These are my complete notes for Capacitance 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 Capacitance (Electromagnetism)
Table Of Contents
XVI. Capacitance.
XVI.I Intro to Capacitance.
When some idiot question asks for the magnitude of the "energy" stored in the electric field of a capacitor (elitist phrasing intentionally confusing the poor Physicist so that universities get more money by you failing and having to retake a class), do not let them fool you: it is only asking for the electric potential energy, which has its own equation described in Rule 218.
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P. Rule .
Parallel Plates:
If not otherwise stated, all Parallel Plates have charges of equal magnitude and opposite signs.
The Parallel Plate is the most basic example of a capacitor - two conductors a distance away from eachother, creating a UNIFORM electric field pointing towards the negative plate. Anywhere in the field, there will electric potential energy stored (zero at the negative plate, and maximum at the positive plate). See Rule 213 for more information.
The electric field between the two plates is always constant due to them being planes that, when considering the perspective of the infinitesimally small electron, can mathematically be considered to be infinite planes, despite having a measurable area.
Thus, the infinite plane equation of Rule 179 can be used, and since the electric fields of the positive and negative plates add together, the electric field of all parallel plates (between the plates, that is - outside of them there is no field) will always end up equalling σ / ε, with ε only being ε0 when there is no dielectric (see end of Rule 216).
If not otherwise stated, all Parallel Plates have charges of equal magnitude and opposite signs.
The Parallel Plate is the most basic example of a capacitor - two conductors a distance away from eachother, creating a UNIFORM electric field pointing towards the negative plate. Anywhere in the field, there will electric potential energy stored (zero at the negative plate, and maximum at the positive plate). See Rule 213 for more information.
The electric field between the two plates is always constant due to them being planes that, when considering the perspective of the infinitesimally small electron, can mathematically be considered to be infinite planes, despite having a measurable area.
Thus, the infinite plane equation of Rule 179 can be used, and since the electric fields of the positive and negative plates add together, the electric field of all parallel plates (between the plates, that is - outside of them there is no field) will always end up equalling σ / ε, with ε only being ε0 when there is no dielectric (see end of Rule 216).
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P. Rule .
Electric Permittivity:
Electric Permittivity, previously famous solely for its association with the great Permittivity of Free Space constant, is a measurement of how much a material is able to be polarized (see Rule 168) when it is placed in an electric field. This is the more appropriate definition for the form of electric permittivity used in relation to capacitance, and specifically the dielectric constant - see Rule 216.
In all circumstances in which an equation uses the permittivity of free space constant, if the object/electric field has a different electric permittivity, then that new permittivity will replace ε0. For Capacitance, capacitors that don't have dielectric materials (see Rule 215) use ε0 as their permittivity, while dielectric capacitors have their own special permittivity that is inherently related to the free space constant - see the blue section prior to Rule 164.
In all cases, do not be afraid of removing a permittivity constant to replace it with another constant when the situation calls for it. The Permittivity of Free Space is no longer that all-powerful ubiquitous entity that it once was.
Electric Permittivity, previously famous solely for its association with the great Permittivity of Free Space constant, is a measurement of how much a material is able to be polarized (see Rule 168) when it is placed in an electric field. This is the more appropriate definition for the form of electric permittivity used in relation to capacitance, and specifically the dielectric constant - see Rule 216.
In all circumstances in which an equation uses the permittivity of free space constant, if the object/electric field has a different electric permittivity, then that new permittivity will replace ε0. For Capacitance, capacitors that don't have dielectric materials (see Rule 215) use ε0 as their permittivity, while dielectric capacitors have their own special permittivity that is inherently related to the free space constant - see the blue section prior to Rule 164.
In all cases, do not be afraid of removing a permittivity constant to replace it with another constant when the situation calls for it. The Permittivity of Free Space is no longer that all-powerful ubiquitous entity that it once was.
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P. Rule .
Capacitance (General): SCALAR.
Units: Farads, e.g. Coulombs / Volts.
Equation:
C ≡ (Q / ∆V)
C = Capacitance. The measure of how much charge must be given to a capacitor (an object that stores electric potential energy - see definition) to produce a certain potential difference. Always Positive.
Q = The magnitude of the charge stored on either plate. Always Positive.
∆V = The electric potential difference between the two plates. Always Positive.
Definition: Capacitance is the measure of how much charge must be given to a capacitor (an object that stores electric potential energy - see definition) to produce a certain potential difference. It is a ratio between these two values.
