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.
#
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 183) 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 171) 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 183) 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 186).
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 183) 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 171) 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 183) 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 186).
#
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 184) and nonuniform (see Rule 187) electric fields.
NEVER, ever assume that a net zero electric flux means net zero electric field - it just means that all the electric field entering into the surface is passing through and out of it, which furthermore just means there is no inner charge. There can still be a net electric field at the individual points inside the surface.
NEVER, ever assume that a net zero electric flux means net zero electric field - it just means that all the electric field entering into the surface is passing through and out of it, which furthermore just means there is no inner charge. There can still be a net electric field at the individual points inside the surface.
#
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 results from the cosine of the uniform electric flux equation (see Rule 184).
XIV.II Charge-Flux Law.
#
P. Rule .
Nonuniform Electric Flux (Charge-Flux Law): SCALAR.
Units: (Newtons × Meters²) / Coulombs
Equation:
$$\Phi_E = \oint \vec{E} d\vec{A} \cos\theta = \frac{q_{\text{enclosed}}}{\varepsilon_0}$$
ΦE = The electric flux through a closed surface. Measured in (Newtons × Meters²) / Coulombs.
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 189).
ε0 = The Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²). (see Rule 164).
Definition: 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 162)).
A new equation, an adapted form of the Uniform Electric Flux equation (see Rule 184), accounts for this previously irreconcilable difference. Again, as was done for Electric Fields with Continuous Charge Distribution (Rule 174), 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 184), and then integrating both sides, produces the new, revolutionary Nonuniform Electric Flux equation (though you could use it for uniform fields if you really wanted to).
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 189 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!
Units: (Newtons × Meters²) / Coulombs
Equation:
$$\Phi_E = \oint \vec{E} d\vec{A} \cos\theta = \frac{q_{\text{enclosed}}}{\varepsilon_0}$$
ΦE = The electric flux through a closed surface. Measured in (Newtons × Meters²) / Coulombs.
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 189).
ε0 = The Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²). (see Rule 164).
Definition: 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 162)).
A new equation, an adapted form of the Uniform Electric Flux equation (see Rule 184), accounts for this previously irreconcilable difference. Again, as was done for Electric Fields with Continuous Charge Distribution (Rule 174), 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 184), and then integrating both sides, produces the new, revolutionary Nonuniform Electric Flux equation (though you could use it for uniform fields if you really wanted to).
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 189 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 .
If the net charge (qenclosed) inside a closed surface, Gaussian or otherwise, is zero, then the net electric flux through that surface/shape is zero. This is because all of the electric field entering into the shape 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.
Never confuse electric field for electric flux! The Charge-Flux Law can just as easily be used to isolate net electric field as it can be used to find electric flux; indeed, the net electric field of an object is as dependent on outside charges as internal ones - 'tis what the net electric field acting upon an object means.
Never confuse electric field for electric flux! The Charge-Flux Law can just as easily be used to isolate net electric field as it can be used to find electric flux; indeed, the net electric field of an object is as dependent on outside charges as internal ones - 'tis what the net electric field acting upon an object means.
#
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, according to legend, they could be real physical surfaces. Such an event, apart from the surface coinciding exactly with an actual physical surface (in which case there would be no point bothering with the Gaussian surface at all), is never used or asked for ever, and thus one should always consider Gaussian surfaces as nonexistent, imagined surfaces without the burden of its own charge conflicting with that of another (as is the case with shells that have internal point charges).
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 imaginary closed surface. It is just a generic surface purpose-built as a template for flux calculations - the electromagnetic industry standard for surfaces that do not carry the usual electric properties of conductors or nonconductors.
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. The spherical Gaussian surface is chosen so that it is concentric with the charge distribution, making the electric field constant for reasons outlined in Rule 191.
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. Gaussian Surfaces also follow the "outside charge = zero electric flux" rule, described in Rule 188.
Generally, these surfaces, when used in problems, are imaginary, though, according to legend, they could be real physical surfaces. Such an event, apart from the surface coinciding exactly with an actual physical surface (in which case there would be no point bothering with the Gaussian surface at all), is never used or asked for ever, and thus one should always consider Gaussian surfaces as nonexistent, imagined surfaces without the burden of its own charge conflicting with that of another (as is the case with shells that have internal point charges).
