r r Φ e = ∫ E ⋅ dS
S
=
1
ε0
∑Q
i
=
1
ε0
∫ dQ
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Φe =
∫
S
r r E ⋅ dS
=
1
ε0
∑
Qi
=
1
ε0
∫ dQ
∫
S
S： The closed surface, i.e. gaussian surface. It is an imaginary surface and need not coincide with any real physical surface. r r E ⋅ dS The close surface integral is over all gaussian surface.
S2
+
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There are many charges inside the guassian surface.
q3
q1
+ +
S
+
q2
+
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A point charge located outside a closed surface. The number of lines entering the surface equals the number leaving the surface.
2. Electric Flux Φe Ring Amount
Flux Amount
∫
L
r r A • d lr
∆S i
r Ei
v v A ⋅ ds ∫∫
r dS
r E
S
S
8

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S
rr nE
1. Uniform electric field
r r E ↑↑ n
Φe = ES
S
r n r E
If the surfaces is closed,
r r E ⋅dA
If the surface is unclosed, the positive normal direction can be free chosen.
φ is a scalar. Its SI unit: N·m2/C.
Example
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n
n
⊕
Φ =0
Φ >0
Φ <0
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Attention
φ = ∫∫ S r According to the convention, dA > 0, outward the r closed surface; dA < 0,inward the closed surface.
§2. 3 ELECTRIC FLUX , GAUSS’S LAW
1. Electric Field Lines A convenient specialized pictorial representation for visualizing electric field patterns is created by drawing lines which are called electric field lines. The electric field lines are related to the electric field in any region of space in the following manner:
r E is the total electric field at any point on the surface due to all
charges.
v ds
Surface element. Its orientation is perpendicular to the surface and points outward from the inside region.
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+
+
∞
+
−
∞
−
For a positive point charge, the lines are directed radially outward. For a negative point charge, the lines are directed radially inward.
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(1) The tangent direction at every point on an electric field line is just the direction of the field intensity at that point or the direction of the force on the positive point charge at that point. (2) The electric field lines are denser in the place where the field intensity is stronger, and the electric field lines are sparser in the place where the field intensity is weaker. (3) The electric field lines start on positive charges and terminate on negative charges , and never intersected each other. It is never interrupted in region without charge; this is called the continuity of electric field line. (4) Keep in mind: electric field lines do not actually exist.
UNIT：Vm ：
r r Φ e = ∫∫ E ⋅ dS＝∫∫ E cos θds
Where
r v Eds = θ
〈 〈 〈 〈
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A closed surface is defined as one that completely divides space into an inside region and outside region, so that movement cannot take place from one region to the other without penetrating the surface. For a closed surface, usually define the normal line at every point on the surface points out of the closed surface A closed surface A open surface
The electric field lines for two positive point charges.
+
+
The electric field lines for a point charge +2q and a second point charge –q.
2q
+
q
++ ++ + + + + +
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The electric field lines for two charges of equal magnitude and opposite sign (an electric dipole)
+
NOTE: the number of lines leaving the positive charge equals the number terminating at the negative charge.
There is a cube surface of edge length a in the uniform r r r electric field E = E 0 ( i + j )（ E 0 is a constant）as shown ） in figure. Find the electric flux of every plane and the cube surface. Z
= 1
q
ε0
A system of charges
Φe =
∫
S
r r E ⋅ dS
ε0
∑
Qi
=
1
ε0
∫ dQ
Continuous distribution of charges
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(2) Proving A spherical gaussian surface of radius r surrounding a point charge q which is at the centre of the sphere. The electric field is normal to the surface and constant in magnitude everywhere on the surface.
2 .Uniform electric field
θ
rr En =θ r r Φe = ES cosθ = E ⋅ S
3. Nonuniform electric field, arbitrary r surface r 〈 〈 〈 〈