Lecture 5: Electric fields and electric flux [Previous lecture: Electric field of a point charge] 𝑞 𝐸 = 𝑘 ! 𝒓& 𝑟 [Previous lecture: Electric field of multiple point charges] 𝐸 = 𝐸" + 𝐸! + 𝐸# 𝑘𝑞" 𝑘𝑞! 𝑘𝑞# = ! 𝒓& 𝟏 + ! 𝒓& 𝟐 + ! 𝒓& 𝟑 𝑟" 𝑟! 𝑟# [Previous lecture: You can divide any object into a bunch of “point charges”] Add up the electric field from each little piece 𝑘𝑞! 𝐸 = # " 𝒓' 𝒊 𝑟! ! In the limit where each piece is infinitesimally small: 𝑑𝑞 𝐸 = ∫ 𝑘 " 𝒓' 𝑟 [Previous lecture: a line of charge] What is 𝐸 at point A? 𝐿/2 𝐿/2 s A The line has a charge per unit length −𝜆 (units of C/m) (Note: I changed the distance that was called “d” in lecture to be called “s”, to avoid confusion in notation) Before you do any integrals: Decide which way 𝐸 needs to point. [Previous lecture: Electric field from a ring of charge] What is 𝐸 at point P? 𝜆 = charge per unit length (units of C/m) Total charge is 𝑄 = 𝜆×2𝜋𝑅 (and sits in the x-y plane) Before you do any integrals: Decide which way 𝐸 needs to point. Example: a really big disk of charge What is 𝐸 at point P? 𝜎 = charge per unit area (units of C/m2) [this solution is provided for your cultural edification] What is 𝐸 at point P? 𝜎 = charge per unit area (units of C/m2) An introduction to Gauss’s Law, starting with point charges: + charges are “sources” of 𝐸 + charges are “sinks” of 𝐸 ? You can tell whether a region of space contains + or – charge by checking whether electric field lines come in or go out Let’s play a game: What is the total charge inside the box? A) + B) C) Zero ? What is the total charge inside the box? A) + B) C) Zero ? What is the total charge inside the box? A) + B) C) Zero -2 ? +1 What is the total charge inside the box? ? A) + B) C) Zero What is the total charge inside the box? A) + B) C) Zero ? Gauss’s law 𝑄./01 = Φ3 𝜖2 You can tell how much charge is inside any region by looking at the electric flux into / out of that region What is electric flux? Φ3 = % 𝐸4 ⋅ 𝐴⃗4 i What is the flux through this piece of surface? Integral definition of electric flux Φ3 = ∫ 𝐸 ⋅ 𝑑𝐴 Electric field has a constant magnitude 𝐸 and points in the vertical direction Shape has a square cross section Which sides of this shape have nonzero electric flux? What is the value of the total electric flux Φ0 across the surface? The electric field next to a disk of charge, revisited How can we quickly find out the magnitude of the electric field 𝐸 above the disk? 𝑅→∞ Use Gauss’s law to figure out the value of the electric field Charge 𝜎 per unit area