V 1 of 9 Voltage Electric Potential An electric charge alters the space around it Throughout the space around every charge is a vector thing called the electric field Also filling the space around every charge is a scalar thing called the voltage or the electric potential Electric fields and voltages are two different ways to describe the same thing Note on terminology The text book uses the term electric potential but it is easy to confuse electric potential with potential energy which is something different So I will use the term voltage instead Voltage overview The voltage at a point in empty space is a number not a vector measured in units called volts V Near a positive charge the voltage is high Far from a positive charge the voltage is low Voltage is a kind of electrical height Voltage is to charge like height is to mass It takes a lot of energy to place a mass at a great height Likewise it takes a lot of energy to place a positive charge at a place where there voltage is high Q lower voltage here higher voltage here Only changes in voltage V between two different locations have physical significance The zero of voltage is arbitrary in the same way that the zero of height is arbitrary We define V in 2 equivalent ways U of q V K K V VB VA E dr q change in potential energy of a test charge divided by the test charge B A K K For constant E field this integral simplifies to V E r r change in position The electric field is related to the voltage in this way Electric field is the rate of change of voltage with position E field is measured in units of N C which turn out to be the same as Phys1120 Dubson 9 14 2009 University of Colorado at Boulder V 2 of 9 volts per meter V m E fields points from high voltage to low voltage Where there is a big Efield the voltage is varying rapidly with distance high voltage E field low voltage In order to understand these strange abstract definitions of voltage we must review work and potential energy Work and Potential Energy U G Definition of work done by a force consider an object pulled or pushed by a constant force F While the force is applied the object moves through a displacement of r rf ri F i f r Notice that the direction of displacement is not the same as the direction of the force in general K K Work done by a force F WF F r F r cos F r constant F F component of force along the direction of displacement If the force F varies during the displacement or the displacement is not a straight line then we K K must use the more general definition of work done by a force WF F dr Work is not a vector but it does have a sign or Work is positive negative or zero depending on the angle between the force and the displacement F r 90 W positive Phys1120 Dubson 9 14 2009 F F r 90 W 0 r 90 W negative University of Colorado at Boulder V 3 of 9 Definition of Potential Energy U Associated with conservative forces such as gravity and electrostatic force there is a kind of energy of position called potential energy The change in potential energy U of a system is defined to be the negative of the work done by the field force which is the work done by an external agent opposing the field U Wext Wfield This is best understood with an example A book of mass vf 0 Forces on book m is lifted upward a height h by an external agent a hand h which exerts a force to oppose the force of gravity The force of gravity is the field In this case the work done g m by the hand is Wext mgh The work done by the field vi 0 Fext Fgrav mg gravity is Wfield mgh The change in the potential energy of the earth book system is U Wext Wfield mgh The work done by the external agent went into the increased gravitational potential energy of the book The initial and final velocities are zero so there was no increase in kinetic energy A conservative force is force for which the amount of work done depends only on the initial and final positions not on the path taken in between Only in the case that the work done by the field in independent of the path does it make any sense to associate a change in energy with a change in position Potential energy is a useful concept because if there is no friction no dissipation K U 0 K U constant no dissipation K kinetic energy m v2 Voltage We define electrostatic potential energy not to be confused with electrostatic potential or voltage in the same way as we defined gravitational potential energy with the relation U Wext Wfield Consider two parallel metal plates a capacitor with equal and opposite charges on the plates which create a uniform electric field between the plates The field will push a test charge q toward the negative plate with a constant force of magnitude F q E The situation is Phys1120 Dubson 9 14 2009 University of Colorado at Boulder V 4 of 9 much like a mass in a gravitational field but there is no gravity in this example Now imagine grabbing the charge with tweezers an external agent and pulling the charge q a displacement r against the electric field toward the positive plate By definition the change in electrostatic potential energy of the charge is K K K K U U f Ui Wext Wfield Ffield r q E r I recommend that you do not try to get the signs from the equations it s too easy to get confused Get the sign of U by asking whether the work done by the external agent is positive or negative and apply U Wext f E q hi PE r F qE lo PE i If the E field is not constant then the work done involves an integral f U U f Ui Wfield i f K K K K Ffield dr q E d r i Now we are ready for the definition of voltage difference between two points in space Notice that the change in PE of the test charge q is proportional to q so the ratio U q is independent of G G Fon q Similarly we define q Recall that electric field is defined as the force per charge E q the voltage difference V as the change in PE per charge f K K U V E dr q i or U q V Remember that the E field always points from high voltage to low voltage K K V E r Phys1120 Dubson 9 14 2009 if E constant University of Colorado at Boulder V 5 of 9 E field high voltage K …
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