Tag Archives: Electromagnetism

Capacitance and Parallel Plate Capacitors

Capacitance relates two fundamental electric concepts: charge and electric potential. The formula that relates the two is Capacitance = charge / electric_potential.

The term equipotential surface refers to how a charge, if moved along a particular path or surface, the work done on the field is equal to zero. If there are many charges along the surface of a conductor (along an equipotential surface), then the potential energy of the charged conductor will be equal to 1/2 multiplied by the electric potential φ and the integral of all charges along this surface.

Ue = ½ φ ∫ dq.

Given a scenario in which both charge and electric potential are related, we may introduce capacitance. The following formula proves important for calculating the energy of a charged conductor:

Ue = ½ φ q = ½ φ2 C = q2 / (2C).

A parallel plate capacitor is a system of metal plates separated by a a dielectric. One plate of the capacitor will be positively charged, while the other is negatively charged. The potential difference and charge on the capacitor places causes a storage of energy between the two plates in an electric field.

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Electric Potential and Electric Potential Energy

Electric potential can be summarized as the work done by an electric force to move a charge from one point to another. The units are in Volts. Electric potential is not dependent on the shape of the path that the work is applied. Being a conservative system, the amount of energy required to move a charge in a full circle, to return it back to where it started will be equal to zero.

The work of an electrostatic field takes the formula

W12 = keqQ(1/r1 – 1/r2),

which is found by integrating the the charge q times the electric field. The work of an electrostatic field also contains both the electric potential and electric potential energy. Electric potential energy, U is equal to the electric potential φ multiplied by the charge q. Electric potential energy is a difference of potentials, while electric potential uses the exact level of electric potential in the given case.

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To calculate electric potential energy, it is convenient to assume that the potential energy is zero at a distance of infinity (and surely it should be). In this case, we can write the electric potential energy as equal to the work needed to move a charge from point 1 to infinity.

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We’ll consider a quick application related to both the dipole moment and the electric potential. The dipole potential takes the formula in the figure below. Dipole potential decreases faster with distance r than it would for a point charge.

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Dipole Moment

Consider we have both a positive and negative charge, separated by a distance. When applying supperposition of the electric force and electric field generated by the two charges on a target point, it is said that the positive and negative charges create an effect called a dipole moment. Let’s consider a few example of how an electric field will be generated for a point charge in the presence of both a positive and negative charge. Molecules also often have a dipole moment.

Here, the target point is at distance b at the center between the negative and positive charges. Where both charges are of the same magnitude, both the vertical attraction and repulsion components are cancelled, leaving the electric field to be generated in a direction parallel to the axis of the two charges.

Capture

Now, we’ll consider a target point along the axis of the two charges. Remember that a positive charge will produce an electric force and electric field that radiates from itself outward, while the force and field is directed inwards towards a negative charge. We can expect then, that the electric field will be different on either side. We can expect that the side of the positive charge will repel and the negative side will attract. This works, because the distance inverse proportionality is squared, making it so that the effect from the other charge will be less. This is a dipole.

Given how a dipole functions, it would be nice to have a different set of formulas and a more refined approach to solving electric field problems with dipoles. The dipole moment p is found using the formula, p=qI with units Couolumb*meter. I is the vector which points from the negative charge to the positive charge. The dipole moment is drawn as one point at the center of the dipole with vector I through it.dipole

In order to treat the two charges as a center of a dipole, there should be a minimum distance between the dipole and the target point. The distance between the dipole and the target should be much larger than the length l of the magnitude of vector I.

dipole2

Finally, the formula for these electric fields using a dipole moment are

E1 = 2kep/b13

E2 = 2kep/b23

Electric Force & Electric Field

While the electric force describes the exertion of one charge or body to another, we also have to remember that the two objects do not need to be touching physically for this force to be applied. For this reason, we describe the force that is being exerted through empty space (i.e. where the two objects aren’t touching) as an electric field. Any charge or body or thing that exerts an electrical force, generated most importantly by the distance between the objects and the amount of charge present, will generate an electric field.

The electric field generated as a result of two charges is directly proportional to the electric force exerted on a charge, or Coulomb force and inversely proportional to the charge of the particle. In other words, if the Coulomb force is greater, then the electric field will be stronger, but it will also be smaller if the charge it is applied to is smaller. Coulomb force as mentioned previously is inversely proportional to the distance between the charges. The electric field, E then uses the formula E = F/q and the units are Volts per meter.