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 imaginary closed surface. It is just a generic surface purpose-built as a template for flux calculations - the electromagnetic industry standard for surfaces that do not carry the usual electric properties of conductors or nonconductors.
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. The spherical Gaussian surface is chosen so that it is concentric with the charge distribution, making the electric field constant for reasons outlined in Rule 191.
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. Gaussian Surfaces also follow the "outside charge = zero electric flux" rule, described in Rule 188.
#
P. Rule .
Gaussian surfaces, for all intents and purposes, are theoretical constructs and are exclusively used as such.
Unlike regular solid shells/spheres/closed surfaces, Gaussian surfaces do not interact with nor influence internal or external charges whatsoever and cannot contribute ANY charge - they only contain/reflect charge from internal/outside sources. Charge upon the surface of a Gaussian sphere from any such outer source is unimpeded, it is just Gaussian Surfaces themselves produce none of it. Conductive and nonconductive solid spheres, which do contain charge which produce electric fields, have their own developed systems for dealing with these intervening forces in calculations (see Rule 193).
This is the result of Gaussian Surfaces purely being constructs solely intended for the calculation of the magnitude of the electric field or flux produced by any charged object. As such, a "Gaussian Shell" is never used, since shells only have purpose in electric field calculations when there is an internal charge to the shell that can affect the net electric field/flux at a point, which is null and irrelevant with regard to Gaussian Surfaces, which can have all spherical distances/calculations accounted for solely using Gaussian Spheres.
In effect, Gaussian Surfaces are mere mathematical, superimposed bodies exploiting symmetry, with no effect of their own on anything: bodies in space whose only purpose is the use of their positioning, whether through symmetry or whatever (see Rule 189), to calculate flux or field at a particular distance from a charge in space, free from the interference of some internal charge.
Unlike regular solid shells/spheres/closed surfaces, Gaussian surfaces do not interact with nor influence internal or external charges whatsoever and cannot contribute ANY charge - they only contain/reflect charge from internal/outside sources. Charge upon the surface of a Gaussian sphere from any such outer source is unimpeded, it is just Gaussian Surfaces themselves produce none of it. Conductive and nonconductive solid spheres, which do contain charge which produce electric fields, have their own developed systems for dealing with these intervening forces in calculations (see Rule 193).
This is the result of Gaussian Surfaces purely being constructs solely intended for the calculation of the magnitude of the electric field or flux produced by any charged object. As such, a "Gaussian Shell" is never used, since shells only have purpose in electric field calculations when there is an internal charge to the shell that can affect the net electric field/flux at a point, which is null and irrelevant with regard to Gaussian Surfaces, which can have all spherical distances/calculations accounted for solely using Gaussian Spheres.
In effect, Gaussian Surfaces are mere mathematical, superimposed bodies exploiting symmetry, with no effect of their own on anything: bodies in space whose only purpose is the use of their positioning, whether through symmetry or whatever (see Rule 189), to calculate flux or field at a particular distance from a charge in space, free from the interference of some internal charge.
# 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 172.
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 187) 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 187) 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.
#
P. Rule .
The individual locations of the charges within an enclosed surface are irrelevant to calculating electric flux or any variable (like Electric Field) using the Charge-Flux Law. All that is needed is the net electric charge within the surface/object - positioning is meaningless, the charges could be in each corner of a cube and it wouldn't matter.
The net Electric Field, however, does require knowledge of the position of the charges, whether they are inside or outside of the enclosed object (due to the nature of the net electric field reflecting the collective influence of all electric fields surrounding the object, which is to say, every charge).
This, of course, is given that you are calculating for electric field in particular, and not for electric flux itself, in which case outside charges contribute zero net flux to an object and thus do not need to be accounted for (see Rule 188).
The net Electric Field, however, does require knowledge of the position of the charges, whether they are inside or outside of the enclosed object (due to the nature of the net electric field reflecting the collective influence of all electric fields surrounding the object, which is to say, every charge).
This, of course, is given that you are calculating for electric field in particular, and not for electric flux itself, in which case outside charges contribute zero net flux to an object and thus do not need to be accounted for (see Rule 188).
