W02D2 Gauss’s Law Class 02

Announcements Math Review Tuesday Week Three Tues from 9-11 pm in 26-152Vector Calculus PS 2 due Week Three Tuesday Tues at 9 pm in boxes outside 32-082 or 26-152 W02D3 Reading Assignment Course Notes: Chapter Course Notes: Sections 3.6, 3.7, 3.10 Make sure your clicker is registered

Outline Electric Flux Gauss’s Law Calculating Electric Fields using Gauss’s Law Class 04

The first Maxwell Equation! Gauss’s Law The first Maxwell Equation! A very useful computational technique to find the electric field when the source has ‘enough symmetry’. Carl Friedrich is so money! Class 04

Gauss’s Law – The Idea The total “flux” of field lines penetrating any of these closed surfaces is the same and depends only on the amount of charge inside Class 04

Gauss’s Law – The Equation Electric flux (the surface integral of E over closed surface S) is proportional to charge enclosed by the volume enclosed by S Class 04

Electric Flux Case I: E is a uniform vector field perpendicular to planar surface S of area A Our Goal: Always reduce problem to finding a surface where we can take E out of integral and get simply E*Area Class 04

Electric Flux Case II: E is uniform vector field directed at angle to planar surface S of area A Class 04

Concept Question: Flux The electric flux through the planar surface below (positive unit normal to left) is: +q -q positive. negative. zero. Not well defined. Class 04

Concept Question Answer: Flux Answer: 2. The flux is negative. +q -q The field lines go from left to right, opposite the assigned normal direction. Hence the flux is negative.

Open and Closed Surfaces A rectangle is an open surface — it does NOT contain a volume A sphere is a closed surface — it DOES contain a volume Class 04

Area Element: Closed Surface Case III: not uniform, surface curved For closed surface, is normal to surface and points outward ( from inside to outside) if points out if points in Class 04

Group Problem Electric Flux: Sphere Consider a point-like charged object with charge Q located at the origin. What is the electric flux on a spherical surface (Gaussian surface) of radius r ? Class 04

Arbitrary Gaussian Surfaces explain True for all surfaces such as S1, S2 or S3 Why? As area gets bigger E gets smaller Class 04

Gauss’s Law Note: Integral must be over closed surface Class 04

Concept Question: Flux thru Sphere The total flux through the below spherical surface is +q positive (net outward flux). negative (net inward flux). zero. Not well defined. Class 04

Concept Question Answer: Flux thru Sphere Answer: 3. The total flux is zero +q We know this from Gauss’s Law: No enclosed charge  no net flux. Flux in on left cancelled by flux out on right

Concept Question: Gauss’s Law The grass seeds figure shows the electric field of three charges with charges +1, +1, and -1, The Gaussian surface in the figure is a sphere containing two of the charges. The electric flux through the spherical Gaussian surface is Positive Negative Zero Impossible to determine without more information.

Concept Question Answer: Gauss’s Law Answer 3: Zero. The field lines around the two charged objects inside the Gaussian surface are the field lines associated with a dipole, so the charge enclosed in the Gaussian surface is zero. Therefore the electric flux on the surface is zero. Note that the electric field E is clearly NOT zero on the surface of the sphere. It is only the INTEGRAL over the spherical surface of E dotted into dA that is zero.

Choosing Gaussian Surface True for all closed surfaces Useful (to calculate electric field ) for some closed surfaces for some problems with lots of symmetry. Desired E: Perpendicular to surface and uniform on surface. Flux is EA or -EA. Other E: Parallel to surface. Flux is zero Class 04

Symmetry & Gaussian Surfaces Desired E: perpendicular to surface and constant on surface. So Gauss’s Law useful to calculate electric field from highly symmetric sources Source Symmetry Gaussian Surface Spherical Concentric Sphere Cylindrical Coaxial Cylinder Planar Gaussian “Pillbox” Class 04

Virtual Experiment Gauss’s Law Applet Bring up the Gauss’s Law Applet and answer the experiment survey questions http://web.mit.edu/viz/EM/visualizations/electrostatics/flux/closedSurfaces/closed.htm Class 04

Applying Gauss’s Law Based on the source, identify regions in which to calculate electric field. Choose Gaussian surface S: Symmetry Calculate Calculate qenc, charge enclosed by surface S Apply Gauss’s Law to calculate electric field: Class 04

Examples: Spherical Symmetry Cylindrical Symmetry Planar Symmetry Class 04

Group Problem Gauss: Spherical Symmetry +Q uniformly distributed throughout non-conducting solid sphere of radius a. Find everywhere. Let students try problem, summarize result at end. Decide how much time you let students work on this while observing what they are doing. Class 04

Concept Question: Spherical Shell We just saw that in a solid sphere of charge the electric field grows linearly with distance. Inside the charged spherical shell at right (r<a) what does the electric field do? a Q Zero Uniform but Non-Zero Still grows linearly Some other functional form (use Gauss’ Law) Can’t determine with Gauss Law Class 04

Concept Question Answer: Flux thru Sphere Answer: 1. Zero Q a Spherical symmetry  Use Gauss’ Law with spherical surface. Any surface inside shell contains no charge  No flux E = 0!

Demonstration Field Inside Spherical Shell (Grass Seeds): Class 04

Worked Example: Planar Symmetry Consider an infinite thin slab with uniform positive charge density . Find a vector expression for the direction and magnitude of the electric field outside the slab. Make sure you show your Gaussian closed surface. This could either be a group problem or a worked example depending on the time left. Class 04

Gauss: Planar Symmetry Symmetry is Planar Use Gaussian Pillbox Note: A is arbitrary (its size and shape) and should divide out Do this one as a worked example Gaussian Pillbox Class 04

Gauss: Planar Symmetry Total charge enclosed: NOTE: No flux through side of cylinder, only endcaps + Class 04

Concept Question: Superposition Three infinite sheets of charge are shown above. The sheet in the middle is negatively charged with charge per unit area , and the other two sheets are positively charged with charge per unit area . Which set of arrows (and zeros) best describes the electric field?

Concept Question Answer: Superposition Answer 2 . The fields of each of the plates are shown in the different regions along with their sum.

Group Problem: Cylindrical Symmetry Week 03, Day 2 Group Problem: Cylindrical Symmetry An infinitely long rod has a uniform positive linear charge density .Find the direction and magnitude of the electric field outside the rod. Clearly show your choice of Gaussian closed surface. 34 Class 07 34

Electric Field for Charged Infinite Plane Dipole: E falls off like 1/r3 Spherical charge: E falls off like 1/r2 Line of charge: E falls off like 1/r (infinite) Plane of charge: E uniform on (infinite) either side of plane