By combining both Coulomb’s Law and our definition for the electric field, the electric field can be written as

E1 = ke * q1/r2 er

where er again is the unit vector direction from charge q1.

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When drawing electric field lines, there are three rules pay attention to:

  1. The direction is tangent to the field line (in the direction of flow).
  2. The density of the lines is proportional to the magnitude of the electric field.
  3. Vector lines emerge from positive charges and sink towards negative charges.

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Adding electric fields to produce a resultant electric field is simple, thanks to the property of superposition which applies to electric fields. Below is an example of how a resultant electric field will be calculated geometrically. The direction of each individual field from the charges is determined by the polarity of the charge.

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Coulomb Force

Electric charge is important in determining how a body or particle will behave and interact electromagnetically. It is also key for understanding how electric fields, electric potentials and electromagnetic waves come into existence. It starts with the atom and it’s number of protons and electrons.

Charges are positive or negative. In a neutral atom, the number of protons in a nucleus is equal to the number of electrons. When an atom loses or gains an electron from this state, it becomes a negatively or positively charged ion. When bodies or particles exhibit a net charge, either positive or negative, an electric force arises. Charges can be caused by friction or irradiation. Electrostatic force functions similar to the gravitational force – in fact the formulas look very similar! The difference between the two is most importantly that electrostatic force can be attraction or repulsion, but gravitational force is always attraction. However for small bodies, the electrostatic force is primary and the gravitational force is negligible.

Charles Coloumb conducted experiments around 1785 to understand how electric charges interact. He devised two main relations that would become Coulomb’s Law:

The magnitude of the force between two stationary point charges is

  1. proportional to the product of the magnitude of the charges and
  2. inversely proportional to the square of the distance between the two charges.

The following expression describes how one charge will exert a force on another:

coulomb

The unit vector in the direction of charge 1 to charge 2 is written as e12 and the position of the two numbers indicates the direction of the force, moving from the first numbered position to the second. Reversing the direction of the force will result in a reversed polarity, F12 = -F21.

The coefficient ke will depend on the unit system and is related to the permittivity:

coulomb2

The permittivity of vacuum, ε0 = 8.85*10^(-12)  C^2N*m^2.

Coulomb forces obey superposition, meaning that a series of charges may be added linearly without effecting their independent effects on it’s ‘target’ charge. Coulomb’s Law extends to bodies and non-point charges to describe an applied electrostatic force on an object; the same first equation may be used in this scenario.

Permittivity

Electric permittivity is an extremely important concept in electromagnetics. This is a material parameter also known as “distributed capacitance” as evidenced by its units [Farads/meter]. The absolute permittivity (∈) is used in the calculation of the capacitance of a parallel plate capacitor and is inherent to a material. The relative permittivity (∈_r) is the ratio of the absolute permittivity of a material to the permittivity of a vacuum (8.85E-12 F/m). This is also known as the dielectric constant.

A good way to understand permittivity is to consider two conductive plates separated by a distance with an equal and opposite amount of charge applied. As you can probably guess, a static electric field exists between the plates due to the charge, since the charges are separated by nonconducting medium.

polarize

As the figure demonstrates, a dielectric material will polarize itself and create a field opposing the external field applied. This is because even though the molecules of a dielectric are mostly stationary due to their lattice structure, they can rearrange a bit due to an externally applied field. In addition, dielectrics can become conducting if a large enough field is applied (dielectric breakdown). The rearranging of these molecules reduces the overall field and increases the “distributed capacitance”.

It is important to know that for normal materials, the electric permittivity is generally a complex number, because permittivity is dependent on the frequency of the field applied. This is because the polarization of a dielectric cannot happen instantaneously due to “causality” (a system’s response depends on past or present inputs, not on future inputs). Permittivity can also be affected by temperature and humidity. The complex permittivity equation can be written as

complexp.PNG

It is seen that the imaginary part of the equation depends on frequency and accounts for conductivity. The response of the material to a static (DC) field is found be decreasing the frequency to zero. The high frequency limit is found by increasing the frequency.

It is important to distinguish between dielectric constant and dielectric strength. Dielectric strength is the ability of the material to resist dielectric breakdown (units V/mil). A high dielectric breakdown means that a high voltage can be applied before the dielectric conducts appreciable current.