It is not so easy to understand what a capacitor is and how it works. Using equations - differentiations and integrations – may help us to understand the mystery of a capacitor and its functions. But you may still not be able to get satisfactory understanding. Let us try to get into this wonderland and come back with some fruits.

1. Explanations from Encyclopedia Wikipedia and hyperphysics

Capacitor (From Wikipedia, the free encyclopedia, April, 2008)

A capacitor is an electric/electronic device that can store energy in the electric field between a pair of conductors (called “plates”). The process of storing energy in the capacitor is known as “charging”, and involves electric charges of equal magnitude, but opposite polarity, building up on each plate

The above explanation can be OK if you already know what a capacitor is. But if you do not, it cannot be well understood unless you understand well enough the terms energy, electric field, charge which may be more difficult to understand than capacitor itself. You can see a capacitor (a real thing) but you cannot see energy, electric field, charge (not real things). And “charging” in the above explanation may have some special meaning since the word in “-----”. One important thing is that energy is stored in the electric field between a pair of conductors (plates) ( later we shall see how ) while electric charges of equal magnitude (opposite polarity) built up on each plate ( also later we shall see how ), that is:

energy in the electric field between a pair of conductors (plates)

electric charges on each plate

Let's continue with Wikipedia.

Stored energy (From Wikipedia)

As opposite charges accumulate on the plates of a capacitor due to the separation of charge(s), a voltage develops across the capacitor due to the electric field of these charges. Ever-increasing work must be done against this ever-increasing electric field as more charge is separated. The energy (measured in joules, in SI) stored in a capacitor is equal to the amount of work required to establish the voltage across the capacitor, and therefore the electric field. The energy (U) stored is given by:

It is very difficult to understand the first sentence as well as the second sentence. And suddenly given the equation of energy (U) in which work is not clearly mentioned. Meanwhile the differentiation of (1/2) C V²with respect to V is CV or the integral of CV is (1/2) C V².

As the energy stored in a capacitor is (defined) equal to the amount of work required to establish the voltage (after a certain time has passed, which means time is involved) across the capacitor, energy and work is supposed to be the integral of CV (C is constant and V is variable) with respect with V (the established voltage), which varies with time.

First sentence

As opposite charges accumulate on the plates of a capacitor due to the separation of charges, a voltage develops across the capacitor due to the electric field of these charges

This sentence does not say correctly how opposite charges accumulate on the plates of a capacitor since 'due to the separation of charge' does not seem to be a real cause of the accumulation of the opposite charges. What does "separation of charge" mean? And where the charges come from? Or how the opposite charges are created? The problem of this explanation is that it does not say anything about the power (or energy or voltage) source. Without a power source a capacitor does not work. Without a power source opposite charges do not accumulate on the plates of a capacitor 'due to the separation of charge' (if it is correct as a cause). So the first sentence should be:

As opposite charges accumulate on the plates of a capacitor due to the separation of charge when an active power (or energy or voltage) source is applied to the capacitor, a voltage develops across the capacitor due to the electric field of these charges. See below.

Then it is more understandable why although the process is not explained in details. The structure of capacitor is quite simple – two parallel palates and the space in between. You may not be able to catch the meaning of "due to the electric field of these charges" since electric field is used here. What is electric field ? And how the electric field relates with these charges or how these electric charges make the the electric field (if they do).

If we talk about electric field here it will leads to a long time of derail from the capacitor argument since electric field is a big topic although an important concept to understand a capacitor more. We regard electric field as just for electric field now.

Second sentence

Ever-increasing 'work' must be done against this ever-increasing electric field as more charge is separated. The energy (measured in joules, in SI) stored in a capacitor is equal to the amount of 'work' required to establish the voltage across the capacitor, and therefore the electric field.

Work here is a physics (or mechanical) term and related with force. Work is a bit 'artificial' definition but is related with the two fundamental physical units, Force and Distance.

Work = Force x Distance

Work done by a variable force (From hyperphysics)

The basic work relationship W=F is a special case which applies only to constant force along a straight line. That relationship gives the area of the rectangle shown, where the force F is plotted as a function of distance. In the more general case of a force which changes with distance, the work may still be calculated as the area under the curve. For example, for the work done to stretch a spring, the area under the curve can be readily determined as the area of the triangle. The power of calculus can also be applied since the integral of the force over the distance range is equal to the area under the force curve:

For any function of x, the work may be calculated as the area under the curve by performing the integral

Analogically, when F(x) = kx is applied to CV, F(V) = CV (V is variable), the integral of CV becomes (1/2) C V². CV=Q (the stored charge). When C is constant (usually is), Energy or work is the integral of CV=Q. Or differentiation of energy or work ((1/2) C V²) by V is CV (=Q).

This can be explained without using work, which is more direct.

