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Spherical Shell Electric Field No Gauss Law, David Griffiths Electrodynamics Problem 2.7 Solution

We calculate the electric field both inside and outside a uniformly charged spherical shell without using Gauss's law. Instead, we employ an explicit two-dimensional integration in spherical coordinates, using the law of cosines and right triangle trigonometry to solve for the field at a point along the z-axis at a distance z from the center of the spherical shell. This follows David Griffiths' "Introduction to Electrodynamics" (Problem 2.7). We carefully derive expressions for different components of the electric field, using concepts like the surface charge density, the differential elements of area, and the geometric relationships between points on the shell. The solution demonstrates that the electric field inside the shell is zero, in agreement with the Faraday cage principle, and shows that the electric field outside behaves like a point charge, confirming the results that one would obtain using Gauss's law. Throughout, we provide detailed step-by-step guidance to help you understand every aspect of the integration process. This video is ideal for advanced electromagnetism to deepen your understanding of electrostatics and integration in spherical coordinates. We also provide the notes from this video as slides: Link to slides: https://drive.google.com/file/d/1RLq5... ------------------------------------ Chapters / Timestamps 00:00 Intro to Griffiths Problem 2.7 00:49 Explicit Integration no Gauss Law 03:03 Spherical shell geometry 3D diagram 07:52 Symmetry argument horizontal components = 0 10:44 Law of Cosines, vertical radial Electric Field Vector 13:58 Differential area, differential charge, Coulomb's Law 16:40 Differential Electric Field 17:11 Setup spherical 2D integral spherical coordinates 18:38 u-substitution to simplify integral 20:11 Additional integral substitution and algebra 26:48 Consider when z less than R and z greater than R 30:04 Integral result in as Electric Field 30:49 Inside the sphere z less than R answer 32:36 Outside the sphere z greater than R answer 34:37 Take action! ------------------------------------ Please support us on Patreon: https://www.patreon.com/highpeakeduca... Please follow us on Facebook:   / high-peak-education-110762917338673   Please follow us on Twitter:   / highpeakeducate   Please follow us on Instagram:   / highpeakeducation   Please follow us on Reddit:   / highpeakeducation   Please follow us on TikTok:   / highpeakeducation   Interact with us on Twitch:   / highpeakeducation   Interact with us on Discord:   / discord   #HighPeakEducation #Physics #Electrodynamics