Lectures 3 4 5 ELECTRIC FIELD ELECTRIC FLUX

Lectures 3, 4 & 5 ELECTRIC FIELD ELECTRIC FLUX a R a 2 R 11/26/2020 1

Last Week…. . Discrete Charges • Magnitude of the Electric Field due to a Point Charge: P • Electric Field due to a Dipole at point P: r+ z r- + q+ d 11/26/2020 - q- Dipole center 2

Today… • More on Electric Field: – Continuous Charge Distributions Text Reference: Chapter 22. 1 Many useful examples 22 -1 -8 etc. 11/26/2020 3

Electric Fields from Continuous Charge Distributions • Principles (Coulomb’s Law + Law of Superposition) remain the same. Only change: + + - - + + + - 11/26/2020 4

Charge Distributions Problems Step 1: Understand the geometry Step 2: Choose dq Step 3: Evaluate d. E 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 11/26/2020 5

Charge Densities • How do we represent the charge “Q” on an extended object? small pieces total charge of charge Q dq Line of charge: = charge per unit length [C/m] Surface of charge: = charge per unit area [C/m 2] Volume of Charge: = charge per unit volume [C/m 3] 11/26/2020 dq = dx dq = d. A = dxdy (Cartesian coordinates) dq = d. V = dx dy dz (Cartesian coordinates) 6

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 11/26/2020 7

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: 11/26/2020 8

Field due to charge on a ring Steps 5 -7 • Integrate over the entire ring with uniform; z and R fixed 11/26/2020 9

Limiting cases Step 8 • z>>R: point-like charge ~q/z 2, Coulomb formula • z<<R: E=0 at the center of the ring, all field elements cancel. 11/26/2020 10

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 11/26/2020 11

Field due to the disk • Integrate over the entire disk • Use substitution of variables 11/26/2020 12

Field due to the disk • Check Limiting Cases z>>R: point-like charge ~Q/z 2 11/26/2020 z>0: upper sign z<0: lower sign 13

Important limiting cases for charged disk z<<R: Ez z 11/26/2020 There is a discontinuity at z= 0. 14

Infinite sheet of positive charge y + + + + x Uniform electric fields generated on both sides of sheet. discontinuity in electric field at x = 0 11/26/2020 15

900 arc of charge y x E In this coordinate system you have to deal with Ex and Ey. Y E X In this coordinate system you have to deal with the horizontal component of E only. uniform charge distribution total charge = Q 11/26/2020 16

900 arc of charge (continued) y rd x d. E 11/26/2020 17