# Electric fields attract everything

## Electric fields¶

In a similar way to how the magnetic field of a permanent or electromagnet can be used to describe the force effect on another magnet, it is also possible to use the electric field of a charge distribution to describe the force effect on other electrical charges. Unlike magnetic fields, however, electric fields do not run on closed lines, but run from positive electric charges to negative charges.

Coulomb's law of force

The basis for the introduction of an electric field is the so-called Colulomb's law, which states that the force between two point charges is proportional to the amount of charge and and inversely proportional to the square of the distance of both charges is:

Here is the electric field constant of the vacuum. The unit of these important natural constants can be due to can also be written as follows:

With the help of the electric field constant, the magnitude of the acting force can be deduced from known charge quantities and their spacing; the entire pre-factor is sometimes also referred to as the “Coulomb constant”. The value of this constant clearly means that two charges of one coulomb each, which are at a distance of one meter from one another, have a force of about would exercise - that would correspond to a weight of about . This example shows, on the one hand, that in many processes, for example with moving electrons, the weight force can usually be completely neglected compared to the Coulomb force. On the other hand, it turns out that 1 coulomb represents a very large amount of charge; Usually only a fraction of this amount of charge occurs in everyday life.

The following applies to the direction of the acting Coulomb force:

• If the signs of both charges are the same, the acting force is positive and the charges repel each other.
• If, on the other hand, the two charges have different signs, the Coulomb force is negative and the charges attract each other.

If several charges are spatially separated, you can first calculate the Coulomb forces in pairs and then determine the total forces acting by adding the partial forces.

### Electric field strength¶

If there is a continuous distribution of many individual charges, it would at least be very difficult to describe the resulting effect on a further test charge as a superposition of the numerous individual Coulomb forces. Instead, the term electric field strength is used ; this indicates what force effect a test charge learns from an already existing charge or charge distribution:

The electric field strength is in the unit specified. [1] As a vector, the electric field strength indicates the direction of the force that acts on a positive test charge. The individual field lines therefore start vertically from positive charges and end vertically from negative charges. The density of the field lines can be viewed as a measure of the strength of the electric field.

The effect of the force on negative test charges results from the fact that the direction of the arrows of the field lines is reversed.

### Electric field of a plate capacitor¶

An electric field with uniformly distributed field lines pointing in the same direction is obtained if two metallic plates arranged parallel to one another are equipped with opposing charge carriers. Inside such a "plate capacitor", the electric field strength is the same at all points ("homogeneous"). [2]

The magnitude of the electric field strength of a plate capacitor depends on how many additional charges there are over the plate surfaces. The ratio of the amount of charge stored and the plate area is also called "electrical flux density" designated. The following applies to your amount:

The electrical flux density stands, as well as the electric field strength , perpendicular to the capacitor plates. The relationship between the electrical flux density , which describes the charge distribution, and the electric field strength , which indicates the force acting on charged particles, can in turn by means of the electric field constant be formulated:

In order to derive an even simpler expression for the electric field strength, a short thought experiment is helpful: Becomes a single positive test charge shifted against the field lines from the negative to the positively charged plate, work must be done for this be performed, whereby denotes the plate spacing. If the charge is then on the positive side, it has an equally large potential energy . As an electrical voltage this is the name given to this potential energy opposite the negative side of the plate, based on the size the test charge:

If you set into the above formula, we get for the electric field of a plate capacitor, the following useful relationship:

As both the electrical voltage as well as the distance are easily measurable quantities between the charged plates, the electric field of a plate capacitor can be determined very easily.

While the electrical field is the same at all points in the plate capacitor, the electrical voltage in the capacitor decreases linearly from the positive to the negative plate to zero.

### Electrical Influence and Faraday Cage¶

There is always a large number of freely moving electrons in metals. In the neutral state, the negative charges of the electrons are balanced by the positive charges of the atomic cores. If a single piece of metal is charged with additional electrons, these are only distributed along the surface, as the additional electrons are also freely movable and repel each other.

If a piece of metal is brought into an electric field, this causes a shift of the freely moving electrons towards the positive plate; on the side oriented towards the negative plate, the positively charged atomic cores remain. This effect, known as “electrical influence”, lasts until an equally strong but oppositely directed field is established in the metal as a result of the charge shift.

Inside the metal, the external and the induced electric field overlap. Since both fields are of the same size, but directed in opposite directions, the interior of the metal remains field-free. This applies not only to solid metallic bodies, but also to metallic hollow bodies. In technology, for example, car bodies represent so-called “Faraday cages” (named after Michael Faraday), which protect the occupants from electrical fields and thus also from current flows such as lightning.

