These are my complete notes for Electric Flux in Electromagnetism.
I color-coded my notes according to their meaning - for a complete reference for each type of note, see here (also available in the sidebar). All of the knowledge present in these notes has been filtered through my personal explanations for them, the result of my attempts to understand and study them from my classes and online courses. In the unlikely event there are any egregious errors, contact me at jdlacabe@gmail.com.
Summary of Electric Flux (Electromagnetism)
Table Of Contents
XIV. Electric Flux
XIV.I Uniform Flux.
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P. Rule .
In defining "Electric Flux", the focus of this section, one must understand a fundamental reconsideration of how one views electric fields themselves - if not, the following definitions and descriptions of Electric Flux will seem nonsensical.
First, note that the previous usage of electrical fields treated them like a structured object, a vector field, in which every point in space around it has a magnitude and a direction in relation to the field. In this mindset, an electric field is limited to just being a function that varies with spacial positioning.
The new, super ultra modern rethinking of electric fields, treats them as a measurement of an amount - something that can be summed up, through integration. By saying "the amount of electric field", what this statement is really referring to is the cumulative influence of the electric field across a given area - in relation to electric flux, the cumulative influence of the electric field passing through a surface.
This use of language is jarring at first, but can be quickly accustomed to by the young physicist. Before, electric fields were solely used in the form of a countable noun (like "apples") - now, they can also be used in the form of an uncountable noun (like "knowledge"). Without further ado:
First, note that the previous usage of electrical fields treated them like a structured object, a vector field, in which every point in space around it has a magnitude and a direction in relation to the field. In this mindset, an electric field is limited to just being a function that varies with spacial positioning.
The new, super ultra modern rethinking of electric fields, treats them as a measurement of an amount - something that can be summed up, through integration. By saying "the amount of electric field", what this statement is really referring to is the cumulative influence of the electric field across a given area - in relation to electric flux, the cumulative influence of the electric field passing through a surface.
This use of language is jarring at first, but can be quickly accustomed to by the young physicist. Before, electric fields were solely used in the form of a countable noun (like "apples") - now, they can also be used in the form of an uncountable noun (like "knowledge"). Without further ado:
#
P. Rule .
Uniform Electric Flux: SCALAR.
Units: (Newtons × Meters²) / Coulombs
Equation:
ΦE = E × A × cos(θ)
ΦE = The magnitude of Electric Flux over a Uniform Electric Field (see definition below).
E = The magnitude of a uniform electric field, the amount of which (see Rule 175) is measured over a surface and not just at an individual point in space (though, being uniform, it will have the same strength at any individual point in space). Given how the existing Electric Field equation (see Rule 168) only specifically calculates the magnitude of an electric force at a point, this equation is overly idealized in assuming a uniform electric field (therefore not a point charge or anything similar).
A = The magnitude of the area of the surface through which the uniform electric field is passing. AREA, not VOLUME (explained in def.).
θ = The angle between the directions of the electric field and the direction of the area (area having its direction perpendicular to the plane, and directed outward for closed surfaces). This value, given that all other values in the equation are positive magnitudes, determines whether the flux is positive or negative.
Definition: Electric flux is the measure of the amount of electric field (see Rule 175) which passes through a defined area. The given equation is specifically for uniform electric fields, something that is totally idealized and never really happens in reality.
This equation in particular is the result of a dot product between the electric field and the surface - this is why the result of the equation is a scalar, and why a cosine is used for the theta.
Because this equation is a dot product, that means the two variables are vectors, both with magnitude and direction. Area, the result of a cross product between two vectors (see Math Rule [[[), has a direction normal to the plane of the area, just like how the direction of angular velocity is normal to the plane in which the object is rotating (see Rule 53).
Note that this equation uses AREA, not VOLUME. When the electric field is passing a multi-sided object, the electric flux must be determined for each side of the object and summed thereafter, forming Φtotal.
Generally, using this equation to determine electric flux is optimal when the flux is passing through some sort of closed surface (see Rule 178).
Units: (Newtons × Meters²) / Coulombs
Equation:
ΦE = E × A × cos(θ)
ΦE = The magnitude of Electric Flux over a Uniform Electric Field (see definition below).
E = The magnitude of a uniform electric field, the amount of which (see Rule 175) is measured over a surface and not just at an individual point in space (though, being uniform, it will have the same strength at any individual point in space). Given how the existing Electric Field equation (see Rule 168) only specifically calculates the magnitude of an electric force at a point, this equation is overly idealized in assuming a uniform electric field (therefore not a point charge or anything similar).