Capacitor function, or the relation of Capacitance, Current (changing) and Voltage (hanging)

$i(t)= \frac{\mathrm{d}q(t)}{\mathrm{d}t}=C\frac{\mathrm{d}v(t)}{\mathrm{d}t}$

Integrate the both side with respect to time

(q/time) x time = C v

q = Cv =C x Joule (Energy)/q

Integrate both sides with respect of q (which changes with time)

(1/2)q x q = C x Joule (Energy)

Then,

Joule (Energy) = (1/2)q x q x 1/C, which is again

-----

Let's look at the process or property of 'charging' or of 'being charged' capacitor.

From hyperphysics

Charging a Capacitor (From hyperphysics)

When a battery is connected to a series resistor and capacitor, the initial current is high as the battery transports charge from one plate of the capacitor to the other. The charging current asymptotically approaches zero as the capacitor becomes charged up to the battery voltage. Charging the capacitor stores energy in the electric field between the capacitor plates. The rate of charging is typically described in terms of a time constant RC.

(Where RC is called , which is defined as the time required to charge the capacitor, through the resistor, to 63.2 (≈ 63) percent of full charge; or to discharge it to 36.8 (≈ 37) percent of its initial voltage. These values are derived from the mathematical constant e, specifically 1 − e ^{− 1} and e ^{− 1} respectively.)

Why e ? (another question) - You must understand e first. Otherwise, you cannot get the meaning of the above "unfriendly" explanation. Very good explanation is found in http://betterexplained.com/articles/an-intuitive-guide-to-exponential-function-e/

If you do not have time, you can treats e as simply the number of

2.71828182845904523536028747135266249775724709369995...

The above explanation includes a resistor R. We neglect this R for capacitor argument or make it 1Ohm (unit). It says "the battery transports charge from one plate of the capacitor to the other" . This description is misleading or wrong since it looks like a charge (comes from where?, from the battery?) being transported one plate of the capacitor and going trough the capacitor (the space between the two palates) and reaches the other plate. This contradict "a direct current does not go through a capacitor or a capacitor blocks a direct current.". Where the charge comes from?, From the battery? We must think about the nature of a dielectric - the material placed the space between the two palates - to get more understanding. But before talking about dielectric, let's look at the above charging (being charged) characteristics.

The above graph shows the two curves – change of the charge on (or in) the capacitor and change of current in the circuit (strange but not through the capacitor) as a function of time (horizontal line). The graph does not show the change of the voltage built up on the capacitor. But since this voltage is proportional to the charge (V=Q/C), the voltage curve is like the Q curve. See below:

Capacitor Energy Integral (From hyperphysics)

Transporting differential charge dq to the plate of the capacitor requires work.

The voltage curve is identical to the charge curve. If we leave out the time, we can get the following curve (straight line with a slope of 1/C) showing the relation between the charge and voltage since V=Q/C.

U = QV/2

Please note that integral form in the above shows the relation between the work (or energy) and charge. The following Energy (U) equations can be obtained from the above charge and voltage curves or equations. Also note that QV/2 is a half of the area (triangle) of Q x V.

One important thing is the rate of the changes the three values (charge, voltage and current) is exponential (- which is why e is used ). By taking Q, it grows quickly at the start and gradually (exponentially) slower and very slow at the end and stops (fully charged – the voltage developed becomes equal to the battery voltage). What makes this (getting slower rate of the change)? Something opposes the charge increasing and the magnitude of this opposing thing relates with the change accumulated on (or in) the capacitor or changes as a function of the accumulated charge, It is analogous to a more well known counter electromotive force (counter emf) of an inductor. It takes a differential form. See below:

Inductor (From hyperphysics)

Inductance is typified by the behavior of a coil of wire in resisting any change of electric current through the coil. Arising from Faraday's law, the inductance L may be defined in terms of the emf generated to oppose a given change in current:

Going back to something opposing of a capacitor. Now we shall talk about electric field. Electric field of capacitor (or around and in a capacitor) is relatively simple since electric field is considered to be uniform. There are some equations of electric field of a capacitor.

Capacitance of Parallel Plates (From hyperphysics)

1) Electric field E and charge density

The electric field between two large parallel plates is given by

2) E and voltage

The voltage difference between the two plates can be expressed in terms of the work done on a positive test charge q when it moves from the positive to the negative plate.

It then follows from the definition of capacitance that

The above explanation says "a positive test charge q when it moves from the positive to the negative plate." Is it true?

Polarization of Dielectric

If a material contains polar molecules, they will generally be in random orientations when no electric field is applied. An applied electric field will polarize the material by orienting the dipole moments of polar molecules.

This decreases the effective electric field between the plates and will increase the

capacitance of the parallel plate structure. The dielectric must be a good electric

insulator so as to minimize any DC leakage current through a capacitor.

