ELECTRIC FIELD ELECTRIC FLUX Lectures 3, 4 & 5 a a R 2R In lectures 3,4 and 5, we will cover Chapter 22 which invovles calculating the electric field for different distribution of charges using 1st Coulomb’s Law and then we will introduce Gauss’ Law which is used to calculate E for symmetric shapes. a R a 2R 11/15/2018

Last Week…..Discrete Charges Magnitude of the Electric Field due to a Point Charge: + - P Dipole center z r+ r- q+ q- d Electric Field due to a Dipole at point P: 11/15/2018

More on Electric Field: Today… More on Electric Field: Continuous Charge Distributions Text Reference: Chapter 22.1 Many useful examples 22-1-8 etc. 11/15/2018

Electric Fields from Continuous Charge Distributions Principles (Coulomb’s Law + Law of Superposition) remain the same. Only change: + - + + + + + + + + + + 11/15/2018

Charge Distributions Problems Step 1:Understand the geometry Step 2:Choose dq Step 3:Evaluate dE contribution from the infinitesimal charge element Step 4:Exploit symmetry as appropriate Step 5:Set up the integral Step 6:Solve the integral Step 7:The Result! Step 8:Check Limiting Cases 11/15/2018

small pieces of charge dq dq =  dA =  dxdy (Cartesian coordinates) Charge Densities How do we represent the charge “Q” on an extended object? small pieces of charge dq total charge Q Line of charge: = charge per unit length [C/m] dq =  dx Surface of charge: dq =  dA =  dxdy (Cartesian coordinates)   = charge per unit area [C/m2] Volume of Charge:  = charge per unit volume [C/m3] dq = dV =  dx dy dz (Cartesian coordinates) 11/15/2018

Charge on a ring Steps 1-3 Problem: calculate the electric field along z-axis due to a (circular) ring (of radius R) of uniform positive charge (with density ). Charge of element The electric field due to this element 11/15/2018

Field due to charge on a ring Step 4:Exploit symmetry as appropriate Symmetry: direction of the field at point P must be along the positive z direction. The z-component of the field: 11/15/2018

Field due to charge on a ring Steps 5-7 Integrate over the entire ring with  uniform; z and R fixed 11/15/2018

Limiting cases Step 8 z>>R: point-like charge  ~q/z2, Coulomb formula z<<R: E=0 at the center of the ring, all field elements cancel. 11/15/2018

Quiz 1 (lecture 3) A nonconducting semi-circular rod has a uniform charge of magnitude Q along its top half and another along its bottom half. What is the direction of the net electric field at point P? y +Q Towards -x Towards +x Towards +y Towards -y Towards negative y. P x -Q 11/15/2018

Charged Disk Problem: calculate the electric field along z-axis due to a (circular) disk (of radius R) of uniform positive charge (with density ). Pick any ring element (of infinitesimal width dr) of the disk. The charge of the element is: The electric field due to the ring element 11/15/2018

Field due to the disk Integrate over the entire disk Use substitution of variables 11/15/2018

Field due to the disk Check Limiting Cases z>>R: point-like charge  ~Q/z2 z>0: upper sign z<0: lower sign 11/15/2018

Important limiting cases for charged disk z<<R: Ez There is a discontinuity at z= 0. z 11/15/2018

Infinite sheet of positive charge y +++++++++++++++ x Uniform electric fields generated on both sides of sheet. discontinuity in electric field at x = 0 11/15/2018

900 arc of charge y In this coordinate system x E In this coordinate system you have to deal with Ex and Ey. Y In this coordinate system you have to deal with the horizontal component of E only. E X uniform charge distribution total charge = Q 11/15/2018

900 arc of charge (continued) y rd  x dE 11/15/2018

Quiz 2 (lecture 3) (A) y (B) (C) Q x -a L (D) (E) A thin rod with length L has a total charge of Q distributed uniformly along it’s length is located on the x-axis with one end at the origin as shown. Which expression represents the electric field due to this rod at a point on the x-axis a distance a to the left of the origin? (A) (B) (C) (D) (E) x y -a L Q Answer C

Quiz 3 (lecture 3) Ea<Eb