calculation of electrostatic forces in presence of dielectrics

8/13/2019 Calculation of Electrostatic Forces in Presence of Dielectrics

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Calculation of electrostatic forces in

presence of dielectricsCreated by Zoltan Losonc ([email protected])

There are two main methods for the calculation of electrostatic forces upon bodies both having

their own advantages and disadvantages. The first is the direct method, when the macroscopic

resultant force upon a body is calculated directly based on Coulombs law by summing up the

elementary electrostatic forces acting on it. Its advantage is that it will give always correct result,

but its disadvantage is that in most cases it is very difficult or even impossible to perform thecalculation.

The second is an indirect method based on the assumed validity of the energy preservation law,

when the forces are derived indirectly from energy correlations. The advantage of this method is

that the forces can be calculated easily even by analytical methods. Its disadvantage is that the

derived results are valid only if the law of energy preservation is also valid for that specific case.

However this condition is not always satisfied and consequently this method can lead us to false

results and conclusions in some cases. Although the representatives of the official science are

firmly convinced that the law of energy preservation has a general validity, and therefore this

method should provide always correct results, I would suggest to use it with due circumspection

and reservation, always verifying the results when suspecting them to be wrong.

Forces on the plates of a flat capacitor calculated with direct

method

As an example let us calculate the force that is pulling the plates of a flat condenser together, with

the direct method (fig.7a).

Calculation of

electrostatic forces

in presence ofdielectricsAccueil

Free energy from Coriolis

pressure

Calculation of electrostatic

forces in presence of

dielectrics

Dielectrophoretic pump

An antigravity propulsion

mechanism

Edition

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Fig. 7.

A simple analytical solution is possible only when the distance between the plates is much smaller

than the size of the plates, because only in that case can we approximate the E-field between the

electrodes to be homogeneous, and neglect the effect of the inhomogeneous E-field at the edges.

With such initial approximation we can assume that there is no E-field outside the condenser, but

only between the plates, where it has the same intensity and direction everywhere. To calculate

the electrostatic forces upon a conductor we use the formula that has been derived

from the Coulomb forces F=qE here.Before calculating the electrostatic pressure, we have to find

the charge density son the inner surfaces of the electrodes:

thus the force density (force per surface area):

We get the total force that pulls one plate towards the other by multiplying the force density

(electrostatic pressure) fwith the surface area of the plate S:

This force can be measured and verified experimentally with a sensitive electrometer (W.

Thomson, Kirchoff around 1880) as shown on fig. 7b; or when the voltage is unknown and the

force is measured, then the voltage can be calculated from the above formula. The two main

electrodesA andB are made of metal discs, and the upper smaller electrodeAis surrounded with

a protective ring GG that is kept at the same potential as the plate Bto eliminate the edge-effect

and ensure a homogeneous E-field between the platesAand B.

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Forces on the plates of a flat capacitor calculated withindirect method

Now lets see how can we derive the same formula with the indirect method by assuming the

validity of energy preservation for this case, and deriving the force from energy correlations. Since

there is an attractive force between the plates, when we increase the distance lbetween the

plates with a very small distance dl, then we will have to perform a work of Fdlagainst the

attractive force F (fig. 8a).

Fig. 8.

According to the energy preservation this invested mechanical work will increase the electrical

energy stored within the condenser with dW=Fdl. Thus from this correlation we get the force:

. Substituting we get:

This result is the same as in the previous case that confirms the validity of energy preservation in

this case. The negative sign means only that when the electric energy of the capacitor increases,

then we will have to invest external mechanical energy to counteract the electric forces.

We have not specified whether the capacitor is in air or surrounded with some other liquid or solid

dielectric, because the obtained formula is valid for all homogeneous and isotropic dielectrics, if

we substitute their absolute dielectric constants. It is interesting to calculate these forces for

different dielectrics and observe that when the voltage is kept constant we get more intense force

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the other electrode is made of metal M. When the precisely polished facing surfaces of the

semiconductor and metal is placed upon each other and a voltage of U~100Vis applied to the

electrodes, they will stick to each other with great force. The reason for this is that the plates

contact each other only in few points and thus only a weak current will flow through these

contacts, thus there will be a very thin air gap (d~10-3

- 10-4

mm) between the electrodes, and the

very intense E-field (E ~ U/d) will create an intense attractive force. Although there is electric

contact between the semiconductor and the metal electrode, the current I will be small due to thehigh contact resistance and high specific resistance of the semiconductor. This phenomenon is

called Johnson Rahbeck effect (1917)and it is utilized in electrostatic clutches and clamps.

Fig. 9.

Two different dielectric layers in series between the plates of

a flat capacitorNow lets examine a flat capacitor containing two layers of different dielectrics, when the boundary

surface between the dielectrics is parallel with the plates. In this case the E-field intensities will not

be the same in both dielectrics. To calculate the E-field intensities we can use the handy rule of

electrostatics that says:

The normal components of the electric displacement vectors D are equal on both sides of the

boundary surface between different dielectrics fig. 10:

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Fig. 10.

In the case of homogeneous and isotropic dielectrics the electric displacement vector is

thus we get the E-field intensities as:

These formulas show that the E-field intensity is greater in the dielectric of lower permittivity e.

