Lecture 4-1 At a point P on axis: At a point P off axis: At point P on axis of ring: ds

Lecture 4-2 Gauss’s Law: Qualitative Statement  Form any closed surface around charges  Count the number of electric field lines coming through the surface, those outward as positive and inward as negative.  Then the net number of lines is proportional to the net charges enclosed in the surface.

Lecture 4-3 Electric flux # of field lines N = density of field lines x “area” where “area” = A 2 x cos  General definition of electric flux: (must specify sense, i.e., which way) To state Gauss’s Law in a quantitative form, we first need to define Electric Flux. Sum over surface 

Lecture 4-4 Electric Flux through Closed Surface The integral is over a CLOSED surface. Since is a scalar product, the electric flux is a SCALAR quantity The integration element is a vector normal to the surface and points OUTWARD from the surface. Out is +, In is -  E proportional to # field lines coming through outward

Lecture 4-5 Why are we interested in electric flux? is closely related to the charge(s) which cause it. Consider Point charge Q If we now turn to our previous discussion and use the analogy to the number of field lines, then the flux should be the same even when the surface is deformed. Thus should only depend on Q enclosed.

Lecture 4-6 Physics 241 –warm-up quiz One point charge Q is placed inside a closed surface with complicated shape. Which of the following statements is correct on the flux through the surface? a)It’s not possible to calculate the flux since the details of the surface shape are not given. b)The flux is Q/ε 0 c)The location of the charge Q inside the surface is need in order to calculate the flux. d)The flux is not zero and must be negative. +Q

Lecture 4-7 Gauss’s Law: Quantitative Statement The net electric flux through any closed surface equals the net charge enclosed by that surface divided by  0. How do we use this equation?? The above equation is TRUE always but it doesn’t look easy to use. BUT - It is very useful in finding E when the physical situation exhibits a lot of SYMMETRY.

Lecture 4-8 Uniformly charged thin shell: outside By symmetry, the electric field must only depend on r and is along a radial line everywhere. Apply Gauss’s law to the blue surface, we get Electric Field Outside

Lecture 4-9 Uniformly charged thin shell: Inside By symmetry, the electric field must only depend on r and is along a radial line everywhere. Apply Gauss’s law to the blue surface, we get E = 0. E = 0 inside Discontinuity in E Equal and opposite contributions from charges on diagonally opposite surface elements.

Lecture 4-10 Electric Field of a Uniformly Charged Sphere Apply Gauss’s Law directly or use superposition of the shell results

Lecture 4-11 Physics 241 –Quiz 2a Two identical point charges are each placed inside a large cube. One is at the center while the other is close to the surface. Which statement about the net electric flux through the surface of the cube is true? a)The flux magnitude is larger when the charge is at the center. b)The flux magnitude is the same (and not zero). c)The flux magnitude is larger when the charge is near the surface. d)Not enough information to tell. e)The flux is zero in both cases. +Q

Lecture 4-12 Physics 241 –Quiz 2b Two identical point charges are each placed inside a large sphere. One is at the center while the other is close to the surface. Which statement about the net electric flux through the surface of the sphere is true? a)The flux magnitude is larger when the charge is at the center. b)The flux magnetude is larger when the charge is near the surface. c)The flux magnitude is the same (and not zero). d)Not enough information to tell. e)The flux is zero in both cases. +Q

Lecture 4-13 Physics 241 –Quiz 2c Two identical point charges are placed, at the center of a large sphere in one case, and outside an identical sphere in the other case. Which statement about the net electric flux through the surface of the sphere is true? a)The flux is larger when the charge is inside. b)The flux is larger when the charge is outside. c)The flux is the same (and not zero). d)Not enough information to tell. e)The flux is zero in both cases. Q>0