# Electrostatic Equilibrium: A state in which all the charges within a conductor are stationary - not moving whatsoever (not just in "equilibrium", despite the name). This is the assumed state for all conducting electrostatic objects (e.g., everything discussed in Sections XII through XVI).
Nonconductors act differently, with internal charges moving about and electric fields being generated by internal and external forces - see Rule 193.
#
P. Rule .
Electric Fields, as influenced by the Placement of Charge within Conductors & Nonconductors:
The placement of charge on the surface and in the thickness of an object as a result of its material (conductor or nonconductor, as previously described in Rule 169) is of considerable effect on the electric field produced by the object.
Note that for the purposes of this section, an "internal electric field" is the electric field contained within the hollow part of an object, whether that object be a perfectly spherical shell or a hollow nonspherical whatever.
Conductors: Since the charge of a conductor is dispersed solely along the surface of the object (see Rule 169), the internal electric field of a charged, isolated conductor is zero, and the external field (at points nearby to the surface) is perpendicular to the surface and has a magnitude that depends on the surface charge density σ.
Within the thickness of a conducting shell, the electric field will also be zero, since the charge will remain concentrated on the surface of the object and thus will not produce anything internally, regardless of whether "internal" means the hollow part of the shell or the material itself. This remains the same for Irregularly shaped conductors, which are detailed in Rule 195.
Furthermore, note that anything inside of a conductor (in electrostatic equilibrium - see the above definition) will be subject to Electrostatic Shielding: The shielding from all external electric fields. Not even external electric fields will force the inside of a conductor to have a field of its own.
Though nonspherical conductors do not have uniform charge density, even at their surface (as described in Rule 169), a general equation for the electric field immediately outside of the surface of the nonspherical shape can be determined quite easily, elaborated upon and derived in Rule 196.
Nonconductors, which have their charge distributed in their thickness wherever it is placed (generally expressed in problems as following a particular formula involving radius, or as simply a matter of uniform volumetric charge density), will NOT have an internal electric field within the hollow portion of the object, regardless of the placement of the charge (whether the charge is concentrated along the surface or uniformly distributed within the thickness).
The reason for this can be visualized simply: if you were to imagine a Gaussian Surface around the hollow part of a shell/object with uniform charge density, you will find that with there being zero electric charge on the inside of the surface, there can be zero electric flux (see Rule 188). However, as noted in Rule 185, zero electric flux does NOT mean zero electric field, just that all electric field going in is leaving. Still - the effect of the electric fields produced by each charge around the hollow portion of the object will be a net zero electric field, a result of the repulsion between charge (because electric fields cannot cross - see #4 of Rule 173) effectively preventing the existence of electric field within the hollow center.
Thus, there would both be a net zero electric flux, and net zero electric field within a nonconductor.
Of course, this also means that if the charge distribution of the object's material is "asymmetric" (defying any logical pattern of symmetry involving radius, see Rule 194), there could theoretically be a nonzero electric field within the hollow portion of a nonconducting shell, since the electric fields produced by the charge would not cancel out nor totally repel eachother.
For the reasons detailed in Rule 194, it is exceedingly unlikely to appear in a problem - note that it is possible under the laws of physics, however.
Within the shell’s thickness (between the hollow interior and the surface), the field is nonzero and varies based on the charge distribution. Outside the shell, a uniformly charged nonconducting shell behaves similarly to a conductive shell, producing an external field equivalent to that of a point charge located at the shell’s center (Shell Theorem #1 - see Rule 170).
The placement of charge on the surface and in the thickness of an object as a result of its material (conductor or nonconductor, as previously described in Rule 169) is of considerable effect on the electric field produced by the object.
Note that for the purposes of this section, an "internal electric field" is the electric field contained within the hollow part of an object, whether that object be a perfectly spherical shell or a hollow nonspherical whatever.
Conductors: Since the charge of a conductor is dispersed solely along the surface of the object (see Rule 169), the internal electric field of a charged, isolated conductor is zero, and the external field (at points nearby to the surface) is perpendicular to the surface and has a magnitude that depends on the surface charge density σ.
Within the thickness of a conducting shell, the electric field will also be zero, since the charge will remain concentrated on the surface of the object and thus will not produce anything internally, regardless of whether "internal" means the hollow part of the shell or the material itself. This remains the same for Irregularly shaped conductors, which are detailed in Rule 195.
