Chapter 23 - Electric Potential

Chapter 23 - Electric Potential ● Gravitational Potential o Gravitational Potential energy: ▪ Technically, it is only defined as a difference in energy, since is a relative measurement. (there's a negative sign in order to ▪ make the ▪ and vectors point in the same direction) Energy is stored in the relationship between objects and the electric field. ▪ Increases when an outside force moves an object against the field (up). ▪ No energy change if you move perpendicular to field lines as per dot product rules. o Gravitational potential (GP) - potential energy per mass ● (defined at a point in the field) ▪ Describes how much energy can be stored at that point in the field. ▪ Objects spontaneously move to lower gp. Electric Potential - we will generate our equations and terms relating to Electric Potential by creating analogies to Gravitational Potential o Electric Potential energy: a difference between two points in a field. ▪ By comparing it to ▪ Similar to , we can say in the sense that charge is a replacement for mass in the formula. ▪ It is a . There is no absolute point ▪ For a positive test charge, you increase electric potential energy by moving the charge against the for energy. field lines. ▪ For a negative charge, reverse it: increase by moving it with the field. o Electric Potential ( ) - an energy per charge (analogous to GP) ▪ ▪ SI unit - Volt (V) - aka Joule per Coulomb ▪ Usually, we talk about a difference in potential between two points: ▪ Scalar quantity, although signs must be used due to the signed nature of charges. ▪ decreases in the direction the field points (doesn't depend on the sign of ). Think of using a positive test charge as the standard. ▪ o Alternative definition of since o Positive charges spontaneously move towards lower . for uniform field and Two charged parallel plates - , then - Voltage changes proportionally to the location of a particle between the plates. o - i.e. - Volts per meter (which is really just another way to say Newtons per Coulomb). It can be used to solve for of o ) between two oppositely charged plates. Electric potential energy often becomes kinetic. Since then (magnitude , (kinetic). (must be negative because it's moving to a negative change in Voltage that is from higher energy to lower energy) (for a positive charge, that is). o Electronvolts (eV) - It is a nonstandard energy unit. (ELEMENTARY CHARGE! - one electron moving through one potential difference of one volt causes a transform of energy equivalent to one electron volt. o Equipotential Diagrams - Essentially a topographic map - showing lines of equal potential (V) like a topographic map shows contours of equal height. When you walk on a contour line, your potential energy stays the same -- when the lines are close together and you move perpendicular, the potential energy changes the fastest. Downhill is perpendicular to the contour line. No work is done by moving along an equipotential line. Equipotential lines are ALWAYS perpendicular to the electric field. ▪ Points on the same equipotential line have the same voltage. Lines closer to a positive source indicate a higher voltage. Lines further indicate a lower voltage. (positive voltage near a positive source charge & negative voltage near a negative source charge). o Non-Uniform Fields ▪ Electric potential energy: Between two point charges the uniform case . Now taking as a function of position. Normally, we'd just use Coulomb's law to calculate . Now: ▪ For two point charges... thus, potential energy for a pair of charges... note how . Also, because of signums of charge, is positive for like charges and negative for opposite charges. o Electric Potential: Pair of charges. Since charges, (where , for a pair of is the source of the field). This is like the situation above except we disregard the charge of the second particle. It gets cancelled out by division, as you're getting the potential energy per charge (i.e. - the definition of Voltage) o Generally, is an integral. thus , so (this is like the potential energy formula, except with no ). o WORK: . Work done by the field is negative. The work that you put on the particle to put it in a location of higher voltage is just . Satisfies conservation of energy. o Conductors - Since electric field is zero inside, and , then is constant everywhere in it. Special Cases for Voltage (4 cases) A]Around a line charge , radius , Since (derived from use of Gauss' law), infinitely long. We will simply define the surface of the curve as of reference) so limit) will make . (reference at zero (i.e. - lower undefined) (because OR B]Above infinite thin sheet, charge density (and potential ) Previously derived: (frame (constant) ) Recall that we incorporated in case A. This time will act like a constant of integration. *C]V at center of charged Since all the charge is the same distance away and V isn't a arc radius vector, the arc acts like a point charge. All infinitesimal elements *D]Find along central are equidistant from the center of the arc. All charge is distance away from point P. Thus, axis of charged ring radius , charge *D and C are relative to , like the point charge. Dielectric Breakdown - in a strong enough electric field, a nonconductor (like air) can be made to conduct. For air, such a breakdown occurs when When two spheres of different radii are connected to each other over a large distance, as there is no electric field at equilibrium, the voltage is constant across the surfaces of everything. We can use this data to calculate charge, charge density, and electric field outside the spheres.