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"property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, B31: The Electric Potential due to a Continuous Charge Distribution, [ "article:topic", "authorname:jschnick", "license:ccbysa", "showtoc:no", "licenseversion:25", "source@http://www.cbphysics.org" ], https://phys.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fphys.libretexts.org%2FBookshelves%2FUniversity_Physics%2FBook%253A_Calculus-Based_Physics_(Schnick)%2FVolume_B%253A_Electricity_Magnetism_and_Optics%2FB31%253A_The_Electric_Potential_due_to_a_Continuous_Charge_Distribution, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\). $$W=\frac{\epsilon_0}{2}\int_\text{all space} E^2d\tau$$ In the next chapter, we exploit the fact that if you know the electric potential throughout a region in space, you can use that knowledge to determine the electric field in that region of space. For 3D applications use charge per unit volume: = Q/V . Electric charges, Conservation of charge, Coulomb's law-force between twopoint charges, forces between multiple charges; superposition principle and continuous charge distribution. $$W=\frac{1}{2}\int_\text{all space}\frac{\rho_1\rho_2}{4\pi\epsilon_0}\frac{1}{r_{12}}d\tau_1d\tau_2$$ The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. When two negatively and positively charged elements come into close proximity, they attract each other. About About this video Transcript. Substituting this into our expression for \(dV\) yields: \[d\phi=\frac{k\lambda(x')dx'}{\sqrt{(x-x')^2+y^2}}], \[\int d\phi=\int_a^b \frac{k\lambda(x')dx'}{\sqrt{(x-x')^2+y^2}}], \[\phi=k\int_a^b \frac{\lambda(x')dx'}{\sqrt{(x-x')^2+y^2}}]. Mathematically, there is a linear charge density - = dq/ dl The unit of the linear load density is C / m. Electric potential energy (part 2-- involves calculus) . The line has to have finite cross-section, otherwise the integral blows up. Electric field may become zero inside it but the electric . We want to calculate the electric potential due to a line of charge. How do we construct \rho from the original distributions and V from the original potentials, and also if \rho accounts for the distribution of charge in all space and V for the potential in all space as well, then aren't we counting the energy of a charge distribution due to its own potential too? Also keep in mind the fact that the various contributions to the electric potential at an empty point in space simply add (like numbers/scalars rather than like vectors). I understand most of the concepts (conservation of energy, electrical potential energy, superposition principle, coulomb's law, etc.) Two charges that are both positive or. Please solve the following example problem and then check your work against my solution which follows the problem statement. . If we can establish the electric potential-energy-per-charge for each point in space in the vicinity of some source charge, it is easy to determine what the potential energy of a victim charge would be at any such point in space. 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