Describe the field lines using the symmetry of the problem

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Reference no: EM13802110

Given a charge distribution of cylindrical, planar or spherical symmetry Gauss's Law may provide a method to calculate the field.

1) Describe the field lines using the symmetry of the problem

2) Can you draw a Gaussian surface which allows you to easily find the total flux, eg.:

E and normal to A are parallel

EAcos(θ) is constant over some part of the surface and maybe zero over other parts

Usually the flux integral Φ =  ? EdA ends up being Φ = E ? dA =EA

3) Calculate the enclosed charge as

Q = ∫ρdV for volume charges if p is constant Q = ρV

Q =  ∫σdA for area chargesif a is constant Q = σA

Q = ∫λdL for linear charges if λ is constant Q = λL

4) Plug into ∫EdA = 4ΠkQenclosed and solve for E

Spherical Symmetry - p(r)

What is E for for a boll of charge of radius R ρ(r)= ar2 (a is constont)

1) Describe the field lines using the symmetry of the problem

Field lines ore radial justdepend on r.

2) Calculate the flux Φ = ? EdA

Draw a Gaussion surface as a sphere at radius r where you want to find field. E is constant at this radius so ? EdA = E ? A =

3) Calculate the Qenclosed

r < Qenclosed = ∫ ρ(r)dV =

r > R Qenclosed = ∫ ρ(r)dV =

4) Plug into ? EdA = 4ΠkQenclosed and solve for E

r < R

r > R

Cylindrical Symmetry -ρ(r)

What is E for a Infinitely long cylinder of charge density ρ(r)= arn, radus R?

393_Cylindrical Symmetry.png

1) Describe the field lines using the symmetry of the problem

Field lines are radial and just depand on r.

2) Calculate the flux Φ = ? EdA

1033_Cylindrical Symmetry1.png

Draw a Gaussion surface as a cylinder coaxial with charge cylinder at the radius, where you want to find the field. E is constant at this radius so

? EdA = E ? dA = EA = E 2ΠrL

3) Calculate the Qenclosed

2377_Cylindrical Symmetry2.png

r < Qenclosed = ∫ ρdV =

r > R Qenclosed = ∫ ρdV =

4) Plug into ? EdA = 4ΠkQenclosed and solve for E

r < R  E(r) =

r > R  E(r) =

Charged Wire - λ

What is E for a infineitely long Wire of constant linear charge densily λ?

2122_Cylindrical Symmetry3.png

1) Describe the field lines using the symmetry the problem Field lines are radial and just depand on r.

2) Calculate  the flux Φ = ? EdA

Drawn Gaussian surfaceas a cylinder coaxial With the charge cylinder the radius r where you motto find the field.

E is consant at this radius so

? EdA = E ? dA = EA = E 2ΠrL

3) Calculate the Qenclosed

Qenclosed = ∫λdL = λ∫dL =

4) Plug into ∫ EdA = 4ΠkQenclosed and solve for E

E(r) =

Planar Symmetry ρ(x)

What is E for a slab of charg ρ(x)= bx2?

1)Describe the field lines using the symmetry the problem Field lines are radial and just depand on r.

2) Calculate  the flux Φ = ? EdA

Draw a Gaussian surface as a cylinder cutting through the slab of length 2x. The flux is the filed going through the ends of

this surface at ± x

? EdA E ? dA= 2A

a) Calculate the Qenclosed

|x| = a/2 Qenclosed = ∫ ρdV = -xx bx2Adx = (Nodete dv = Adx)

|x| = a/2 Qenclosed = ∫ ρdV =

4) Plug into ? EdA = 4ΠkQenclosed and solve for E

|x| = a/2 E(x) =

|x| = a/2 E(x) =

E Due to an Infinite sheet of charge by Gauss's Law charge density σ C/m2

What is E for a sheet of charge density σ C/m2

1) Describe the field lines using the symmetry of the problem field lines point pendiculor to the slab of the lenght 2x.

2) Calculate the flux Φ = ? EdA

Draw a Gaussion surface as a cylirlder cutting through the slab of length 2x.

The flux is the filed going through the ends of this surface at ± x

? EdA = E?dA = 2A

3) Calculate the Qenclosed

Qenclosed = ∫σdA = σA

4) Plug into ?EdA = 4ΠkQenclosed and solve for E

E(x) =

Reference no: EM13802110

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