# Electric Potential, Capacitance, and Capacitors [Electrostatics (1958)]

 AMAZON multi-meters discounts AMAZON oscilloscope discounts 25. Potential and Potential Difference As mentioned in Section 14, the potential difference between two points may be defined as the ratio of energy to charge, or: Potential Difference = Energy /charge (38) It remains to determine the units that may be properly applied to this relation and to visualize the physical meaning that underlies the concept of potential difference. The absolute potential of a given body is measured by the number of ergs or joules required to bring a unit charge q of like sign (so that work must be done against the repelling force) up to the body from infinity. Absolute potential is virtually impossible to determine, but for convenience the earth is assumed to be at zero potential and all other potentials are measured relative to it. It may be shown by the integral calculus that the work done in moving any point charge q from infinity to a distance r from another point charge q (in the mks system) is given by the equation W = qq' / 4 pi e_0 r joules (39) The equivalence of work and energy makes it possible to state that the work done according to this equation is also the potential energy given to the point charge as a result of moving it from infinity to a distance r from another point charge of similar sign. Since electrical potential is the ratio of energy to charge, (40) (41) According to the definition of the volt given in Section 2, however, it is fundamentally the same unit as the joule per coulomb, hence the expression above is said to be the potential in volts of a point charge q with reference to some zero level taken at a great distance away from the point charge. Determination of the potential around a charged conducting sphere will demonstrate several interesting points. We have already shown that the electric field around a charged sphere may be computed by considering that the entire charge is concentrated at the center of the sphere. The potential of such a sphere is, therefore, also given by Equation 41, in which r must be taken equal to or greater than the radius of the sphere. Now consider such a sphere being charged in air by some electrostatic method. (Such methods will be discussed in Chap. 5.) When the electric field intensity becomes sufficiently great, the air in the vicinity of the sphere becomes conductive and the charge leaks off as fast as it is generated. The question arises as to what factor or factors determine the maximum potential to which the sphere can be charged. This question may be answered by reasoning as follows: designating the maximum electric field intensity by Em.., we can determine the largest charge that the sphere can hold by solving Equation 19 for q: (42) in which r. is the radius of the sphere. Substituting this value of q_max into Equation 41, (43) Experimental evidence shows that air becomes conducting when the field intensity approaches 3 X 10^6 volts per meter, hence the maximum potential in volts (the voltage) that can be built up on a con-ducting sphere of given radius regardless of the charging method is: Vmax = 3 X 10^6 (r) volts (44) …where r. is measured in meters. This development provides a supplementary explanation of the phenomenon of discharge from points discussed in Chap. 3. Since the maximum voltage is a function of the radius of the charged body, a surface that has an extremely small radius of curvature cannot sustain a high potential without causing ionization of the air and consequent discharge. In addition to this, it also illustrates the reason for the use of extremely large spheres in million-volt Van
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