A = The magnitude of the area of the surface through which the uniform electric field is passing. AREA, not VOLUME (explained in def.).
θ = The angle between the directions of the electric field and the direction of the area (area having its direction perpendicular to the plane, and directed outward for closed surfaces). This value, given that all other values in the equation are positive magnitudes, determines whether the flux is positive or negative.
Definition: Electric flux is the measure of the amount of electric field (see Rule 175) which passes through a defined area. The given equation is specifically for uniform electric fields, something that is totally idealized and never really happens in reality.
This equation in particular is the result of a dot product between the electric field and the surface - this is why the result of the equation is a scalar, and why a cosine is used for the theta.
Because this equation is a dot product, that means the two variables are vectors, both with magnitude and direction. Area, the result of a cross product between two vectors (see Math Rule [[[), has a direction normal to the plane of the area, just like how the direction of angular velocity is normal to the plane in which the object is rotating (see Rule 53).
Note that this equation uses AREA, not VOLUME. When the electric field is passing a multi-sided object, the electric flux must be determined for each side of the object and summed thereafter, forming Φtotal.
Generally, using this equation to determine electric flux is optimal when the flux is passing through some sort of closed surface (see Rule 178).
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P. Rule .
If the surface of which the electric field is passing through is perpendicular to the electric field, then since the angle created between the surface and the field will be 90° (and as cos(90) = 0), there will be a net zero electric flux acting upon that surface. This is true for both uniform (see Rule 176) and nonuniform (see Rule 179) electric fields.
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P. Rule .
When an electric field is going into a closed surface (like an object), the electric flux is negative. When an electric field is coming out of a closed surface, the electric flux is positive. This is the rule of thumb that emerges from the cosine of the uniform electric flux equation (see Rule 175).
XIV.II Charge-Flux Law.
#
P. Rule .
Nonuniform Electric Fields are the standard form of Electric Fields: a field emanating outward (from a point charge or CCD), decreasing in strength with distance (symbolized in the 'r' of the Law of Electric Force (see Rule 161)).
A new equation, an adapted form of the Uniform Electric Flux equation (see Rule 176) will account for this previously irreconcilable difference. Again, as was done for Electric Fields with Continuous Charge Distribution (Rule 171), this adaptation/rethinking will be done using Calculus.
Infinitesimally small areas of the surface plane, dA, will replace A. Correspondingly, infinitesimally small fluxes dΦE (created by the electric field) will pass through dA.
Substituting these values into the original uniform equation (see Rule 176), and then integrating both sides, will produce the new, revolutionary Nonuniform Electric Flux equation (though you could use it for uniform fields if you really wanted to):
$$\Phi_E = \oint \vec{E} d\vec{A} \cos\theta = \frac{q_{\text{enclosed}}}{\varepsilon_0}$$ ΦE = The electric flux through a closed surface.
E = The magnitude of a uniform electric field, measured in relation to its strength at the infinitesimally small area (essentially a point) dA.
dA = An infinitesimally small area of the surface plane, each of which is collectively represented by dA in the integral.
θ = The angle between the directions of the electric field and the direction of the infinitesimally small area dA (area having its direction perpendicular to the plane, and directed outward for closed surfaces). This value, given that all other values in the equation are positive magnitudes, determines whether the flux is positive or negative.
qenclosed = The charge enclosed in the Gaussian Surface (see Rule 180).
ε0 = The Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²). (see Rule 163).
This equation is known as Gauss's Law, hereafter referred to as the Charge-Flux Law. This law, in serving nonuniform electric fields (all real ones), relates electric flux through a "Gaussian surface" to the charge enclosed by said surface.
The shapes of these surfaces can be specifically chosen for characteristics that significantly simplify calculations in specific problems. See Rule 180 for what these characteristics entail.
The loop on the integral sign denotes a closed surface integral (see Math Rule [[[), meaning that one must integrate over the entire closed surface to get the net flux through the surface.
The only major difference from this to integrating normally is that you essentially have to sum the normal integrals of each side of the object. This must be done every time!
A new equation, an adapted form of the Uniform Electric Flux equation (see Rule 176) will account for this previously irreconcilable difference. Again, as was done for Electric Fields with Continuous Charge Distribution (Rule 171), this adaptation/rethinking will be done using Calculus.
Infinitesimally small areas of the surface plane, dA, will replace A. Correspondingly, infinitesimally small fluxes dΦE (created by the electric field) will pass through dA.
