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Electric Potential Module Overview

University Physics Volume 2 Acknowledgments This presentation is based on and includes content derived from the following OER resource: University Physics Volume 2 An OpenStax book used for this course may be downloaded for free at: https://openstax.org/details/books/university-physics-volume-2

Electric Potential Energy The electric force is conservative, so we can define an electrical potential energy, 𝑈. The change in energy is equal to the work done to move a test charge 𝑄 from 𝑟 1 to 𝑟 2 through the field of a source charge 𝑞, given by the equation ∆𝑈=𝑘𝑞𝑄 𝑟 1 𝑟 2 1 𝑟 2 𝑑𝑟 =𝑘𝑞𝑄 1 𝑟 2 − 1 𝑟 1 . By convention, the potential energy is taken to be zero infinitely far away, so at a distance 𝑟, we have 𝑈 𝑟 =𝑘 𝑞𝑄 𝑟 . (University Physics Volume 2. OpenStax. Fig. 7.3.)

Electric Potential Energy 2 If there are many source charges, the principle of superposition may be applied. In this case, the work done to assemble a system of charges is equal to the sum of the work of bringing each charge in one at a time, written symbolically as 𝑊 12…𝑁 = 𝑘 2 𝑖 𝑁 𝑗 𝑁 𝑞 𝑖 𝑞 𝑗 𝑟 𝑖𝑗 for 𝑖≠𝑗. This is also the sum of the individual potential energies between each pair of charges. (University Physics Volume 2. OpenStax. Fig. 7.11.)

Electric Potential and Electric Potential Difference The electric potential 𝑉 is the potential energy 𝑈 per unit charge 𝑞, 𝑉= 𝑈 𝑞 . It is a measure of potential energy that is independent of the test charge. The electric potential difference between two points 𝐴 and 𝐵, or voltage, is the change in potential energy between the points, 𝑉 𝐵 − 𝑉 𝐴 , which has units of joules per coulomb, or volts. In terms of potential energy, ∆𝑉= ∆𝑈 𝑞 . (University Physics Volume 2. OpenStax. Fig. 7.12.)

The Electron-Volt and Conservation of Energy The electron-volt (eV) is a unit of energy that is more convenient for the description of energies of individual particles, like electrons and protons. It is defined as the energy of a fundamental charge accelerated by a 1 V potential difference, 1 eV=1.60 × 10 −19 J. Using conservation of mechanical energy, written 𝐾+𝑈=constant, the electron gun accelerates stationary electrons to significant velocities. (University Physics Volume 2. OpenStax. Fig. 7.13.)

Voltage and Electric Field The voltage at a point 𝑃 with respect to a reference point 𝑅 is related to the electric field by 𝑉 𝑃 =− 𝑅 𝑃 𝐄 ∙𝑑 𝐥 , which may be evaluated along any path between 𝑃 and 𝑅 to get the same result. This relation may be applied to find the difference in potential between any two points, ∆𝑉 𝐴𝐵 =− 𝐴 𝐵 𝐄 ∙𝑑 𝐥 . (University Physics Volume 2. OpenStax. Fig. 7.15.)

Systems of Multiple Point Charges The electric potential 𝑉 of a point charge is given by 𝑉= 𝑘𝑞 𝑟 , where the potential at infinity is taken to be zero. The electric potential obeys the superposition principle, so the voltage of a system of point charges is simply the sum of the potentials due to each individual particle, 𝑉=𝑘 𝑖 𝑁 𝑞 𝑖 𝑟 𝑖 . (University Physics Volume 2. OpenStax. Fig. 7.19.)

The Electric Dipole An electric dipole is a pair of equal and opposite charges a fixed distance 𝑑 apart. The potential due to the dipole is 𝑉=𝑘 𝑞 𝑟 + − 𝑞 𝑟 − . At large distances 𝑟 and angle 𝜃 with respect to the dipole, the potential is reduced to 𝑉=𝑘 𝑞𝑑cos𝜃 𝑟 2 . We can define the electric dipole moment, 𝐩 ≡𝑞 𝐝 , and rewrite the potential as 𝑉=𝑘 𝐩 ∙ 𝐫 𝑟 2 . (University Physics Volume 2. OpenStax. Fig. 7.21.)

Potential of Continuous Charge Distributions Like a system of point charges, the potential of a continuous charge distribution is given by the sum of contributions from each charge. In the limit of a continuous distribution, the charges become infinitesimal and the sum becomes an integral, giving the relation, 𝑉=𝑘 𝑑𝑞 𝑟 . For a line charge, the charge distribution is described by a linear charge density 𝜆, so 𝑑𝑞=𝜆𝑑𝑙. In two dimensions, the distribution is described by a surface charge density 𝜎, so 𝑑𝑞=𝜎𝑑𝐴. In three dimensions, the distribution is described by a volume charge density 𝜌, so 𝑑𝑞=𝜌𝑑𝑉.