Furthermore, note that anything inside of a conductor (in electrostatic equilibrium - see the above definition) will be subject to Electrostatic Shielding: The shielding from all external electric fields. Not even external electric fields will force the inside of a conductor to have a field of its own.
Though nonspherical conductors do not have uniform charge density, even at their surface (as described in Rule 169), a general equation for the electric field immediately outside of the surface of the nonspherical shape can be determined quite easily, elaborated upon and derived in Rule 196.
Nonconductors, which have their charge distributed in their thickness wherever it is placed (generally expressed in problems as following a particular formula involving radius, or as simply a matter of uniform volumetric charge density), will NOT have an internal electric field within the hollow portion of the object, regardless of the placement of the charge (whether the charge is concentrated along the surface or uniformly distributed within the thickness).
The reason for this can be visualized simply: if you were to imagine a Gaussian Surface around the hollow part of a shell/object with uniform charge density, you will find that with there being zero electric charge on the inside of the surface, there can be zero electric flux (see Rule 188). However, as noted in Rule 185, zero electric flux does NOT mean zero electric field, just that all electric field going in is leaving. Still - the effect of the electric fields produced by each charge around the hollow portion of the object will be a net zero electric field, a result of the repulsion between charge (because electric fields cannot cross - see #4 of Rule 173) effectively preventing the existence of electric field within the hollow center.
Thus, there would both be a net zero electric flux, and net zero electric field within a nonconductor.
Of course, this also means that if the charge distribution of the object's material is "asymmetric" (defying any logical pattern of symmetry involving radius, see Rule 194), there could theoretically be a nonzero electric field within the hollow portion of a nonconducting shell, since the electric fields produced by the charge would not cancel out nor totally repel eachother.
For the reasons detailed in Rule 194, it is exceedingly unlikely to appear in a problem - note that it is possible under the laws of physics, however.
Within the shell’s thickness (between the hollow interior and the surface), the field is nonzero and varies based on the charge distribution. Outside the shell, a uniformly charged nonconducting shell behaves similarly to a conductive shell, producing an external field equivalent to that of a point charge located at the shell’s center (Shell Theorem #1 - see Rule 170).
#
P. Rule .
Asymmetric Charge Distributions:
An asymmetric charge distribution is an uneven distribution of electric charge within a system, which results in an uneven distribution of electrical field.
The effect of an asymmetric charge distribution is profound in objects with hollow interiors, spherical or nonspherical.
An asymmetric charge distribution cannot be produced by an equation involving radius, since even an object whose charge density changes with respect to radius is still considered a symmetric distribution (regardless of whether the object is a perfect sphere or not).
Instead, it can only be expressed formulaically by creating a relation between the density and a particular point or position rather than radius, with density depending on distance from said point (such as from one end of the object). This, however, is itself a small fringe case of possible asymmetric distributions:
Asymmetric distributions which can be described by equation, only reflect an tiny, idealized fraction of all possible asymmetric distributions - imagine a simply random pattern of charge placements and concentrations, for example.
These forms of charge distributions, when applied to a nonconducting object with a hollow center, result in a nonzero internal electric field, a special case of the nature of nonconductors as described in Rule 193, and an absolute pain to deal with. The problem must really hate you if it is going through the trouble of creating a position-based charge distribution equation just to have an internal electric field.
An asymmetric charge distribution is an uneven distribution of electric charge within a system, which results in an uneven distribution of electrical field.
The effect of an asymmetric charge distribution is profound in objects with hollow interiors, spherical or nonspherical.
An asymmetric charge distribution cannot be produced by an equation involving radius, since even an object whose charge density changes with respect to radius is still considered a symmetric distribution (regardless of whether the object is a perfect sphere or not).
Instead, it can only be expressed formulaically by creating a relation between the density and a particular point or position rather than radius, with density depending on distance from said point (such as from one end of the object). This, however, is itself a small fringe case of possible asymmetric distributions:
Asymmetric distributions which can be described by equation, only reflect an tiny, idealized fraction of all possible asymmetric distributions - imagine a simply random pattern of charge placements and concentrations, for example.
