Electric Flux and Gauss's Law: Study Notes

Electric Flux in Uniform Electric Fields

Definition and Calculation of Electric Flux

Electric flux is a measure of the number of electric field lines passing through a given surface. It is a key concept in electromagnetism, especially in the application of Gauss's Law.

• Electric Flux ($\Phi_E$): Defined as the product of the electric field and the area projected perpendicular to the field.

• Formula: For a uniform electric field $\vec{E}$ and a flat surface of area $A$ with normal vector $\hat{n}$ making an angle $\theta$ with the field: $\Phi_E = \vec{E} \cdot \vec{A} = EA \cos\theta$

• Application: If a disk of area $A$ is placed in a uniform electric field $E$, and the normal to the disk makes an angle $\theta$ with the field, the flux through the disk is: $\Phi_E = EA \cos\theta$

• Example: If $E = 5\ \mathrm{N/C}$, $A = 2\ \mathrm{m}^2$, and $\theta = 30^\circ$, then $\Phi_E = 5 \times 2 \times \cos(30^\circ) = 10 \times 0.866 = 8.66\ \mathrm{N\cdot m^2/C}$

Gauss's Law and Electric Flux Through Closed Surfaces

Gauss's Law: Statement and Implications

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. It is fundamental for calculating electric fields in symmetric situations.

• Gauss's Law: The total electric flux through a closed surface is proportional to the net charge enclosed: $\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{enc}}}{\varepsilon_0}$

• Key Point: The total flux depends only on the enclosed charge, not on the size or shape of the surface.

• Example: If two Gaussian surfaces A and B enclose the same point charge, the total electric flux through each is equal, regardless of their areas.

• Comparison Table:

Surface	Area	Enclosed Charge	Total Electric Flux
A	2 × B	q	$\frac{q}{\varepsilon_0}$
B	Reference	q	$\frac{q}{\varepsilon_0}$

Additional info: This demonstrates that the total flux is independent of the surface area when the enclosed charge is the same.

Electric Flux Through Cube Faces: Point Charge at a Corner

Distribution of Flux Through Cube Faces

When a point charge is placed at the corner of a cube, the flux through each face can be determined using symmetry and Gauss's Law.

• Situation: A charge $+q$ is placed at one corner of a cube.

• Enclosed Charge: The cube does not fully enclose the charge, but the flux through its faces can be calculated by considering how many cubes would share that corner.

• Key Point: Eight cubes can be arranged so that the charge is at the shared corner; thus, each cube 'gets' $1/8$ of the total flux from the charge.

• Total Flux from Charge: $\Phi_E = \frac{q}{\varepsilon_0}$

• Flux Through Each Face Forming the Corner: The three faces meeting at the corner each receive equal flux. Since there are three such faces per cube, and eight cubes share the corner, the flux through each of these faces is: $\Phi_{\text{corner face}} = \frac{1}{24} \frac{q}{\varepsilon_0}$

• Flux Through Each Other Face: The other three faces of the cube do not touch the charge and thus receive no flux from the charge at the corner: $\Phi_{\text{other face}} = 0$

• Example: For $q = 1\ \mathrm{C}$, $\varepsilon_0 = 8.85 \times 10^{-12}\ \mathrm{C^2/(N\cdot m^2)}$, the flux through each corner face is $\frac{1}{24} \times \frac{1}{8.85 \times 10^{-12}} \approx 4.7 \times 10^{9}\ \mathrm{N\cdot m^2/C}$

Face Type	Flux Multiple	Flux Value
Corner face (touching charge)	$\frac{1}{24}$	$\frac{1}{24} \frac{q}{\varepsilon_0}$
Other face	0	0

Additional info: This result is derived from symmetry and the fact that the total flux from the charge is distributed among all faces of the eight cubes sharing the corner.