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17.1 Electric Potential and Potential Difference Monday, July 20, 2015 5:46 PM The notion of energy allows us to solve many dynamics problems without dealing with the detailed trajectories that Newton's Laws would require. Similarly, the notion of energy can be used to solve problems involving electric forces. A charged particle in an electric field has potential energy because, under the influence of the field, there will be a force on the particle and it will move, thereby gaining kinetic energy. Energy is of course conserved so if the kinetic energy changes, the potential energy must also change. Thus, we need to know how to calculate potential energy. We define the change in electric potential energy to be the negative of the work done by an electric force to move a charge from one point to another (this parallels the definition of gravitational potential energy). Positive plate Negative plate An electric field exists between the charged plates. A positive charge near the positive plate will be acted upon by the electric force which will move it towards the negative plate. The work done on the positive charge is f x d, and this work results in the particle gaining kinetic energy (f x d). Thus it must lose potential energy equal to -(f x d). The electric field was previously defined as the (electric) force per unit charge. Electric potential is defined as the potential energy per unit charge. The symbol for electric potential (or simply, potential) is V. Suppose a point charge has electric potential energy PE(a) at point a, then the electric potential (the potential energy per unit charge), or V, is Ch. 17 Page 1 then the electric potential (the potential energy per unit charge), or V, is V(a) = PE(a)/q. Since the reference point for potential energy is arbitrary it is only differences in electric potential (the potential difference) that are significant/measureable. We can write V(a)(b) = V(a) - V(b) = - W(b)(a)/q. The units for potential difference are joules/coulomb. One joule/coulomb is given the name volt. Only differences in potential energy are significant. For gravitational potential energy, ground level is often used as the zero reference. For electric potential energy, the ground (or any conductor connected to the ground) is often used as the zero level. Remember that electric potential is defined as the potential energy per unit charge so when a charge q moves from point (a) to point (b) the change in potential energy is PE(a) - PE(b) = q x V(b)(a). That is, just as two different rocks can have the same gravitational potential (same height), the more massive rock will have greater potential energy, so too, two different charges can be at the same potential but the larger charge will have greater potential energy. Note: positive charges move naturally from locations of high potential to low potential. Negative charges move naturally from locations of low potential to high. Do Problems 1-4, page522. Ch. 17 Page 2 17.2 Relation Between Electric Potential and Electric Field Thursday, July 30, 2015 11:25 AM A bunch of distributed charges will have an effect on any other charge placed in space. The effect of the distributed charges can be described using either the idea of electric field or the idea of electric potential. Since the electric potential is a scaler quantity it is often easier to do calculations using the potential rather than the more complicated (vector) quantity electric field. What is the relation between electric potential and the electric field? This question is answered for the simple case of the uniform electric field that exists between parallel plates when there is a potential difference between them. Suppose there is a positive charge Q at the positively charged plate. The potential energy lost as the charge moves to the negative plate is the work done by the field (on the charge). That is ΔPE = W = q x V(b)(a) [see sec. 17.1]. But we can also write work as W = f x d = (q x E) x d where d is the distance between the plates and E is the field strength. Thus, W = qV = qEd, so V = Ed, or E = V/d. • Note that E, the electric field strength, has the units Newtons/colomb. E= V/d shows us the units could also be Volts/m. Ch. 17 Page 3 E 17.3 Equipotential Lines Sunday, August 2, 2015 9:11 AM Just as topographic maps have contour lines to connect places of equal heights (in effect, equal gravitational potential) so too we can draw lines connecting places of equal electric potential around charges. Note that it takes no work to move a charge along an equipotential line. Also, the equipotential lines must be perpendicular to the direction of the electric field at each point. Ch. 17 Page 4 17.4 The Electron Volt Sunday, August 2, 2015 9:32 AM The Joule is the unit of energy appropriate for much of physics. A Joule is a Newton-meter, which is to say a force of 1 Newton acting over a distance of one meter. For charged particles, such as protons and electrons, the associated electric forces are very small. A unit of energy more appropriate for dealing with small charges is the Electron-Volt. The electron volt (eV) is defined as the amount of energy acquired by a particle carrying a charge equal to that of an electron through a potential difference of 1 volt. So 1 eV = (1.6 x C)(1 V) = 1.6 x Joules Note that when doing calculations electron-volts should be converted to standard S.I. units (i.e. Joules) Ch. 17 Page 5 17.5 Electric Potential Due to Point Charges Sunday, August 2, 2015 10:09 AM Recall (sec 16.7) that the electric field strength produced by a point charge is . Using calculus it is easy to show that the electric potential around the point charge is V = [this presumes the electric field, and electric potential, is zero at infinity]. Note: the electric field decreases with the square of the distance while the electric potential decreases directly with the distance Note: the potential near a positive charge is large and decreases towards zero with distance from the charge; the potential near a negative charge is negative and increases towards zero with distance from the charge Ch. 17 Page 6