These forms of charge distributions, when applied to a nonconducting object with a hollow center, result in a nonzero internal electric field, a special case of the nature of nonconductors as described in Rule 193, and an absolute pain to deal with. The problem must really hate you if it is going through the trouble of creating a position-based charge distribution equation just to have an internal electric field.
# Radius of Curvature: The distance from the center of mass to a particular point on the surface of the object. When this radius is not constant for all points on the surface, that means that the object is not a sphere.
#
P. Rule .
Irregularly Shaped Conductors:
All conductors that lack uniform symmetry and do not conform to simple geometric definitions are irregular. Luckily, all attributes of conductors relating to internal electric fields (e.g., net-zero) and the distribution of charge (e.g., along the surface), see Rule 193 and Rule 169 respectively, remain the same for irregularly shaped conductors, with one exception: Local Surface Charge Density (σ).
The distribution of the charge on the surface of the irregular object will not be constant, as it is for regular, symmetrical objects like spheres. Thus, the local surface charge density (σ) of an irregular conductor will fluctuate according to a simple pattern:
The local σ of the object will be at its maximum where the Radius of Curvature (see definition) is at its minimum.
The reverse is also applicable:
The local σ of the object will be at its minimum where the Radius of Curvature is at its maximum.
All conductors that lack uniform symmetry and do not conform to simple geometric definitions are irregular. Luckily, all attributes of conductors relating to internal electric fields (e.g., net-zero) and the distribution of charge (e.g., along the surface), see Rule 193 and Rule 169 respectively, remain the same for irregularly shaped conductors, with one exception: Local Surface Charge Density (σ).
The distribution of the charge on the surface of the irregular object will not be constant, as it is for regular, symmetrical objects like spheres. Thus, the local surface charge density (σ) of an irregular conductor will fluctuate according to a simple pattern:
The local σ of the object will be at its maximum where the Radius of Curvature (see definition) is at its minimum.
The reverse is also applicable:
The local σ of the object will be at its minimum where the Radius of Curvature is at its maximum.
XIV.III New Electric Fields.
There are several new equations for electric fields created by particular shapes/circumstances that can only be derived using the Charge-Flux Law. They are presented below.
#
P. Rule .
Electric Field on an Object's Surface (Conductor & Nonconductor!): VECTORS.
Units: Newtons / Coulombs. They are types of electric fields.
Equation:
Conductor: E = (σ / ε0)
Nonconductor: E = (σ / 2ε0)
E = The electric field on the surface of a conductor.
σ = The charge per unit area on the surface of the conductor.
ε0 = The Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²).
Definition: The equations for the electric field on the surface of any object, spherical or nonspherical, can be easily determined using the Charge-Flux Law. Although the equations are derived in the same fashion, as a result of internal differences relating to charge distribution, the equations differ between conductors and nonconductors.
Imagine two objects, one a conductor and the other a nonconductor. By the laws explained in Rule 169 (and noted again in Rule 193), the conductor, if nonspherical, will not have uniform charge density at its surface, though its charge will still be concentrated at its surface under the laws outlined in Rule 169.
Through the creation of an extremely small cylinder perpendicularly bisecting the surface of either object at a particular point (the size of bisection being small enough that, there, the object's surface can be considered flat, and the electric field thus perfectly perpendicular), one can discover that the only electric flux being produced is upon the cap of the cylinder, since the curved sides of the cylinder will be perpendicular to the object (nullifying the cosine of the Charge-Flux equation - see Rule 193).
Thus, in using the uniform Charge-Flux Theorem (uniform because the cylinder is small enough and directly upon the surface that the electric field can be considered perfectly perpendicular and constant at exactly that point/scale), all one needs to do is find the electric flux occurring on the cap of the cylinder, which will lead directly to an equation for the electric field.
The flux of the cap of the cylinder, being such a small distance away from the surface itself, can be taken to be the flux of that cap-shaped portion of the surface of the object itself. From there, the derivation of the equations (for a conducting object and a nonconducting object) simply follows these logical movements:
The charge qenc enclosed by the Gaussian surface lies on the object’s surface in an area A. If σ is the charge per unit area, then qenc is equal to σA. In the uniform Charge-Flux Law, substituting EA in for flux (since the cosine is nullified by the perpendicular electric field - see Rule 185), and σA in for qenc, will produce the above equations (after simplification).