Capacitance is always a positive value, and thus so are charge and voltage. None of that "two negatives cancelling out" nonsense either, only the magnitude of the charge is being considered, so it literally has to be a positive number (and thus voltage as well).
The above equation functions for ALL forms of capacitance, for any object that is capable of storing and maintaining an electric potential difference. However, the individual types of capacitors possess their own type-specific equations that are useful in their specific circumstances. For example, see Rule 213 for the equation of Parallel Plates, the simplest and most common type of capacitor.
Units: Farads, e.g. Coulombs / Volts.
Equation:
C ≡ (Q / ∆V)
C = Capacitance. The measure of how much charge must be given to a capacitor (an object that stores electric potential energy - see definition) to produce a certain potential difference. Always Positive.
Q = The magnitude of the charge stored on either plate. Always Positive.
∆V = The electric potential difference between the two plates. Always Positive.
Definition: Capacitance is the measure of how much charge must be given to a capacitor (an object that stores electric potential energy - see definition) to produce a certain potential difference. It is a ratio between these two values.
Capacitance is always a positive value, and thus so are charge and voltage. None of that "two negatives cancelling out" nonsense either, only the magnitude of the charge is being considered, so it literally has to be a positive number (and thus voltage as well).
The above equation functions for ALL forms of capacitance, for any object that is capable of storing and maintaining an electric potential difference. However, the individual types of capacitors possess their own type-specific equations that are useful in their specific circumstances. For example, see Rule 213 for the equation of Parallel Plates, the simplest and most common type of capacitor.
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P. Rule .
Capacitance (Parallel Plates): SCALAR.
Units: Farads, e.g. Coulombs / Volts.
Equation:
Cplates = (ε × A) / d
C = Capacitance. The measure of how much charge must be given to two parallel plates to produce a certain potential difference between them. Always Positive.
ε = Permittivity (see Rule 211) of the capacitor. If there is no Dielectric (see Rule 215), then it is the Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²).
A = The total surface area of either plate (imagined to be infinitely thin).
d = The distance between the two plates. This is also the distance upon which the electric potential difference of the general equation is taken.
Definition: Parallel Plate Capacitors are the most common form of capacitor in Physics, and are amongst spherical and cylindrical capacitors as the only capacitor forms worth knowing in baby freshman Physics (alongside some additional features and tricks - see dielectrics in Rule 215 & Rule 216).
The capacitance of a parallel plate measures how much charge must be given to the plates to produce a certain potential difference between them. The given equation was found by substituting in an equation specific to the characteristics of the parallel plates (the electric field at the surface of a conductor - Rule 196) into the uniform voltage equation of Rule 200, and then substituting in that entire mess into the generalized capacitance equation of Rule 212.
A parallel plate capacitor functions by separating the charge onto the two plates, the very act of which is performing work to keep them in place (since the plates are attracted toward eachother) and putting electric potential energy into the system.
The net charge on a capacitor, being composed of a negative and positive plate of equal charge magnitudes, will be zero. However, not to worry: all that is necessary for any capacitance-related calculations is the q of a single plate.
The Electric Field between the plates is always uniform, as described in Rule 210. It is always equal to σ / ε, also derived in Rule 210.
Note that through usage of the permittivity of free space constant, all parallel plate capacitors will be considered as being within a vacuum (for reasons explained in Rule 164). However, if a dielectric material (see Rule 215) is added between the plates, then the vacuum is rendered null and a different permittivity is used.
Units: Farads, e.g. Coulombs / Volts.
Equation:
Cplates = (ε × A) / d
C = Capacitance. The measure of how much charge must be given to two parallel plates to produce a certain potential difference between them. Always Positive.
ε = Permittivity (see Rule 211) of the capacitor. If there is no Dielectric (see Rule 215), then it is the Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²).
A = The total surface area of either plate (imagined to be infinitely thin).
d = The distance between the two plates. This is also the distance upon which the electric potential difference of the general equation is taken.
Definition: Parallel Plate Capacitors are the most common form of capacitor in Physics, and are amongst spherical and cylindrical capacitors as the only capacitor forms worth knowing in baby freshman Physics (alongside some additional features and tricks - see dielectrics in Rule 215 & Rule 216).
The capacitance of a parallel plate measures how much charge must be given to the plates to produce a certain potential difference between them. The given equation was found by substituting in an equation specific to the characteristics of the parallel plates (the electric field at the surface of a conductor - Rule 196) into the uniform voltage equation of Rule 200, and then substituting in that entire mess into the generalized capacitance equation of Rule 212.
