Coulomb’s Law• Coulomb’s Law• Electric Field• Electric Field from Multiple Charges• Integration of Volume charge• Electric Field near Infinite Wire• Electric Field near Infinite Sheet• Electric Field between two Infinite Sheets• Field Lines• Streamlines

Coulomb’s Law• Coulomb’s Law with k = 9 x 109 Nm2/C2

εo= 8.85 x 10-12 C2 / Nm2

• Unit vector from r1 to r2

• Combining

• (Action reaction F1 = -F2)

Example of Coulomb’s Law• Force of charge 1 on charge 2

– Charge 1 - 3 x 10-4 C at M(1,2,3)– Charge 2 - -1 x 10-4 C at N(2,0,5)

• Coulomb’s Law

• R magnitude

• Unit vector

• Result

Electric Field• Electric Field

– Coulomb’s Law without 2nd charge– Separates Problem into “Background” and “Test Charge”– Units newtons/coulomb (volts/meter)

– For source charge at r’ observed at r’

• For source charges at r1and r2, observed at r

Electric Field from Multiple Charges• 2 source charges at 1 and 2, observed at r

• Multiple source charges at m, observed at r

• Infinite # source charges, observed at r

• We’re going to spend some time on the last one!

Example – Electric Field from 4 charges• Sources charges at P1(1,1,0), P2(-1,1,0), P3(-1,-1,0), P4(1,-1,0). Each 3 nC.

• Observation point r at P(1,1,1)

– P1(1,1,0)

– P2(-1,1,0)

– P3(-1,-1,0)

– P4(1,-1,0)

• Total field is:

Continuous Charge - Integration of Charge

• Differential charge element

• Integrate for total charge

• Example – charge density – Find total charge over region 0 <ρ<1 cm, 2cm <z< 4cm

• Comments– Dependence on ρ and z in negative exponential causes rapid fall-off in ρV

– Concentrated near z= 0 plane where exponential is small– Concentrated near ρ = 0 z-axis where exponential is small

• Integral

Integration of Charge (cont)• Integration on φ

• Integration on z

Continuous Charge - Other examples• Setup Cartesian

– Integrate volume – Subtract volume 1– Q will be zero from integration of odd function.

• Setup cylindrical– )

– Differential volume

• Universe

Continuous Charge -Middle Example• Integral is

Continuous Charge -Field near infinite line charge

• Will do in cylindrical coordinates– Observation on y axis, z = 0 plane – Source distributed along z axis – Linear charge density constant

• Source to observation vector

• Differential Field Contribution

Field near infinite line of charge (cont)• ρ and z components

(odd - integrates to zero)

• Integration for a long wire is thus

Field near infinite sheet of charge• Given an infinite line charge and surface density ρs

• x and y components

(odd - integrates to zero)

(symmetry)

• x component

Field near infinite sheet (cont)• Integration for a sheet is thus

• Field points away toward

• Field is independent of distance r<<width

Electric Field between 2 Infinite Sheets

(-Q) charge sign and unit vector reversed)

Field Lines• Field lines– Point in direction of electric field

– Direction + test charge moves

– Originates on +Q terminates on -Q

– Cross-sectional density proportional to E magnitude

Streamlines• Equation of line which follows field line at x, y, z

– slope of this line y=f(x) – should equal field ratio

– Set

– Solve for equation y=f(x) as function of x

Streamlines• Vector field are Ax, Ay, and Az function of x,y,z• From geometry

• Example

• Plugging in

• Result

• Plug in x and y at particular pointto evaluate C

Streamline Example

• Find streamlines of following in rectangular coordinates

• Transforming to rectangular

• Plugging in streamline equation

• Solution

• At P(-2,7,10) y = -3.5 x

Example problem 11. 3 point charges are in xy plane; with 5 nC at y= 5 cm, -10 nC at y =-5 cm, and 15 nC

at x=-5cm. Find position of 20 nC that exactly cancels field at origin.

- Add first 3 fields to get resultant as function of ax , ay (like example 2.2)

- 4th charge must exactly cancel field with same combination of ax , ay

- Write in general field form as magnitude times unit vector

- Equate magnitudes

Example problem 27. A 2uC charge is located at A(4,3,5) in free space. Find Eρ, Eφ, and Ez at

P(8,12,2)- Get field in rectangular coordinates as function of ax, ay, az

- translate rectangular variables to cylindrical variables- translate rectangular unit vectors to cylindrical variables.

Example problem 313. A uniform charge density extends throughout a spherical shell from r=3

cm to r=5 cm. Find the total charge and the radius containing half the charge.

Example problem 4• Find the electric field on the z-axis produced byan annular ring z= 0, a <ρ

<b, 0 < φ < 2π