Electric field (From Wikipedia)

1) Physics

In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field (that can also be equated to electric flux density). This electric field exerts a force on other electrically charged objects. The concept of electric field was introduced by Michael Faraday.

2) Electronics

The electric field is a vector field with SI units of newtons per coulomb (N C⁻¹) or, equivalently, volts per meter (V m⁻¹). The strength of the field at a given point is defined as the force that would be exerted on a positive test charge of +1 coulomb placed at that point; the direction of the field is given by the direction of that force. Electric fields contain electrical energy with energy density proportional to the square of the field intensity. The electric field is to charge as gravitational acceleration is to mass and force density is to volume.

A moving charge has not just an electric field but also a magnetic field, and in general the electric and magnetic fields are not completely separate phenomena; what one observer perceives as an electric field, another observer in a different frame of reference perceives as a mixture of electric and magnetic fields. For this reason, one speaks of "electromagnetism" or "electromagnetic field." In quantum mechanics, disturbances in the electromagnetic fields are called photons, and the energy of photons is quantized.

The maximum energy that can be (safely) stored in a particular capacitor is limited by the maximum electric field that the dielectric can withstand before it breaks down. Therefore, capacitors made with the same dielectric have about the same maximum energy density (joules of energy per cubic meter), if the dielectric volume dominates the total volume.

When there is a difference in electric charge between the plates, an electric field is created in the region between the plates that is proportional to the amount of charge that has been moved from one plate to the other. This electric field creates a potential difference V = E·d between the plates of this simple parallel-plate capacitor.

where V is the voltage across the capacitor.

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Duality of Electric and Magnetic Fields (1)

(ref: http://info.ee.surrey.ac.uk/Workshop/advice/coils/terms.html)

Magnetomotive force (in amperes) F_m = H (Magnetic field strength)× l _e (effective length)

Electromotive force (in volts) V = E (Electric field strength) × d (distance)

Magnetic field strength amperes per metre H = F _m/_l (or I/d)

Electric field strength volts per metre E = V/d

Magnetic flux Φ = V × time (volt second)

Electric charge (= coulomb) Q = I × time (amp second)

Note:

F_{m =}_{Magnetomotive force}₍_MMF_{) (see below; wiki)}

Magnetomotive force (MMF) (SI Unit: Ampere) is any physical force that produces magnetic flux. In this context, the word "force" is used in a general sense of "work potential", and is analogous to, but distinct from mechanical force measured in newtons. The name came about because in magnetic circuits it plays a role analogous to the role electromotive force (voltage) plays in electric circuits.

SI versus CGS units

The SI unit of magnetomotive force is the ampere (A), represented by a steady, direct electric current of one ampere flowing in a single-turn loop of electrically conducting material in a vacuum.

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Electrostatic energy (From Wikipedia 'Energy', June, 2009)

The electric potential energy of given configuration of charges is defined as the work which must be done against the Coulomb force to rearrange charges from infinite separation to this configuration (or the work done by the Coulomb force separating the charges from this configuration to infinity). For two point-like charges Q₁ and Q₂ at a distance r this work, and hence electric potential energy is equal to:

$E_{\rm p,e} = {1\over {4\pi\epsilon_0}}{{Q_1Q_2}\over{r}}$

where ε₀ is the electric constant of a vacuum, 10⁷/4πc₀² or 8.854188…×10⁻¹² F/m. If the charge is accumulated in a capacitor (of capacitance C), the reference configuration is usually selected not to be infinite separation of charges, but vice versa - charges at an extremely close proximity to each other (so there is zero net charge on each plate of a capacitor). The justification for this choice is purely practical - it is easier to measure both voltage difference and magnitude of charges on a capacitor plates not versus infinite separation of charges but rather versus discharged capacitor where charges return to close proximity to each other (electrons and ions recombine making the plates neutral). In this case the work and thus the electric potential energy becomes

$E_{\rm p,e} = {{Q^2}\over{2C}}$

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Duality of Electric and Magnetic Fields (2)

$i(t)= \frac{\mathrm{d}q(t)}{\mathrm{d}t}=C\frac{\mathrm{d}v(t)}{\mathrm{d}t}$

By integrating with time

i(t) . sec = C v(t) (C in Farand)

Electric charge (= coulomb) Q = I × time (amp second)

while

$v(t) = L \frac{di(t)}{dt}$

By integrating with time

v(t) . sec = L i(t) (L in Henry)

Magnetic flux Φ = V × time (volt second)

Energy/Q = V ---> Energy = QV

Energy/ Φ = I ---> Energy = ΦI

Energy x Energy =QV x ΦI = VI x QΦ

= VI x <I x time> x <V x time>

Therefore

Energy² = (VI x time)²

Energy = +/- VI x time

This helps us to what Power (VI) means in terms of Electric and Magnetic Fields.

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sptt

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