Determining Field from Potential The electric field can be calculated from the potential of a system. The field over a displacement Δ𝑠 is 𝐸=− Δ𝑉 Δ𝑠 . The “grad” or “del” operator is defined by 𝛻 = 𝐢 𝜕 𝜕𝑥 + 𝐣 𝜕 𝜕𝑦 + 𝐤 𝜕 𝜕𝑧 . Using this operator, we can write the electric field as 𝐄 =− 𝛻 𝑉= 𝐢 𝜕 𝜕𝑥 + 𝐣 𝜕 𝜕𝑦 + 𝐤 𝜕 𝜕𝑧 . The grad operator can also be written in other coordinate systems. Cartesian coordinates 𝛻 = 𝐢 𝜕 𝜕𝑥 + 𝐣 𝜕 𝜕𝑦 + 𝐤 𝜕 𝜕𝑧 Cylindrical coordinates 𝛻 = 𝐫 𝜕 𝜕𝑟 + 𝛗 1 𝑟 𝜕 𝜕𝜑 + 𝐳 𝜕 𝜕𝑧 Spherical coordinates 𝛻 = 𝐫 𝜕 𝜕𝑟 + 𝜽 1 𝑟 𝜕 𝜕𝜃 + 𝛗 1 𝑟sin𝜃 𝜕 𝜕𝜑

Equipotential Surfaces and Conductors We can represent voltage graphically by drawing equipotential surfaces or equipotential lines, which are perpendicular to the field lines. In static situations, the electric field is perpendicular to the surface of a conductor and the conductor acts as an equipotential surface. The conductor can be brought to the same potential as the Earth by directly connecting the two, called grounding. (University Physics Volume 2. OpenStax. Fig. 7.30.)

Distribution of Charges on Conductors Consider a conducting sphere with excess charge 𝑞. The charge on the sphere arranges itself on the surface. Applying Gauss’s law, we can show that the electric field inside the sphere is zero and outside the sphere, the field is 𝐸= 1 4𝜋 𝜀 0 𝑞 𝑟 2 𝐫 . The potential inside the sphere is a constant, 𝑉= 1 4𝜋 𝜀 0 𝑞 𝑅 , and outside the sphere, 𝑉= 1 4𝜋 𝜀 0 𝑞 𝑟 . (University Physics Volume 2. OpenStax. Fig. 7.38.)

Distribution of Charges on Conductors 2 If two spherical conductors of different radii 𝑅 1 and 𝑅 2 are connected, the charge distributes itself over the two spheres such that the potentials inside the spheres are the same. As a result, the relationship between the charge distributions 𝜎 1 and 𝜎 2 on the surface of the spheres is 𝜎 1 𝑅 1 = 𝜎 2 𝑅 2 . This result extends to any shape, indicating that the charge density is highest where the radius of curvature is lowest. (University Physics Volume 2. OpenStax. Fig. 7.40.)

The Van de Graaff Generator A Van de Graaff generator is a device that uses static electricity to create a large static charge and voltage. A battery supplies charge to a nonconductive belt that carries the charge to a conductive sphere, which holds the charge on its surface. Voltages of up to 15 million volts have been achieved with Van de Graaff generator. (University Physics Volume 2. OpenStax. Fig. 7.41.)

Xerography Xerography takes advantage of static electricity to transfer toner onto paper. The material selenium is used because it is a photoconductor, meaning it is a conductor when exposed to light and an insulator otherwise. A drum coated with selenium is positively charged and an image is made on its surface. Negatively charged toner is attracted to the image, and positively charged paper pulls the toner off the drum. (University Physics Volume 2. OpenStax. Fig. 7.42.)

Laser Printers Laser printers use xerography to make prints. However, instead of using conventional imaging optics, a laser printer scans a laser across the surface of a drum to create the image. The ability to precisely control the laser makes it possible to make extremely high-quality prints. (University Physics Volume 2. OpenStax. Fig. 7.43.)

Ink Jet Printers and Electrostatic Painting The ink jet printer uses electrostatics to charge a fine stream of ink droplets. The ink droplets are then deflected by an electric field that can be precisely controlled to form images on paper. This method can also be applied to put an even coating of paint onto oddly shaped conducting surfaces because the charged droplets tend to approach perpendicular to the surface along the electric field lines. (University Physics Volume 2. OpenStax. Fig. 7.44.)

Smoke Precipitators and Electrostatic Air Cleaning Electrostatic precipitators are used to remove pollutants and irritants from the smoke created in industrial applications and in the air in our homes. Filters charge incoming neutral particles and then capture them in a grid with the opposite charge. (University Physics Volume 2. OpenStax. Fig. 7.45.)

How to Study this Module Read the syllabus or schedule of assignments regularly. Understand key terms; look up and define all unfamiliar words and terms. Take notes on your readings, assigned media, and lectures. As appropriate, work all questions and/or problems assigned and as many additional questions and/or problems as possible. Discuss topics with classmates. Frequently review your notes. Make flow charts and outlines from your notes to help you study for assessments. Complete all course assessments.

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