Net charge enclosed inside the cube \(=8\left(\frac{q}{8}\right)=q\)
net electric flux through cube \(=\frac{q}{\varepsilon_{0}}\)
\(\therefore\) charges are symmetrically placed so the flux through each face \(=\frac{q}{6 \varepsilon_{0}}\)
KEY CONCEPTS
Electric flux
(a) \(\mathrm{d} \phi=\overrightarrow{\mathrm{E}} \cdot \overrightarrow{\mathrm{dS}}\)
(b) \(\phi \int \quad \overrightarrow{\mathrm{E}} \cdot \overrightarrow{\mathrm{dS}}=E A \cos \theta\) (If electric field is constant over the whole surface)
(c) Unit of \(\phi=\left(\mathrm{Nm}^{2} / \text { Coulomb }\right)=\mathrm{J} . \mathrm{m} /\) Coulomb
(d) \([\phi]=\left[\mathrm{M}^{1} \mathrm{L}^{3} \mathrm{T}^{-3} \mathrm{A}^{-1}\right]\)
Electric Flux In electromagnetism, electric flux is the measure of flow of electric field through a given area. Electric flux is proportional to the number of electric field lines going through a normally perpendicular surface. If the electric field is uniform, the electric flux passing through a surface of vector area is
\(\Phi_{E}=E . S=E S \cos \theta\)
Where \(\mathrm{E}\) is the electric field (having units of \(\frac{V}{m}\) ), \(E\) is its magnitude, \(S\) is the area of the surface, and \(\theta\) is the angle between the electric field lines and the normal (perpendicular) to
For a non-uniform electric field, the electric flux \(d \Phi \boldsymbol{E}\) through a small surface area \(d \boldsymbol{S}\) is given by
\(d \Phi_{E}=E . d S\)
(the electric field, \(E\), multiplied by the component of area perpendicular to the field). The electric flux over a surface \(S\) is therefore given by the surface integral:
\(\Phi_{E}=\iint_{S} E \cdot d S\)
Where \(E\) is the electric field and \(d S\) is a differential area on the closed surface \(S\) with an outward facing surface normal defining its direction.
For a closed Gaussian surface, electric flux is given by:
\(\Phi_{E}=\iint_{S} E . d S=\frac{Q}{\epsilon_{0}}\)
where
\(\mathrm{E}\) is the electric field
S is any closed surface,
\(\mathrm{Q}\) is the total electric charge inside the surface \(\mathrm{S}\)
\(\varepsilon_{0}\) is the electric constant (a universal constant, also called the "permitivity of free space")
\(\left(\varepsilon_{0} \approx 8.854187817 \ldots x 10-12 \text { faradays per meter }\left(F \cdot m^{-1}\right)\right)\)
This relation is known as Gauss law for electric field in its integral form and it is one of the four Maxwell's equations. While the electric flux is not affected by charges that are not within the closed surface, the net electric field, \(E\), in the Gauss' Law equation, can be affected by charges that lie outside the closed surface. While Gauss' Law holds for all situations, it is only useful for "by hand" calculations when high degrees of symmetry exist in the electric field. Examples include spherical and cylindrical symmetry.Electrical flux has SI units of volt meters (V \(\mathrm{m}\) ), or, equivalently, newton metres squared percoulomb \(\left(\mathrm{N} \mathrm{m}^{2} \mathrm{C}^{-1}\right)\) Thus, the SI base units of electric flux are \(k g \cdot m^{3} \cdot s^{-3} \cdot A^{-1}\). Its dimensional formula is \(\left[L^{3} M T^{-3}\right]\)
Gauss Law Gauss's law can be stated using either the electric field E or the electric displacement field D. This section shows some of the forms with \(\mathrm{E}\); the form with \(\mathrm{D}\) is below, as are other forms with E.
Integral form
Gauss's law may be expressed as:
\(\Phi_{E}=\frac{Q}{\epsilon_{0}}\)
where \(\Phi_{E}\) is the electric flux through a closed surface \(\$\) enclosing any volume \(V\), \(Q\) is the total charge enclosed within \(S\), and \(\varepsilon_{0}\) is the electric constant. The electric flux \(\Phi E\) is defined as a surface integral of the electric field:
\(\Phi_{E}=\iint_{S} E . d A\)
where \(\mathrm{E} \text { is the electric field, } \mathrm{d} \mathrm{A} \text { is a vector representing an infinitesimal element of area, [note } 1]\) and \(\cdot\) represents the dot product of two vectors.
since the flux is defined as an integral of the electric field, this expression of Gauss's law is called the integral form.
Applying the integral form
If the electric field is known everywhere, Gauss's law makes it quite easy, in principle, to find the distribution of electric charge:
The charge in any given region can be deduced by integrating the electric field to find the flux.
However, much more often, it is the reverse problem that needs to be solved: The electric charge distribution is known, and the electric field needs to be computed. This is much more difficult, since if you know the total flux through a given surface, that gives almost no information about the electric field, which (for all you know) could go in and out of the surface in arbitrarily complicated patterns.
An exception is if there is some symmetry in the situation, which mandates that the electric field passes through the surface in a uniform way. Then, if the total flux is known, the field itself can be deduced at every point. Common examples of symmetries which lend themselves to Gauss's law include cylindrical symmetry, planar symmetry, and spherical symmetry. See the article Gaussian surface for examples where these symmetries are exploited to compute electric fields.
Differential form
By the divergence theorem, Gauss's law can alternatively be written in the differential form:
\(\nabla \cdot E=\frac{\rho}{\varepsilon_{0}}\)
where \(\nabla \cdot E\) is the divergence of the electric field, \(\varepsilon 0\) is the electric constant, and \(\rho\) is the total electric charge density (charge per unit volume).
Equivalence of integral and differential forms
The integral and differential forms are mathematically equivalent, by the divergence theorem. Here is the argument more specifically.
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