Physics 1120 Review What follows is a rough outline of physics 1120 material it is by no means complete but is meant to highlight basic ideas and topics It is certainly not meant to be used as a crib sheet for your exam As usual the equations by themselves are meaningless so please read the text of this document to aid in understanding their meaning Where vector arrows are omitted for vector quantities it denotes the amplitude only The reader must determine the direction 1 Electrostatics A Electric Fields and Forces The electric field is a vector field which obeys the laws of superposition i e each contributing charge or charge distribution can be considered separately and then the vector sum of all the contributions can be taken to give the total field Electric field lines are a tool for visualizing a field The strength of the field is proportional to the density of field lines Lines originate at positive charges and terminate on negative charges kq r Electric Field of a point charge E r2 Force on a charged particle due to an electric field F e q E q1 q2 r Force between two charged particles combine last two eqns F e k r2 In the last equation the plus or minus sign is chosen so that like charges repel and opposite charges attract A couple of examples of electric fields for specific charge distributions follow Electric field for a single infinite sheet of charge E 2 o Electric field of two infinite planes of opposite charge between the plates E o This last result can be seen from the superposition of two infinite plate fields where each plate has an opposite charge but equal magnitudes Of course the field outside of two oppositely charged parallel plates is zero why All of the expressions for electric fields that you will see can be derived from Gauss s Law see below Electric flux is defined to be dA e E where since we are integrating over an area the double integral is implied Flux is interpreted as the amount of field lines penetrating a surface If the surface is closed ie a sphere then the outward normal is traditionally taken as the positive For a non closed surface the choice for the direction of the area vector lies with you For a uniform electric field and a flat surface the above equation reduces to A EA cos uniform field and flat area only e E Here is the angle between the area vector and the electric field To find the electric field for a charge distribution with high symmetry a useful tool is Gauss s law e dA qenc E o Here qenc is of course the amount of charge enclosed by our closed imaginary Gaussian surface The most common Gaussian surfaces are spheres cylinders and boxes Without high symmetry Gauss s law is essentially useless It is only useful in cases where we can choose a surface that is perpendicular to the E 1 field everywhere and everywhere along that surface the field has the same magnitude Then we can back the electric field out of the integral leaving a simple integral over dA which just gives the area of our gaussian surface know your area formulas It is then a simple matter to solve for the electric field Conductors in electrostatic equilibrium have a few properties The electric field inside a conductor is zero make sure to know why On the surface of a conductor the electric field points perpendicular to the surface or parallel to the normal This is due to the fact that the entire conductor is an equipotential see below B Electric potential and Electric potential energy Since the electric field is a vector quantity it is mathematically a pain in the ass to deal with at times A very useful concept which is both convenient and offers a bridge into the world of electronics is electric potential field As with gravitational potential with which you are hopefully more familiar the zero point of the potential can be chosen anywhere we want as long as we are consistent after that choice Since this means that the absolute value of the potential has no meaning the only thing that is meaningful are changes in potential In terms of the electric field change in electric potential is defined to be d l V E This is a path integral the dot product serves to pick out components of our path that are along the electric field through the cosine dependence Thus if we move perpendicularly to the field we move along an equipotential line Instead of drawing field lines we can draw equipotential lines or surfaces if you are a talented artist to represent a field By the preceding argument field lines are always perpendicular to equipotential surfaces In the gravity analogy the gravitational force acts downward and equipotential lines surfaces are then planes of constant height If we are in a uniform field like in a parallel plate capacitor and choose a path along the direction of the field the above integral reduces to V E l For non infinite charge distributions we normally take the zero point for the potential to be at infinity With this convention we can write Electric potential due to a point charge V kq r Notice that potential can be either positive or negative depending on the sign of the charge Since we integrate the field the electric potential for a point charges falls off like 1 r in contrast to the electric field that falls off like 1 r2 Just as an object with mass in a gravitational field can have gravitational potential energy a object with charge can have electric potential energy EPE EP E qV for a point charge This is the potential energy associated with a charged particle q being placed in a electric potential field created by other charge s Moving a charged particle in an electric field can change its potential The work required and the corresponding change in potential are related by the following V Workext EP E Workelec q q q The subscripts elec and ext represent the works done by the field itself and an external agent respectively The minus sign is important convince yourself why Notice that if we move along an equipotential EP E the dot is zero and so no work is done by either the field or the external agent Recall that Work F d product of the force and the displacement Another useful result is the work kinetic energy theorem Worknet KE 2 where net means the total work done on an object by both conservative and non conservative forces The energy density of an electric field is given by uE o E 2 2 To find the energy stored in an electric field you have to integrate this energy density over the volume in question C Capacitors Capacitors are objects that store
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