Lecture 5
Electric flux
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The product of the magnitude of the electric field (𝐸) and surface area (𝐴) perpendicular to the
field is called the electric flux (𝜙𝐸 ).
o 𝜙𝐸 = 𝐸𝐴 → 𝑁 ∙ 𝑚 2 /𝐶
The electric flux is proportional to the number of electric field lines penetrating some surface.
The field lines may make some angle with the perpendicular to the surface:
o 𝜙𝐸 = 𝐸𝐴⊥ = 𝐸𝐴𝑐𝑜𝑠𝜃
The electric flux through a surface of fixed area (𝐴) has maximum value (𝐸𝐴) when the surface is
perpendicular to the field.
o When the normal to the surface is parallel to the field (𝜃 = 0°)
The electric flux is zero when the surface is parallel to the field
o When the normal to the surface is perpendicular to the field (𝜃 = 90°)
(𝜙𝐸 = 𝐸𝐴𝑐𝑜𝑠𝜃) is only valid for a small element of the area.
o 𝜙𝐸 = 𝐸𝑖 𝐴𝑖 𝑐𝑜𝑠𝜃𝑖
⃗⃗⃗𝑖 ∙ Δ𝐴
⃗⃗⃗⃗𝑖
o 𝜙𝐸 = lim ∑𝐸
Δ𝐴𝑖→0
⃗⃗⃗⃗𝑖 (General definition of electric flux)
𝜙𝐸 = ∫𝑠 ⃗⃗⃗
𝐸𝑖 ∙ d𝐴
▪ The surface integral means that the integral must be evaluated over the surface
in question.
The value of the electric flux depends on both the field pattern and on the surface.
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Gauss’s law
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The magnitude of the electric field everywhere on the surface of the sphere:
o The field lines are directed radially outward and hence are perpendicular to the surface
at every point on the surface.
o At each surface point, 𝐸⃗ is parallel to the vector Δ𝐴𝑖 :
⃗⃗⃗𝑖 = 𝐸Δ𝐴𝑖
▪ 𝐸⃗ ∙ Δ𝐴
o The net flux through a gaussian surface (a closed surface):
⃗⃗⃗𝑖 = ∮ 𝐸⃗ ∙ d𝐴 = 𝐸 ∮ dA
▪ 𝜙𝐸 = lim ∑𝐸⃗ ∙ Δ𝐴
Δ𝐴𝑖→𝑜
▪
▪
𝜙𝐸 = 𝐸∫ 𝑑𝐴
𝑞
𝜙𝐸 = 𝑘𝑒 𝑟 2 (4𝜋𝑟 2 )
▪
▪
𝜙𝐸 = 𝑘𝑒 𝑞4𝜋
𝜙𝐸 = 4𝜋𝑘𝑒 𝑞
▪
𝜙𝐸 = 4𝜋𝑞 ∙ 4𝜋𝜖
▪
𝜙𝐸 =
𝑞
𝜖0
1
0
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The net number of electric field lines through a closed surface is proportional to the charge
enclosed and to the flux.
𝑞
The net flux through any closed surface surrounding a point charge 𝑞 is given by 𝜖 and is
0
independent of the shape of that surface.
The net electric flux through a closed surface that surrounds no charge is zero.
The electric field due to many charges is the vector sum of the electric fields produced by the
individual charges:
⃗ = ∮(𝐸⃗1 + 𝐸⃗2 + ⋯ ) ∙ 𝑑𝐴
o ∮ 𝐸⃗ ∙ dA
▪ 𝐸⃗ is the total electric field at any point on the surface produced by the vector
addition of the electric fields at that point due to the individual charges.
Gauss’s law states that the net flux through any enclosed surface is:
𝑞
o 𝜙𝐸 = ∮ 𝐸⃗ ∙ 𝑑𝐴 = 𝑖𝑛
𝜖0
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▪ 𝐸⃗ represents several electric field at any point on the surface.
▪ 𝑞𝑖𝑛 Represents the net charge inside the surface.
The Gauss’s law is a very powerful method, which can be used to find the electric field especially
in situations where there is a high degree of symmetry.
o It can be shown that the electric field calculated by Gauss’s law is identical to that
calculated by Coulomb's law.
𝑞
▪ 𝐸𝐴 =
𝜖0
▪
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𝑞
𝐸4𝜋𝑟 2 = 𝜖
0
𝑞
▪
𝐸 = 4𝜋𝑟2 𝜖
▪
𝑞
0
𝐸 = 𝑘𝑒 2
𝑟
To solve a problem using Gauss’s law, choose the closed surface (gaussian surface)
Examples
1. Consider a uniform electric field oriented in the 𝑥 direction in empty space. Find the net electric
flux through the surface of a cube of edge length ℓ.
2. Starting with Gauss's law, calculate the electric field due to an insulated point charge 𝑞.
3. An insulating solid sphere of radius 𝑎 has a uniform charge density 𝜌 and carries a total positive
charge 𝑄.
a. Calculate the magnitude of the electric field at a point outside the sphere.
b. Find the magnitude of the electric field at a point inside the sphere.
4. A thin spherical shell of radius 𝑎 has a total charge 𝑄 distributed uniformly over its surface.
a. Find the electric field at points outside the shell.
b. Find the electric field at points inside the shell.
5. Find the electric field a distance 𝑟 from a line of positive charge of infinite length and constant
charge per unit length 𝜆
6. Find the electric field due to an infinite plane of positive charge with uniform surface charge
density 𝜎.
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