Substituting these values into the original uniform equation (see Rule 176), and then integrating both sides, will produce the new, revolutionary Nonuniform Electric Flux equation (though you could use it for uniform fields if you really wanted to):
$$\Phi_E = \oint \vec{E} d\vec{A} \cos\theta = \frac{q_{\text{enclosed}}}{\varepsilon_0}$$ ΦE = The electric flux through a closed surface.
E = The magnitude of a uniform electric field, measured in relation to its strength at the infinitesimally small area (essentially a point) dA.
dA = An infinitesimally small area of the surface plane, each of which is collectively represented by dA in the integral.
θ = The angle between the directions of the electric field and the direction of the infinitesimally small area dA (area having its direction perpendicular to the plane, and directed outward for closed surfaces). This value, given that all other values in the equation are positive magnitudes, determines whether the flux is positive or negative.
qenclosed = The charge enclosed in the Gaussian Surface (see Rule 180).
ε0 = The Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²). (see Rule 163).
This equation is known as Gauss's Law, hereafter referred to as the Charge-Flux Law. This law, in serving nonuniform electric fields (all real ones), relates electric flux through a "Gaussian surface" to the charge enclosed by said surface.
The shapes of these surfaces can be specifically chosen for characteristics that significantly simplify calculations in specific problems. See Rule 180 for what these characteristics entail.
The loop on the integral sign denotes a closed surface integral (see Math Rule [[[), meaning that one must integrate over the entire closed surface to get the net flux through the surface.
The only major difference from this to integrating normally is that you essentially have to sum the normal integrals of each side of the object. This must be done every time!
#
P. Rule .
A Gaussian surface is a three-dimensional closed surface in which flux of any kind is calculated. Generally, these surfaces, when used in problems, are imaginary, though they could be real physical surfaces.
In most cases, all references to a "Gaussian Surface" can be replaced by a simple description of a closed surface - there is nothing particularly unique or individual the distinguishes a ""Gaussian"" surface from any other , it is just a generic surface purpose-built as a template for flux calculations.
Typically, the shapes of these surfaces are chosen such that the electric field generated by the enclosed charge is either perpendicular or parallel to the sides of the Gaussian surface. This greatly simplifies the surface integral because all the angles are multiples of 90 degrees and the cosine of those angles have a value of -1, 0, or 1.
As long as the amount of charge enclosed in a Gaussian surface is constant, the total electric flux through the Gaussian surface does not depend on the size of the Gaussian surface.
If the net charge inside a closed Gaussian surface (qenclosed) is zero, then the net electric flux through the Gaussian surface is zero. This is because all of the electric field entering into the object exits right through the other side. There must be some charge coming from the inside of the object for there to be any electric flux.
In most cases, all references to a "Gaussian Surface" can be replaced by a simple description of a closed surface - there is nothing particularly unique or individual the distinguishes a ""Gaussian"" surface from any other , it is just a generic surface purpose-built as a template for flux calculations.
Typically, the shapes of these surfaces are chosen such that the electric field generated by the enclosed charge is either perpendicular or parallel to the sides of the Gaussian surface. This greatly simplifies the surface integral because all the angles are multiples of 90 degrees and the cosine of those angles have a value of -1, 0, or 1.
As long as the amount of charge enclosed in a Gaussian surface is constant, the total electric flux through the Gaussian surface does not depend on the size of the Gaussian surface.
If the net charge inside a closed Gaussian surface (qenclosed) is zero, then the net electric flux through the Gaussian surface is zero. This is because all of the electric field entering into the object exits right through the other side. There must be some charge coming from the inside of the object for there to be any electric flux.
# Concentric: The state of objects sharing the same center. Objects are said to be concentric to one another when they share the same center.
#
P. Rule .
With respect to point charges, the electric field will have the same magnitude as it passes through each area dA at a constant distance from the point charge - see Rule 169.
Thus, for all concentric entities surrounding a point charge, where each point on the circle or sphere are at a constant distance from the point charge, the strength of the point charge at each point along the entity will be constant.
In such cases (limited specifically to circular and circularly-shaped object), the E variable of the Nonuniform Electric Flux equation (see Rule 179) can be moved outside of the integral, since it will be a constant.
Additionally, in these cases, the cos(θ) will equal one and thus be removed, since the angle between the electric field and every area dA will equal zero, since they are both outward and in the same direction.
In such cases (limited specifically to circular and circularly-shaped object), the E variable of the Nonuniform Electric Flux equation (see Rule 179) can be moved outside of the integral, since it will be a constant.
Additionally, in these cases, the cos(θ) will equal one and thus be removed, since the angle between the electric field and every area dA will equal zero, since they are both outward and in the same direction.