### Orientation polarization¶

If there are two charges with different signs, but the same amount of charge at a distance to each other, one speaks of an electric dipole. Such a dipole has a so-called dipole moment , which is proportional to the amount of charge and the distance between the charges and points in the direction of the positive charge:

The unit of the dipole moment is .

In reality, electrical dipoles exist in the form of certain molecules that have a permanent dipole moment, such as water.

If you bring an electric dipole into a homogeneous electric field, it aligns itself parallel to the direction of the field. For the torque acting here applies:

The torque is maximum when the electric dipole is oriented perpendicular to the electric field lines and becomes zero when both directions are identical.

The alignment of electric dipoles by electric fields is called orientation polarization. In real applications, the statistically evenly distributed thermal movement of the particles prevents the dipoles from fully aligning; the orientation polarization therefore increases with decreasing temperature. When the electric field is switched off, the orientation polarization disappears again.

### Displacement polarization and dielectric¶

If you introduce a non-conductive material (“dielectric”) without electric dipoles into a homogeneous electric field, the centers of charge in all atoms are slightly shifted, and each atomic nucleus gets somewhat out of the center of its electron shell. All atoms thus become electric dipoles, even if they originally had no dipole character. This form of polarization is called displacement polarization.

With both forms of polarization, the dipoles in the dielectric itself generate a comparatively weak electrical field that is directed in the opposite direction to the external field. If the dielectric fills the entire area between the capacitor plates, the value of the electric field strength becomes compared to the original value by a factor lowered. For a plate capacitor with a dielectric, the following generally applies:

The numerical value is a material constant called the relative permittivity. Strictly speaking, air must already be regarded as a dielectric, but its value is only slightly different from the dielectric constant of the vacuum.

If a capacitor is charged by a power source connected to the plates, this continues until the electrical voltage between the capacitor plates is the same as the external voltage applied. By introducing a dielectric, however, the electric field and thus also the voltage between the capacitor plates is lowered; thus further charge continues to flow onto the capacitor plates until the voltage within the capacitor (with dielectric) is again as great as the external voltage applied. A plate capacitor can therefore store a larger amount of charge with a dielectric than without it.

### Capacity of a plate capacitor¶

The capacity of a plate capacitor indicates how much the amount of charge is that the capacitor at an applied voltage can accommodate in total:

The unit of capacitance is farad . Since one coulomb represents a very large amount of charge, a capacitance amount of one farad is also very large. Capacitors commonly used in practice are therefore in picofarads , or microfarads specified.

The above formula (6) applies in general to all types of capacitors. In the case of a plate capacitor, the capacitance depends on the area of the two capacitor plates, from their distance as well as the dielectric, which is located between the two capacitor plates. If the dielectric is vacuum or air, then the following applies to the capacitance of the plate capacitor:

Here referred to again the electric field constant. If the dielectric is a different material, instead of the value can be substituted into the above equation, where is the dielectric constant of the respective material. With a suitable dielectric between the capacitor plates, the capacitance of the capacitor can be increased many times over with the same size.

Electrical energy in a plate capacitor

If a charge becomes positive charge in a plate capacitor if moving against the electric field lines, work must work against the electric force be performed. If you move the charge from the negative to the positive plate, the distance between the plates is the same then applies to the work done :

When a capacitor is charged, the electrical work performed can be imagined as the step-by-step transport of electrical charge from one capacitor plate to the other - not via the air between the capacitor plates, but via the connecting wires. As a result of the charge separation, an electrical voltage increasingly builds up in the capacitor.

Does the voltage between the capacitor plates have the value , voltages between zero and to be overcome; the average charge voltage has be.

With and surrendered:

If you write in addition , we get for the total electrical work done during charging:

This amount of work is stored in the capacitor in the form of electrical energy.

### The Millikan experiment¶

In 1910, Robert Millikan was able to determine the size of the elementary charge for the first time using a plate capacitor determine experimentally. The basic idea of ​​his experiment was to use an atomizer to bring tiny oil droplets, at least partially charged by friction, between the plates of the capacitor.

If there is no electrical voltage on the capacitor, the droplets sink due to their weight slowly down; due to the small droplet size, this is the static buoyancy of the droplets in air as well as the frictional force not to be neglected.

If, on the other hand, an electric field is applied, the electric force (only on electrically charged oil droplets) can act balance the weight; if the electrical voltage is sufficiently high, the charged particles can even rise again.

