Electric Energy and Potential 15 In the last chapter we discussed the forces acting between electric charges. Electric fields were shown to be produced by all charges and electrical interactions between charges were shown to be mediated by these electric fields. As we’ve seen in our study of mechanics, conservation of energy principles can often be used to under- stand the interactions and dynamics of a system. In this chapter we introduce the concept of electric potential energy and electric potential, and apply these consid- erations to a variety of situations. The fundamental electric interactions in atomic, macroscopic, and macromolecular systems are each presented. Biological mem- branes are discussed in some detail, with emphasis on their ability to act as capac- itors, energy storage devices. Membrane channels are introduced, focusing on sodium channels: how they work and how they are selective. We return to a more detailed description of the electrical properties of channels in the next chapter. This chapter concludes with a discussion of the mapping of the electric potential pro- duced by various organs of the human body including muscles, heart, and brain (EMG, EKG, and EEG, respectively). These medical techniques are often used for diagnostic purposes. 1. ELECTRIC POTENTIAL ENERGY The electric force is a conservative force. As we saw in Chapter 4, this means that the work done by the electric force in moving a particle (in this case, charged) between two points is independent of the path and depends only on the starting and ending locations. Furthermore, there is an electric potential energy function that we can write down, whose negative difference at those two locations is equal to the work done by the electrical forces Ϫ Ϫ ϭ -¢ ϭ (PEE,final PEE,inital ) PEE W. (15.1) Recall that two expressions we have used for potential energy functions in mechanics, gravitational (mgy) and spring potential energy 1kx2 , followed from 12 2 the general definition of work and the particular form of the force. In a similar way, if Coulomb’s law for the force due to a point charge q1, on a second point charge q2, separated by a distance r is substituted into the general definition of work (see the box below), one obtains the electric potential energy of the two point charges q q ϭ 1 2 PEE (r) . (15.2) 4pe0 r J. Newman, Physics of the Life Sciences, DOI: 10.1007/978-0-387-77259-2_15, © Springer Science+Business Media, LLC 2008 E LECTRIC P OTENTIAL E NERGY 373 Here we derive an expression for the electric Note that, just as in the mechanical energy cases, we need to define potential energy between two point charges. the location of zero potential energy because only potential energy differ- We imagine that there is a point charge, say ences have meaning. For springs, the natural choice was to reference the Ͼ spring potential energy to a zero value for an unstretched spring that exerts q1 0, located at the origin and bring a sec- Ͼ no force. For gravitational potential energy near the Earth’s surface, we ond point charge, q2 0, from infinitely far away where it does not feel any electric force were free to define the location of zero potential as we chose because the gravitational force on a mass is constant in the approximation we used. to some distance r away from q1. Because both charges are positive, there is a repulsive For other more general situations using gravity, the zero of gravitational force between them and positive external potential energy occurs when all masses are infinitely far apart so as not work must be done to bring q toward the to be interacting. Similarly, in the case of electrical forces, when the 2 → ϱ charge at the origin. This work is equal and charges are infinitely far apart (r ) they do not interact and it is there- opposite to the (then, negative) work done on fore natural to choose this situation to correspond to zero electric poten- tial energy. Equation (15.2) already satisfies this convention. q2 by the electric force from q1. According to Equation (15.1) the change in potential energy The electric potential energy for charges of like sign that repel one will then be positive as might be expected, another is positive according to Equation (15.2), whereas for unlike charges because if the external force is removed, the that attract each other it is negative. Example plots for both cases are given repulsive force will change the positive elec- in Figure 15.1. We recall that the negative of the slope of such a plot is tric potential energy of q into kinetic energy equal to the force acting at position r. In this case, with one charge at the 2 Ͼ as it accelerates away from the origin. origin and the other at r, when PEE(r) 0 because the energy decreases From the general definition of work and with increasing r, the negative of the slope is always positive, consistent Equation (15.1), the electric potential with a repulsive force acting. The steeper the curve is, the larger the force energy change is given by (and therefore acceleration) acting. We can imagine a charged particle sit- ting on the energy curve and falling down its hill with a decreasing accel- r eration (but still increasing velocity) as it moves toward larger r values. A ¢PE ϭ PE(r)ϪPE(q) ϭ - [F cos u]ds Lq charge projected toward the origin with some initial kinetic energy will travel up the PE hill as far as corresponds to the conversion of all its where F is the electric force on charge q E 2 kinetic energy to potential before falling back down the energy hill. and ␪ is the angle between the force vector : Similarly, when PE (r) Ͻ 0, because increasing r leads to less neg- and the displacement vector ds. The path E ative, or increasing PE , the negative of the slope is itself negative, con- taken by the charge does not matter, there- E firming that the force is attractive. A charged particle placed on this fore we choose it to be inward along the curve will also fall down the potential hill ever more rapidly (with radial direction. In this case ␪ is equal to increasing acceleration) as its distance from the other charge at the ori- 180°, so that cos ␪ is equal to Ϫ1, and the gin decreases. We discuss electric potential energies for other situations displacement ds is equal to Ϫdr. We substi- later in this chapter in connection with molecular bonding. tute Coulomb’s law for the force to find Having found an expression for the electric potential energy of a Ϫ r q1 q2 1 pair of point charges, we can write an expression for the total energy of ¢PE ϭ dr. 2 this two-particle system. We include the kinetic energy of each particle, 4peo Lq r the electric potential energy, and any other mechanical potential ener- Remembering that gies, PEmech, appropriate to the situation. The conservation of energy 1 Ϫ 1 principle then states that dr ϭ , Lr2 r q q ϭ ϩ ϩ ϩ 1 2 ϭ we do the integration and evaluate the E KE1 KE2 PEmech constant. (15.3) resulting expression at the limits to find that 4peo r the potential energy at a distance r from the origin is given by Equation (15.2). As we have seen in applications in mechanics, energy conservation is a powerful concept that has a great degree of practical utility as well. FIGURE 15.1 Electric potential 4 energy for two point charges of 3 1 ␮C magnitude with the upper 2 curve for like sign charges and the 1 lower curve for opposite charges. 0 PE (J) –1 0 0.2 0.4 0.6 0.8 –2 –3 –4 r (m) 374 E LECTRIC E NERGY AND P OTENTIAL Example 15.1 Find an expression for the total energy of a hydrogen atom treat- ing the electron as traveling in a circular orbit around the stationary proton. Find an answer in terms of only the radius of the circular orbit. Solution: The total energy consists of the kinetic energy of the electron, travel- ing in a circle, and the electric potential energy of the electron–proton pair. We can write this as 1 1 (ϩe)(Ϫe) E ϭ mv2 ϩ . 2 4pe0 r To express the velocity of the electron in terms of its orbital radius, we use the fact that the only force on the electron is the Coulomb force and this must supply the centripetal acceleration according to 1 e2 v2 F ϭ ϭ m , 2 4peo r r where both the force and centripetal acceleration are radially directed. Solving for mv2 and substituting into the expression for the energy, we have 1 1 e2 1 e2 1 e2 E ϭ Ϫ ϭϪ . 2 4peo r 4peo r 8peo r This result says that the energy of a hydrogen atom is solely determined by the radius at which the electron orbits the proton. Note that the total energy is neg- ative. This is the signature of a bound system, with the negative potential energy term dominating over the positive kinetic energy term. We show in Chapter 25 that although this is a correct statement, the electron cannot orbit the proton at any radius, but only at certain allowed radii. This fact of nature leads to a dis- crete set of allowed energy levels for the hydrogen atom from the above equa- tion relating E to r, as first derived by Neils Bohr in 1913.