii i iQ q CV V C= = == = =� � �So we can write the equivalent capacitance Cequiv as:equivequiv1NiiQ C VC C===�In other words, the equivalent capacitance of N capacitors in parallel is the sum of the individual capacitances. Considering the example of parallel plate capacitors, adding several in parallel is equivalent to extending the area of the plates. Since the capacitance is proportional to the area, it increases in direct proportion.Capacitors in SeriesFor N capacitors in series, the magnitude of the charge q on each plate must be the same. Consider the electric conductor connecting any 2 capacitors, and suppose that a charge +qis on the plate of one of the capacitors the conductor is connected to. Since the conductor was originally uncharged, a charge –q must exist on the plate of the second capacitor. Now a capacitor has the same charge magnitude on each plate, so by inference we can determine that the magnitude of charge on each plate in the series of capacitor must be the same.D. Acosta Page 4 1/13/2019PHY2061 Enriched Physics 2 Lecture Notes CapacitanceThe potential difference across any capacitor is given by iiqVC=The total potential difference must add up to electric potential supplied by the battery or power supply:1equivNiiq qVC C== =�So the equivalent capacitance of capacitors connected in series is given by:1equiv1 1NiiC C==�The potential difference across any capacitor can be determined by:11NjjiiVqCqVC===�Energy Stored in a CapacitorLet’s calculate the work required of a battery or power supply to move an infinitesimal charge dq� onto the plate of a capacitor already containing a charge q�. This is the same as finding the change in the potential energy of the capacitor.Recall that the electric potential difference across a device is equal to the potential energydifference per unit charge:UVqDD =The potential energy difference is equal to the negative of the work done by the electric field to set up the configuration, or in other words equal to the work done by the power supply or battery to move the charge (the charge must move against the direction of the electric field):appW U q V=D = DSo the work done to move an infinitesimal charge dq� onto the plate of a capacitor is given by:D. Acosta Page 5 1/13/2019PHY2061 Enriched Physics 2 Lecture Notes CapacitanceappdW dq V�= DIf the capacitor already has a charge q�, then qVC�D =So appqdW dqC��=So to charge up a capacitor initially uncharged to a total charge q will require integrating over the above expression:2app app02app1 122qqW dW q dqC CqW UC� �= = ==D =� �Since q C V= D for a capacitor, the electric potential energy stored in a capacitor can be expressed in 2 ways:( )2212 2qU C VCD = = DThis potential energy can be used to perform work if the capacitor is disconnected from the power supply and connected to an electrical circuit. For example, a flash bulb on a camera works in this way.Using both forms of the relation for the energy in a capacitor, we can see which capacitor has a greater energy when two are connected in series or parallel. When two capacitors are
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