A parallel plate capacitor functions by separating the charge onto the two plates, the very act of which is performing work to keep them in place (since the plates are attracted toward eachother) and putting electric potential energy into the system.
The net charge on a capacitor, being composed of a negative and positive plate of equal charge magnitudes, will be zero. However, not to worry: all that is necessary for any capacitance-related calculations is the q of a single plate.
The Electric Field between the plates is always uniform, as described in Rule 210. It is always equal to σ / ε, also derived in Rule 210.
Note that through usage of the permittivity of free space constant, all parallel plate capacitors will be considered as being within a vacuum (for reasons explained in Rule 164). However, if a dielectric material (see Rule 215) is added between the plates, then the vacuum is rendered null and a different permittivity is used.
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P. Rule .
A Charged Capacitor is one in which charge flow is enabled (and thus has an extant electric potential difference), allowing for all properties of capacitance to be applied. This is the state that every problem relating to capacitance will place the capacitor in, and it should always be assumed for a given capacitor.
An Uncharged Capacitor (also known by idiots as a "Discharged Capacitor") has no charge whatsoever. For parallel plates, this means that neither plate will have charge. Thus, there will be no electric potential/potential energy nor any electric field between the plates. In such a capacitor, the electric potential difference between the plates is zero.
An Uncharged Capacitor (also known by idiots as a "Discharged Capacitor") has no charge whatsoever. For parallel plates, this means that neither plate will have charge. Thus, there will be no electric potential/potential energy nor any electric field between the plates. In such a capacitor, the electric potential difference between the plates is zero.
XVI.II Dielectrics.
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P. Rule .
Full Dielectric Analysis (for Parallel Plates).
A dielectric is an insulating material placed between the plates of a capacitor, helping to physically separate the two plates and increase the capacitance through a process known as polarization (the very same that was described in Rule 168, that makes balloons stick on walls).
When a dielectric is placed within the (uniform) electric field between the plates, the electric field arranges the molecules in the dielectric according to the law of charges.
The positive sides of the molecules in the dielectric material are attracted to the negative plate of the capacitor, while the negative sides of the molecules are attracted to the positive plate.
As a result, the dielectric becomes polarized, with the arranged molecules forming what is known as an induced field opposite the direction of the original electric field. This new, induced field is inherently nonuniform, and while it does not outright replace the original field, it changes the scheme of what the net electric field within the parallel plates IS, and thus changes any and all calculations involving such.
As a result of the induced field being opposite in direction of the original, the net electric field of a capacitor with a dielectric is smaller than the original electric field:
Enet = Eoriginal - Einduced
Thus, a smaller electric field will lead to a lower electric potential difference (because of the uniform equation of Rule 200). This net electric field will still retain the σ / ε equation, despite the influence of the nonuniform induced electric field.
If the capacitor is not part of a circuit (see Rule [[[), then the dielectric will not change the charge on the capacitor. Thus, capacitance will actually increase when there is a dielectric in place, since the electric potential difference is in the denominator of the capacitance equation (see Rule 212).
If the capacitor IS part of a circuit however, then its a totally different story.
Mathematically, the effect of the dielectric is represented using its own constant (which can be then used to create a new version of the capacitance equation), described in detail in Rule 216.
A dielectric is an insulating material placed between the plates of a capacitor, helping to physically separate the two plates and increase the capacitance through a process known as polarization (the very same that was described in Rule 168, that makes balloons stick on walls).
When a dielectric is placed within the (uniform) electric field between the plates, the electric field arranges the molecules in the dielectric according to the law of charges.
The positive sides of the molecules in the dielectric material are attracted to the negative plate of the capacitor, while the negative sides of the molecules are attracted to the positive plate.
As a result, the dielectric becomes polarized, with the arranged molecules forming what is known as an induced field opposite the direction of the original electric field. This new, induced field is inherently nonuniform, and while it does not outright replace the original field, it changes the scheme of what the net electric field within the parallel plates IS, and thus changes any and all calculations involving such.
As a result of the induced field being opposite in direction of the original, the net electric field of a capacitor with a dielectric is smaller than the original electric field:
Enet = Eoriginal - Einduced
Thus, a smaller electric field will lead to a lower electric potential difference (because of the uniform equation of Rule 200). This net electric field will still retain the σ / ε equation, despite the influence of the nonuniform induced electric field.
If the capacitor is not part of a circuit (see Rule [[[), then the dielectric will not change the charge on the capacitor. Thus, capacitance will actually increase when there is a dielectric in place, since the electric potential difference is in the denominator of the capacitance equation (see Rule 212).