Units: Newtons / Coulombs. They are types of electric fields.
Equation:
Conductor: E = (σ / ε0)
Nonconductor: E = (σ / 2ε0)
E = The electric field on the surface of a conductor.
σ = The charge per unit area on the surface of the conductor.
ε0 = The Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²).
Definition: The equations for the electric field on the surface of any object, spherical or nonspherical, can be easily determined using the Charge-Flux Law. Although the equations are derived in the same fashion, as a result of internal differences relating to charge distribution, the equations differ between conductors and nonconductors.
Imagine two objects, one a conductor and the other a nonconductor. By the laws explained in Rule 169 (and noted again in Rule 193), the conductor, if nonspherical, will not have uniform charge density at its surface, though its charge will still be concentrated at its surface under the laws outlined in Rule 169.
Through the creation of an extremely small cylinder perpendicularly bisecting the surface of either object at a particular point (the size of bisection being small enough that, there, the object's surface can be considered flat, and the electric field thus perfectly perpendicular), one can discover that the only electric flux being produced is upon the cap of the cylinder, since the curved sides of the cylinder will be perpendicular to the object (nullifying the cosine of the Charge-Flux equation - see Rule 193).
Thus, in using the uniform Charge-Flux Theorem (uniform because the cylinder is small enough and directly upon the surface that the electric field can be considered perfectly perpendicular and constant at exactly that point/scale), all one needs to do is find the electric flux occurring on the cap of the cylinder, which will lead directly to an equation for the electric field.
The flux of the cap of the cylinder, being such a small distance away from the surface itself, can be taken to be the flux of that cap-shaped portion of the surface of the object itself. From there, the derivation of the equations (for a conducting object and a nonconducting object) simply follows these logical movements:
The charge qenc enclosed by the Gaussian surface lies on the object’s surface in an area A. If σ is the charge per unit area, then qenc is equal to σA. In the uniform Charge-Flux Law, substituting EA in for flux (since the cosine is nullified by the perpendicular electric field - see Rule 185), and σA in for qenc, will produce the above equations (after simplification).
# Electric Field Near an Infinite Charged Line:
The electric field at a point near (read: not infinitely away from) an infinite line of charge (or cylindrical charged rod) with uniform linear charge density 𝜆 is perpendicular to the line, and is defined by the following equation: $$E_{rod} = \frac{\lambda}{2\pi \varepsilon_0 r}$$ E = The magnitude of the electric field near (at a distance r from) a charged rod.
λ = Linear Charge Density. See treatise here.
ε0 = The Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²).
r = The perpendicular distance from the line to the point.
The derivation is fairly simple, using a cylinder to find the electric field of the radial point in space (the infinite line running along the central axis of the cylinder), from there using the Charge-Flux Law as one normally would, nullifying the caps of the cylinder as being perpendicular to the electric field produced by the line (and thus giving no flux - see Rule 185) and only using the area of the curved side (2πrh) in the calculations. The enclosed charge being substituted for λh (since the linear charge density is uniform), this all simplifies down to the equation given.
This equation also approximates the field of a finite line of charge at points that are not too near the ends, in relation to the distance r from the line.
If the rod has a uniform volume charge density ρ, a similar procedure can be used to find the magnitude of the electric field inside the rod (given the rod is not purely a flat axial line). All that would need to be done is to shrink the cylinder until it itself is inside the rod: as a result, the charge qenc enclosed by the cylinder would be proportional to the volume of the rod enclosed by the cylinder, since the charge density is uniform. From there, apply the Charge-Flux Law as before, the caps of the cylinder again getting nullified, r still being the radius of the cylinder itself (effectively the distance between the center of the rod to the edge of the cylinder).
#
P. Rule .
Electric Field outside and inside a Sphere:
The electric field due to a shell within and outside of a sphere with uniform charge density (any nonconductor, or a conductor specified as such - see Rule 169) can be determined using the Charge-Flux Law. In the process, the two Shell Theorems (see Rule 170) can be mathematically proven, whereas before they were simply stated without proof.
For the following equations, assume the variables are those depicted in the image below: 'b' for the full radius of the shell, center to exterior, and 'a' for the internal radius of the shell, center to interior.
An example uniformly charged spherical shell, with a full radius of b and an internal radius of a.
Outside a spherical shell a uniform charge density q, the electric field due to the shell is radial (the lines themselves point inward or outward, depending on the sign of the charge, while the strength of the lines is constant along a ring/radius, as a result of the shell have u.c.d.). For all electric fields of a radius r ≥ b, the electric field of that radius can be determined simply using the Charge-Flux Law, and has the following magnitude: $$E_{outside} = \frac{1}{4\pi \varepsilon_0} \frac{q}{r^2}$$ Eoutside = The magnitude of the outward electric field created by a shell with uniform charge density.
ε0 = The Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²).
q = The total enclosed charge, namely that within the shell's material.
r = The distance to the point of measurement from the center of the shell, retricted to where r ≥ b.
Through a small amount of effort, one can find that this field is the very same as the field created by a shell with all of its charge concentrated in a particle of charge q in the center. Thus, the first theorem is proved.
Through an extraordinarily simple process, one can furthermore find through the Charge-Flux Law how the electric field at radius a is equal to 0, since the Gaussian surface encloses no charge. Thus, if a charged particle were enclosed by the shell, the shell would exert no net electric force on the particle. This proves the second shell theorem.
Inside a sphere with a uniform volume charge density, the electric field is radial, and any radius r (where a ≤ r ≤ b) will have an electric field of the following magnitude: $$E_{within} = \frac{1}{4\pi \varepsilon_0} \frac{q}{b^3} r$$ Ewithin = The magnitude of the electric field within the material of a shell, between a and b.
ε0 = The Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²).
q = The total charge WITHIN the specified radius; the charge between the inner surface of the shell and the specified radius r. All charge within the shell but outside of the radius r is irrelevant and not included in this subsphere.
b = The full radius of the shell, from center to external surface.
r = The radius from the center of the sphere to the point of measurement between a and b. This is akin to creating a smaller shell (cutting out the material between r and b) and measuring the electric field at its surface.
The electric field due to a shell within and outside of a sphere with uniform charge density (any nonconductor, or a conductor specified as such - see Rule 169) can be determined using the Charge-Flux Law. In the process, the two Shell Theorems (see Rule 170) can be mathematically proven, whereas before they were simply stated without proof.
For the following equations, assume the variables are those depicted in the image below: 'b' for the full radius of the shell, center to exterior, and 'a' for the internal radius of the shell, center to interior.
Outside a spherical shell a uniform charge density q, the electric field due to the shell is radial (the lines themselves point inward or outward, depending on the sign of the charge, while the strength of the lines is constant along a ring/radius, as a result of the shell have u.c.d.). For all electric fields of a radius r ≥ b, the electric field of that radius can be determined simply using the Charge-Flux Law, and has the following magnitude: $$E_{outside} = \frac{1}{4\pi \varepsilon_0} \frac{q}{r^2}$$ Eoutside = The magnitude of the outward electric field created by a shell with uniform charge density.
ε0 = The Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²).
q = The total enclosed charge, namely that within the shell's material.
r = The distance to the point of measurement from the center of the shell, retricted to where r ≥ b.
Through a small amount of effort, one can find that this field is the very same as the field created by a shell with all of its charge concentrated in a particle of charge q in the center. Thus, the first theorem is proved.
Through an extraordinarily simple process, one can furthermore find through the Charge-Flux Law how the electric field at radius a is equal to 0, since the Gaussian surface encloses no charge. Thus, if a charged particle were enclosed by the shell, the shell would exert no net electric force on the particle. This proves the second shell theorem.
Inside a sphere with a uniform volume charge density, the electric field is radial, and any radius r (where a ≤ r ≤ b) will have an electric field of the following magnitude: $$E_{within} = \frac{1}{4\pi \varepsilon_0} \frac{q}{b^3} r$$ Ewithin = The magnitude of the electric field within the material of a shell, between a and b.
ε0 = The Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²).
q = The total charge WITHIN the specified radius; the charge between the inner surface of the shell and the specified radius r. All charge within the shell but outside of the radius r is irrelevant and not included in this subsphere.
b = The full radius of the shell, from center to external surface.
r = The radius from the center of the sphere to the point of measurement between a and b. This is akin to creating a smaller shell (cutting out the material between r and b) and measuring the electric field at its surface.