Now lets turn to our main interest, and examine the electrostatic forces upon the capacitor shown

on fig. 11. The formula for the electrostatic pressure can be also written in the

following forms: . Thus the force upon the left electrode is:

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The force upon the right electrode can be calculated in similar way to get the result:

These formulas show that the force is greater on the plate, which is covered with the dielectric of

less permittivity, and we might think that consequently there would be a resultant unidirectional

reaction-less force upon the capacitor. However this is not true, since there are non-compensated

bound surface charges sbon the boundary surface between the two dielectrics, and we have to

take into account also the force acting on this boundary surface. To calculate this force we take

advantage of a simple principle of electrostatics that says:

When a very thin conductive surface is placed upon an equipotential surface, the E-field will

remain the same and undisturbed.

Fig. 11.

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An equipotential surface is always perpendicular to the E-field lines. Since the boundary surface

between the dielectrics is also perpendicular to E-field lines, it is also an equipotential surface.

Now if we place a very thin metal plateon that surface between the two dielectrics, we practically

get two capacitors connected in series, both filled with different dielectric as shown on fig. 11b.It

is easy to recognize that the force on the left side of the boundary surface will be the same as F1

and on its right side the same as F2as calculated above. Thus the resultant force upon thisboundary layer will be F

b=F

1 F

2:

that has the same intensity but opposite orientation as the resultant force upon the two main

electrodes. Consequently there will be no unidirectional resultant force upon the whole condenser.

The direction of the resultant force upon the boundary layer is such that it wants to fill the whole

condenser with the dielectric of greater permittivity. The same results can be derived with the

indirect method from energy correlations.

Two different dielectric layers in parallel between the plates

of a flat capacitor

Lets take another example, when there are two different dielectrics between the plates, but the

boundary surface between the dielectrics is perpendicular to the plates (fig. 12.)

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Fig. 12.

In this case we can use the following rule:

The tangential components of the E-field intensities are identical on both sides of the boundary

surface between two dielectrics.

It is easy to realize that this must be so, since U=E1l=E

2l thus E

1=E

2. Although the E-field

intensities are identical, the electric displacement Dand surface charge densities son the plates

touching the different dielectrics will be different (fig. 12a):

Since the E-field lines are parallel with the boundary surface, there can be no induced bound

surface charges there, and consequently no electric force can act on that boundary surface.Examining the forces upon the plates we can see that there can not be any resultant unidirectional

force inxdirection upon the capacitor, because the forces in xdirection are the same on both

plates having opposite directions, thus canceling each other (fig. 12b).

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After concluding that there is no electric force upon the boundary layer, we might get the idea to

disconnect the capacitor from the power supply and move this surface in such direction to fill the

whole capacitor with the dielectric of lower permittivity without mechanical resistance. Then the

capacitance will be decreased and the electric energy stored in the condenser increased. If we

could do this without mechanical resistance, then free energy could be produced in this way.

However if we take a look at the shape of the E-field at the edges (fig. 12b), then we can

recognize the force effects upon the dielectrics due to the arced inhomogeneous E-field,discussed on the previous pages. These forces will tend to pull both dielectrics into the capacitor,

but since the elementary dipoles are bigger in the dielectric of higher permittivity, greater force will

act on this dielectric than on the other. The macroscopic effect is again that the forces at the

edges tend to fill the capacitor with the dielectric of higher permittivity.

Let us calculate the resultant force upon the dielectrics in ydirection with the indirect method (fig.

13a).

Fig. 13.

This type of capacitor can be considered as two capacitors with different dielectrics connected in

parallel. Thus the resultant capacitance of the whole condenser will be the sum of these two

partial capacitances:

The electric energy contained in the condenser is:

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The electric energy change per distance of movement when the dielectrics are moved in y

direction:

substituting we get finally that:

The resultant force upon the dielectrics is:

When e2>e

1then F

yis directed upwards trying to pull the dielectric of higher permittivity into the

capacitor.

Electrostatic pump

This phenomenon can be nicely demonstrated with an experiment as shown on fig. 13b. If the

bottom edge of a charged flat capacitor with air between the plates is merged in a liquid dielectric,

then the above-discussed forces will push the liquid up between the plates against the force of

gravity. When the electrical and gravitational forces are in equilibrium, then the upward movement

of the dielectric between the plates stops, and the surface of the liquid within the capacitor willremain at a higher level than the liquid in the container. It is interesting to see how high can we lift

up the liquid column for different parameters of the experimental setup. The level of the dielectric

will move upwards as long as the weight of the liquid column does not become equal with the

above-calculated electric forces. The weight of the dielectric column is: G =rh b l g, (r- is the

density of the liquid; g is the gravitational acceleration). When the two forces are equal:

Lets calculate the numerical values for two examples. When the dielectric is transformer oil:

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r=860 kg/m3; e

r=2.2; l=5mm; U=30 kV; then the height of the liquid column: h=2.26 cm.

When the dielectric is glycerin:

r=1270 kg/m3; er=56 (at 15 deg. Celsius); l=5mm; U=30 kV; then the height of the liquid column:

h=70.4 cm.

The higher dielectric constant of the glycerin naturally produces greater electric pressure at the

edges, and therefore the liquid goes much higher between the plates than in the case of oil. After

proper calibration this device can be also used for measuring high voltage.

The dielectrophoretic pump violates the law of energy preservation using the discussed

basic principles.

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calculation of electrostatic forces in presence of dielectrics

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