If the capacitor IS part of a circuit however, then its a totally different story.
Mathematically, the effect of the dielectric is represented using its own constant (which can be then used to create a new version of the capacitance equation), described in detail in Rule 216.
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P. Rule .
Dielectric Constant: SCALAR.
Units: None! They cancel out.
Equation:
κ = (εdielectric / ε0)
κ = (Evacuum / Edielectric)
κ = Dielectric Constant. Represents the effect of the dielectric on the capacitor, as well as the relationship between the capacitor's permittivity and ε0.
εdielectric = The Electric Permittivity (see Rule 211) of the dielectric material.
ε0 = Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²).
Evacuum = The magnitude of the Uniform Electric Field of the capacitor when there is no dielectric. Always σ / ε, with ε being ε0 for non-dielectric capacitors.
Edielectric = The magnitude of the (net & still uniform) Electric Field of the capacitor when there is a dielectric. Always σ / ε, but ε ≠ ε0.
Definition: The Dielectric Constant, also known as the Relative Permittivity of a dielectric, isn't really a constant in the traditional constant sense, in that it fluctuates from dielectric to dielectric. While it is inherent to all capacitors that have an internal, insulating dielectric material, it depends on the characteristics and electric permittivity of that dielectric.
Do not confuse relative permittivity as the actual permittivity of a capacitor with a dielectric - relative permittivity is simply the constant that, when multiplied by the permittivity of free space, gives the special dielectric permittivity value εdielectric.
The dielectric constant represents the effect and power of the dielectric on the capacitor, and is used in several equations, most notably the Dielectric Capacitance equation (see Rule 217). The easier it is for electrons to change configuration in a material (e.g., for them to move into polarized positions within a dielectric, represented by the dielectric's permittivity), the larger the dielectric constant.
The equation given for the electric fields is contingent on there being different electric permittivity values for the non-dielectric and dielectric electric fields, since if they were the same then the equation would equal 1. IF THE CAPACITOR IS PARALLEL PLATES, then the electric field equations for both values (as noted in Rule 210) is equal to σ / ε, and since charge and area will remain the same regardless of a dielectric, then ε must be the only thing that changes. Note that the electric permittivity of the non-dielectric vacuum electric field is, indeed, the Permittivity of Free Space.
Units: None! They cancel out.
Equation:
κ = (εdielectric / ε0)
κ = (Evacuum / Edielectric)
κ = Dielectric Constant. Represents the effect of the dielectric on the capacitor, as well as the relationship between the capacitor's permittivity and ε0.
εdielectric = The Electric Permittivity (see Rule 211) of the dielectric material.
ε0 = Permittivity of Free Space, equal to 8.85 × 10⁻¹² (C²) / (N × m²).
Evacuum = The magnitude of the Uniform Electric Field of the capacitor when there is no dielectric. Always σ / ε, with ε being ε0 for non-dielectric capacitors.
Edielectric = The magnitude of the (net & still uniform) Electric Field of the capacitor when there is a dielectric. Always σ / ε, but ε ≠ ε0.
Definition: The Dielectric Constant, also known as the Relative Permittivity of a dielectric, isn't really a constant in the traditional constant sense, in that it fluctuates from dielectric to dielectric. While it is inherent to all capacitors that have an internal, insulating dielectric material, it depends on the characteristics and electric permittivity of that dielectric.
Do not confuse relative permittivity as the actual permittivity of a capacitor with a dielectric - relative permittivity is simply the constant that, when multiplied by the permittivity of free space, gives the special dielectric permittivity value εdielectric.
The dielectric constant represents the effect and power of the dielectric on the capacitor, and is used in several equations, most notably the Dielectric Capacitance equation (see Rule 217). The easier it is for electrons to change configuration in a material (e.g., for them to move into polarized positions within a dielectric, represented by the dielectric's permittivity), the larger the dielectric constant.
The equation given for the electric fields is contingent on there being different electric permittivity values for the non-dielectric and dielectric electric fields, since if they were the same then the equation would equal 1. IF THE CAPACITOR IS PARALLEL PLATES, then the electric field equations for both values (as noted in Rule 210) is equal to σ / ε, and since charge and area will remain the same regardless of a dielectric, then ε must be the only thing that changes. Note that the electric permittivity of the non-dielectric vacuum electric field is, indeed, the Permittivity of Free Space.
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P. Rule .
Dielectric Capacitors (Parallel Plates): SCALAR.
Units: Farads, e.g. Coulombs / Volts.
Equation:
Cdielectric = (κ × ε × A) / d
Cdielectric = Capacitance of a dielectric capacitor. The measure of how much charge must be given to two parallel plates to produce a certain potential difference between them. Always Positive.
κ = Dielectric Constant. Represents the effect of the dielectric on the capacitor, as well as the relationship between the capacitor's permittivity and ε0.
ε = Permittivity (see Rule 211) of the capacitor.
A = The total surface area of either plate (imagined to be infinitely thin).
d = The distance between the two plates. This is also the distance upon which the electric potential difference of the general equation is taken.
Definition: Derived by just plugging in the equation for the dielectric constant (Rule 216) into the parallel plates capacitance equation (see Rule 213), the capacitance of a dielectric parallel plates capacitor is simply that of a regular parallel plates capacitor with the dielectric constant attached. Fascinating.
Units: Farads, e.g. Coulombs / Volts.
Equation:
Cdielectric = (κ × ε × A) / d
Cdielectric = Capacitance of a dielectric capacitor. The measure of how much charge must be given to two parallel plates to produce a certain potential difference between them. Always Positive.
κ = Dielectric Constant. Represents the effect of the dielectric on the capacitor, as well as the relationship between the capacitor's permittivity and ε0.
ε = Permittivity (see Rule 211) of the capacitor.
A = The total surface area of either plate (imagined to be infinitely thin).
d = The distance between the two plates. This is also the distance upon which the electric potential difference of the general equation is taken.
Definition: Derived by just plugging in the equation for the dielectric constant (Rule 216) into the parallel plates capacitance equation (see Rule 213), the capacitance of a dielectric parallel plates capacitor is simply that of a regular parallel plates capacitor with the dielectric constant attached. Fascinating.
XVI.III Energy & Work.
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P. Rule .
Work & Potential Energy within a Capacitor: SCALAR.
Units: Joules.
Equation:
Uplates = (1/2) × Q × ∆V
W = (1/2) × Q × ∆V
Uplates = The electric potential energy stored in the electric field of the capacitor.
W = Work required to move charges from one plate to the other.
Q = The magnitude of the charge stored on either plate. Always Positive.
∆V = The electric potential difference between the two plates. Always Positive.
Definition: The derivation of this equation is a simple multi-step process: The first step is to use the derivative form of the work of a charged particle equation (see Rule 207), justified by imagining, for the sake of argument, that an infinitesimally small charge is moving from one plate to the other. This allows us to find the work of said charge, which just so happens (inherently) to be the electric potential energy of the capacitor as a whole. $$W = \int_{0}^{Q} \Delta V \, dq$$ Then, the variables of the capacitance equation (see Rule 212) can be substituted in and the integral solved. $$\frac{Q^2}{2C} - \frac{\theta^2}{2C}$$ Which, of course, leads right back where one started after some further simplification and substitution.
The energy (read: electric potential energy) stored in the electric field of the capacitor is equal to the amount of work needed to move the charges from one plate to the other.
Units: Joules.
Equation:
Uplates = (1/2) × Q × ∆V
W = (1/2) × Q × ∆V
Uplates = The electric potential energy stored in the electric field of the capacitor.
W = Work required to move charges from one plate to the other.
Q = The magnitude of the charge stored on either plate. Always Positive.
∆V = The electric potential difference between the two plates. Always Positive.
Definition: The derivation of this equation is a simple multi-step process: The first step is to use the derivative form of the work of a charged particle equation (see Rule 207), justified by imagining, for the sake of argument, that an infinitesimally small charge is moving from one plate to the other. This allows us to find the work of said charge, which just so happens (inherently) to be the electric potential energy of the capacitor as a whole. $$W = \int_{0}^{Q} \Delta V \, dq$$ Then, the variables of the capacitance equation (see Rule 212) can be substituted in and the integral solved. $$\frac{Q^2}{2C} - \frac{\theta^2}{2C}$$ Which, of course, leads right back where one started after some further simplification and substitution.
The energy (read: electric potential energy) stored in the electric field of the capacitor is equal to the amount of work needed to move the charges from one plate to the other.
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P. Rule .
(No) Kinetic Energy in Capacitors:
Since Capacitors are stationary objects that only store electric potential energy, they do not store kinetic energy, not even when they are placed within a circuit (see Rule [[[). Any and all allegations to the contrary are worthless nonsense.
Since Capacitors are stationary objects that only store electric potential energy, they do not store kinetic energy, not even when they are placed within a circuit (see Rule [[[). Any and all allegations to the contrary are worthless nonsense.