The following applies to the forces acting:

Here referred to the location factor, the density of the oil and the density of the air. The following applies to the volume of the spherical oil droplets , in which indicates the radius of the oil droplets.

If the oil droplets are floating in the air, the following equilibrium must apply:

For the cargo of a floating oil droplet must therefore apply:

In this equation, apart from the radius of the oil droplets, all sizes constant or easily measurable. The greatest difficulty lies in precisely measuring the radius (made even more difficult by the Brownian molecular movement), whereby measurement errors due to the third power can have a considerable influence on the result. Millikan therefore also determined the speeds of individual droplets as they sink, which he achieved by switching off the applied voltage in the meantime.

The droplets reach a constant speed as they sink , the following equilibrium of forces applies:

Denoted in the above equation the viscosity of the air; at is this . Solve the equation on, one obtains:

By measuring the viscosity of the air and the rate of descent of the droplet without an electric field can be the radius of the droplets can be determined with good accuracy.

Millikan found that the resulting charge values ​​were always integer multiples of an "elementary charge". He assigned the value of this cargo what with the value known today of already matched very well.

### Movement of charged particles in electric fields¶

If you bring a particle with an electric charge into an electric field with a field strength so it learns accordingly a force effect. If the particle is an electron or a proton, the weight of the particle can be compared to the electrical force mostly neglected.

The movement of electrons in electrical fields is particularly important for technical applications.

Movement in the direction of the electric field

Assume that a freely moving electron is initially in the immediate vicinity of the negatively charged side of a plate capacitor. The electric field strength accelerates it towards the positively charged side. This movement is similar to the free fall of an object in the earth's gravitational field: potential energy is converted into kinetic energy.

According to the definition of voltage (4), the potential energy of the electron can be expressed as follows: [3]

The potential energy of the electron is therefore exclusively dependent on the voltage in the plate capacitor as the charge of the electron is constant.

Example:

• Lies on the plates of a capacitor an electron that is in close proximity to the negative plate has the following amount of energy:

The unit results from and to .

Since the amounts of energy in individual electrons are quite small, it is common to express them in the unit “electron volt”. Here with denotes the charge of a single electron; If you multiply this value by the value of the applied voltage, you immediately get the amount of energy in electron volts. For the above example it would be the same be valid.

When the electron reaches the positively charged plate, the entire potential energy of the electron has been converted into kinetic energy. The following must therefore apply here:

The electron thus reaches the following speed immediately before the impact on the positive circuit board:

This equation can be used not only for electrons but also for other charged particles (e.g. ions). These usually only carry a single elementary charge or a small multiple of it, but have a much higher mass; this results in much lower speed values ​​than with electrons.

Example:

• As in the example above, there is a voltage of at the capacitor, it results with the following impact speed for an electron:

The unit results from the following relationship:

Despite the seemingly small amount of energy of the electron already reaches a speed of over ; this already corresponds to around the speed of light. [4] If the voltage, as is common in Braun tubes, for example, is increased by a factor on increased, the speed on impact increases by the factor at.

In classic oscilloscopes and Braun tubes, the free electrons are emitted by a spiral-shaped heating wire ("glow-electric effect"). Without a further effective electric field, the wire would become positively charged due to the remaining atomic cores, and the electrons would be accelerated back in the direction of the wire. As a result, there would be an “electron cloud” only a few millimeters thick around the heating wire. If, on the other hand, an electric field is applied by means of a (positively charged) anode, the electrons are accelerated along the field lines in the direction of the anode.

Movement perpendicular to the electric field

If a charged particle, for example an electron, moves (initially) perpendicular to the direction of the electric field of a plate capacitor, the path followed by the charged particle is the same as that which a horizontally thrown object moves in the gravitational field of the earth.

If the electric field runs in a vertical direction, the horizontal component of the speed of the charged particle remains unchanged. The particle occurs at the time at the point into the electric field, then the following must apply:

In the vertical direction, the charged particle initially has a velocity of . If the particle occurs in the middle (in height ) enters the electric field, it is constantly accelerated by the electric field. So the following must apply:

The acceleration , which a charged particle experiences in an electric field, can be due to as write. If the charged particle is a free electron, then is equal to the elementary charge , it results:

About the speed or the position of the particle not as a function of time, but as a function of the horizontal distance to express one can see the context use:

In the case of a beam of free electrons, these enter with the same entry speed into the electric field. By varying the voltage on the capacitor plates and thus influencing the electrical field the path of the electrons can thus be directly influenced. This effect is used, for example, in tube oscilloscopes to make the temporal progression of one or two voltage signals visible on a screen.

Remarks: