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Chapter 2, , Electrostatic Potential, and Capacitance, , Chapter Contents, , Introduction, , Electrostatic Potential, Energy, Electrostatic Potential, , Potential due to a Point, Charge, , Potential due to an Electric, Dipole, , Potential due to System of, Charges, , Equtpotential Surfaces, , Potential Energy of a, System of Charges, , Potential Energy in an, External Field, , Electrostatics of Conductors, Dielectrics and Polarization, Capacitors and Capacitance, The Parallel Plate Capacitor, Effect of Dielectric on, Capacitance, , Combination of Capacitors, , Energy Stored ina, Capacitor, , Van de Graaff Generator, Some Important Definitions, Formulae Chart, , Quick Recap, , Introduction, , We saw in Chapter on Work, Energy and Power in class XI that the, concept of energy was extremely valuable in dealing with the subject, of mechanics., , For one thing, energy is a conserved quantity and is thus an important, tool for understanding nature. Furthermore, we saw that many, Problems could be solved using the energy concept even though a, detailed knowledge of the forces involved was not possible, or when a, calculation involving Newton's laws would have been too difficult., , The energy point of view can be used in electricity and it is especially, useful. It not only extends the law of conservation of energy. but it, gives us another way to view electrical phenomena. Energy is also a, tool in solving Problems more easily in many cases then by using forces, and electric fields. Electric energy can be stored in a common device, called a capacitor, which is found in nearly all electronic circuits., A capacitor is used as a storehouse for energy. Capacitors store the, energy in common photoflash units. The rapid release of this energy is, evident in the short duration of the flash of camera used to take, , photographs., , ELECTROSTATIC POTENTIAL ENERGY, , The concept of potential energy is extremely useful. For example, in thinking, about gravity we learned that a mass m at a height h (much less than Earth's, fadius) above Earth's surface has a potential energy that can be written as, U(h) = mgh. This helps us to determine the object’s speed at any other, height, if we know its speed at one height. Any conservative force has a, potential energy associated with it. This potential energy is a function of, position, and it can be converted to kinetic energy in accordance with the, conservation of mechanical energy., , The total energy is E = K + U, where K is the kinetic energy. Conservation of, mechanical energy means that the change in Eis zero,so AE=0=AK+ AU, or AK = — AU. This means that any change in U will be equal but opposite to, change in K., , , , Aakash Educational Services Pvt. Ltd. - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456

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74 Electrostatic Potential and Capacitance Board & Competitive Exams., , , , Let us now look at Coulomb's law. The electric force on charge q, due to charge q, separated by a distance r, is, F = 2% 41;, , 4ne, rr’, , Where 7 is the unit vector that points radially outward from the position of q to the position of go. Recall, , that the gravitational force between a mass m, and a mass m separated by a distance ris, , , , F =-Gmm, af, This is just the same form as Coulomb's law, although the gravitational force is always attractive. Whereas, the electric force is attractive or repulsive according to whether q, is negative or positive. Each of the above, , forces is conservative, so a potential energy is associated with each of them. This potential energy must of, course take the same form for both cases., , Expression for Electrostatic Potential Energy, Consider an electrostatic field E due to some charge configuration. First, for simplicity, consider the field E, due to a charge Q placed at the origin. Now, imagine that we bring a test charge q from a point R to a point, P against the repulsive force on it due to the charge Q. (See the following figure), A test charge q ( > 0 ) is moved from the point R to the point P against the repulsive force on it by the charge, Q (> 0) placed at the origin., , e, ”, , Q, , With reference to figure, this will happen if Q and q are both positive or both negative. For definiteness, let, us take Q, q > 0. Two remarks may be made here. First, we assume that the test charge q is so small, , that it does not disturb the original configuration, namely the charge Q at the origin (or else, we keep Q fixed, at the origin by some unspecified force). Second, in bringing the charge q from R to P, we apply an external, , , , force F,,; just enough to balance the repulsive electric force F, (i.e, F., =— FF, ). This means there is, , no net force on or acceleration of the charge q when it is brought from R to P, i.e., it is brought with, infinitesimally slow constant speed. In this situation, work done by the external force is the negative of the, work done by the electric force, and gets fully stored in the form of potential energy of the charge q. If the, extemal force is removed on reaching P, the electric force will take the charge away from Q — the stored, energy (potential energy) at P is used to provide kinetic energy to the charge q in such a way that the sum, of the kinetic and potential energies is conserved. Thus, work done by external forces in moving a charge q, from R to P is, , e, Was ae, , Fear, , R, This work done is against electrostatic repulsive force and gets stored as potential energy. At every point in, electric field, the particle with charge q possesses a certain electrostatic potential energy. This work done, increases the potential energy by an amount equal to potential energy difference between points R and P., Thus, potential energy difference, , , , , , AU = Up— U gp = Wrap, (Note here that the displacement is in an opposite sense to the electric force and hence work done by electric, field is negative, i.e., —-Wpp .), , Aakash Educational Services Pvt. Ltd. - Regd. Office : Aakash Tower, 8, Pusa Road, New Dethi-110005 Ph. 011-47623456

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Board & Competitive Exams. Electrostatic Potential and Capacitance 75, , ——, Mlustration: (i) A proton is fixed and an electron is moved from = to a point near the proton, W,,, < 0, , (-- external force and displacement are in opposite direction). This means AU < 0, i.e., potential energy decreases., , , , (ii) A proton is fixed and another proton is moved from = to a point near it, W,,, > 0 (-.- external, force and displacement are in same direction). This means AU > 0, i.e. potential energy increases., , ae z, eevee seneeet SF, > E. P, , , , , , 1. Three charged particles (their sign are shown) move along the path in a uniform electric field, as shown. In each of three cases, state whether the potential energy increases or decreases?, , —, , —++—O (A), , a, , ———_O®), , ——————— ee, , (c)@—>—, , —, , 2. What is the work done by the field of a nucleus in a complete circular orbit of the electron and, what if the orbit is elliptical?, , , , Definition of Electrostatic Potential Energy, , We can define electric potential energy difference between two points as the work required to be done by, an external force in moving (without accelerating) charge q from one point to another in electric field of any, arbitrary charge configuration., , —__ = —E a — — = = =, , Example 1: A charge is moved in an electric field of a fixed charge distribution from point A to another point, B slowly. The work done by external agent in doing so is 100 J. What is the change in potential, energy of the charge as it moves from A to B? What is the work done by the electric field of, the charge distribution as the charge moves from A to B?, , Solution : Wy = AU = Uz — U, = 100 J, As, F,q =— Fz for the charge to move slowly, so, Ww, =— W,,, =— 100 J., , Aakash Educational Services Pvt. Ltd. - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456

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76 = Electrostatic Potential and Capacitance Board & Competitive Exams., , , , , , 3. ‘In the given question, let the charge be released from rest at point 8. What will be its kinetic, energy when it crosses point A?, , , , , , Reference for Potential Energy, The above equation defines potential energy difference in terms of the physically meaningful quantity work., Clearly, potential energy so defined is undetermined to within an additive constant. What this means is that, the actual value of potential energy is not physically significant; it is only the difference of potential energy, that is significant. We can always add an arbitrary constant to potential energy at every point, since this will, not change the potential energy difference:, (Up + a) — (Up + a) = Up—, Put it differently, there is a freedom in choosing the point where potential energy is zero. A convenient choice, , is to have electrostatic potential energy zero at infinity. With this choice, if we take the point R at infinity,, we get from the previous result,, , W_ p= Up-U_=U,, , Since the point P is arbitrary, the above equation provides us with a definition of potential energy of a charge, q at any point., , Definition of potential energy of a point charge, , Potential energy of charge q at a point (in the presence of field due to any charge configuration) is defined, as the work done by the external force (equal and opposite to the electric force) in bringing the charge q from, infinity to that point., , ELECTROSTATIC POTENTIAL, , To understand electric potential, consider the particle with the small positive charge located at some distance, from a positively charged sphere in the following figure., , & F, O- a=, , If you push the particle closer to the sphere, you will expend energy to overcome electrical repulsion;, i.e., you will do work in pushing the charged particle against the electric field of the sphere. This work done, in moving the particle to its new location increases its energy. We call the energy the particle, possesses by virtue of its location as electric potential energy. If the particle is released, it accelerates in a, direction away from the sphere, and its electric potential energy changes to kinetic energy. If we instead, push a particle with twice the charge, we do twice as much work pushing it, so the doubly charged, particle in the same location has twice as much electric potential energy as before. A particle with three, times the charge has three times as much potential energy; ten times the charge, ten times the, potential energy; and so on. Rather than dealing with the potential energy of a charged body, it is convenient,, when working with charged particles in an electric field, to consider the electric potential energy per unit, of charge. We simply divide the amount of potential energy in any case by the amount of charge., , , , , , , , Aakash Educational Services Pvt. Ltd. - Regd. Office : Aakash Tower, 8, Pusa Road, New Dethi-110005 Ph. 011-47623456

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Board & Competitive Exams. Electrostatic Potential and Capacitance 77, , For example, a particle with ten times as much charge as another in the same location will have ten times, as much potential energy but having ten times as much charge means that the energy per unit charge is, the same. The concept of potential energy per unit charge is called electric potential (V) i.e.,, , Electric potentialenergy, , Electric potential =, poten Charge, , Work done by an external force in bringing a unit positive charge from infinity to point in electrostatic field of, some another charge without acceleration is equal to electrostatic potential (V) at that point., , Unit for Electric Potential, , The unit of measurement for electric potential is the volt, so electric potential is often called voltage. A potential, of 1 volt (V) equals 1 joule (J) of energy per 1 coulomb (C) of charge., , joule, , ‘Tek =f coulomb, , Electric Potential Difference, Similar to electric potential, the electric potential difference is the work done by external force in bringing a, unit positive charge from point R to point P. i.e.,, , Vp —V,_= Ye—Ye, q, Here V, and Vp are the electrostatic potentials at P and R, respectively and Up, and U, are the potential, energies of a charge q when it is at P and at R respectively., , Note : As before, that it is not the actual value of potential but the potential difference that is physically, significant. If, as before, we choose the potential to be zero at infinity, the above equation implies., , =, Example 2: Find the change in electric potential energy, AU as a charge of (a) 2.20 x 10-* C., (b) —1.10 x 10-6 C moves from a point A to a point B, given that the change in electric potential, between these points is AV = V, — V, = 24.0 V., Solution : (a) Solving AV = AU / q, for AU, we find, AU = q,AV = (2.20 * 10 C) (24.0 V) = 5.28 « 1055, (6) Similarly, using gq, = —1.10 = 10-* C, we obtain, AU = q,AV = (-1.10 * 10-* C) (24.0 V) = -2.64 x 10° J, , , , , , , , -_, , Example 3: A particle of mass m and positive charge q is released from point A. Its speed is found to be v, when it passes through a point B. Which of the two points is at higher potential? What is the, potential difference between the points?, , Solution : The point A is at higher potential than B as the particle has gained kinetic energy while moving, from A to B., , The kinetic energy acquired by the particle is equal to loss in potential energy. i.e.,, 1 mv?, , AK =q(V,-Vs) => mv? = q(Va—Vs) 3 GaVes a=, , , , Aakash Educational Services Pvt. Ltd. - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456