Electric current is dangerous for people and animals for various reasons. Fluids of the human body conduct electricity; almost all organs function with the help of electrical impulses coming from the brain. These impulses control our movements and organs with the power of about 50 mV.
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Different medical devices are able to measure these currents: EKG (electrocardiogram), for example, provides information about the electric activity of the heart. The heart also functions with electrical impulses which it itself generates.

But: If, for instance, an alternating current flows through the heart, our natural pump will try to follow the stronger and faster external impulses, which can eventually cause cardiac arrhythmia and ventricular fibrillation. The definition of alternating current and the quantities that are used to describe it will be clarified later in this article.

In order to understand different quantities of electrical engineering, please read the article ‘Electricity I’ in our magazine.

## Electric Field

An electric field is a space around a charged particle in which another charged particle experiences a force. An electric field can be represented by electric field lines along which an isolated positive charge would move if it were free to do so. They run from the positive pole to the negative pole and set each point of the electric field towards the force produced by a positive charge.

Electric field lines always emit vertically from the surface area of the conductive body. They point from positive charges and towards negative charges. Depending on the course of the field lines, radial, homogeneous, and inhomogeneous fields are defined. Lines of the homogeneous field run parallel with each other and are of equal weight and density. If the lines are not parallel, then the field should be defined as inhomogeneous.

Electric field lines

## Electric Field Intensity

### E: Electric Field Intensity [N/C]

Electric field intensity at a point of an electric field is the ratio of the force experienced by a charge at that point and the quantity of that charge. It is a vector quantity with the direction of a force. The movements of the charge in a homogeneous field require work,

W = F * s

which runs along the electric field lines. On the basis of the following equations,

E = F / Q
U = E * s

we have:

W = F * s
W = E * Q * s
W = U * Q

The electronvolt [eV] is a common unit among these types of work. The charge is thereby given by a multiple of the elementary charge, and voltage is given by volts. The force that exerts upon the electric charge in the electric field results from the electric field density E and the electric charge Q:

F = E * Q

## Electrostatic Induction

Charges act on other charges in their environment. Experiencing the force of the electric field, evenly distributed electrons of the uncharged conductor of the two oppositely charged plates collect on one side of the conductor, while the other side is deficient of electrons. The charges of the conductor are separated by the influence of the nearby charge.

Electrostatic Induction

Due to the separation of charges, the uncharged body can be attracted by a charged body. Repulsion of likely charges thus proves the existence of the electric charge.

Now, let’s imagine an electrostatic field in a medium (air) and take an uncharged metal sphere which consists of two parts, as demonstrated in the picture below.

As a result of electrostatic forces, the electrons inside the ball start to separate in such a way that negative charges will accumulate opposite to the negatively charged plate because unlike charges attract each other. If we separate the two hemispheres from each other and then remove them from the field, we will get two oppositely charged bodies. This influence of the electrostatic force upon the charges is called electrostatic induction.

## Capacitance

### C: Capacitance [F]

ε0: the electric constant
εR: the relative static permittivity of the material between the plates

A capacitor consists of two closely adjacent conductors. The two conductors are oppositely charged. Because of the mutual attraction of the charges, the amount of the charges collected on both plates at the same voltage will be bigger than if the conductors were separate. A capacitor has a large capacity (capacitance). If both conductors have plates, they are called parallel-plate capacitors. The capacitance is a quotient of the charge Q and the voltage U between the conductors.

C = Q / U

A capacitance of 1 farad produces 1 V of voltage for an electric charge of 1 C. The capacitance of a parallel-plate capacitor is directly proportional to the surface area of the conductor plates A and inversely proportional to the separation distance between the plates d.

In case there is air or vacuum between the plates, the following applies:

C = ε0 * A / d

If there is the dielectric material between the plates, the following applies:

C = ε0 * εR * A / d

The more charges are collected on the plates, the bigger the electrical voltage is between them, i.e., the relation Q ∼ U applies.

Note: The larger the capacitance is, the more charges can be accumulated at the plates at a given voltage.

### Series Capacitors

For the series capacitors applies the following:

Q = Q1 = Q2 = Q3 = … = Qn
1/C = 1/C1 = 1/C2 = 1/C3 = … = 1/Cn
U = U1 + U2 + U3 + … + Un
U = Q / C

Series Capacitors

### Parallel Capacitors

For the parallel capacitors applies the following:

U = U1 = U2 = U3 = … = Un
C = C1 + C2 + C3 + … + Cn
Q = Q1 + Q2 + Q3 + … + Qn
Q = C * U

Parallel Capacitors

## Parallel Plate Capacitors

### C: Capacitance [F]

ε0: the electric constant
εR: the relative static permittivity of the material between the plates

A capacitor consists of two closely adjacent conductors. The two conductors are oppositely charged. Because of the mutual attraction of the charges, the amount of the charges collected on both plates at the same voltage will be bigger than if the conductors were separate. A capacitor has a large capacity (capacitance). If both conductors have plates, they are called parallel-plate capacitors. The capacitance is a quotient of the charge Q and the voltage U between the conductors.

C = Q / U

A capacitance of 1 farad produces 1 V of voltage for an electric charge of 1 C. The capacitance of a parallel-plate capacitor is directly proportional to the surface area of the conductor plates A and inversely proportional to the separation distance between the plates d.

In case there is air or vacuum between the plates, the following applies:

C = ε0 * A / d

If there is the dielectric material between the plates, the following applies:

C = ε0 * εR * A / d

The more charges are collected on the plates, the bigger the electrical voltage is between them, i.e., the relation Q ∼ U applies.

Note: The larger the capacitance is, the more charges can be accumulated at the plates at a given voltage.

### Series Capacitors

For the series capacitors applies the following:

Q = Q1 = Q2 = Q3 = … = Qn
1/C = 1/C1 = 1/C2 = 1/C3 = … = 1/Cn
U = U1 + U2 + U3 + … + Un
U = Q / C

Series Capacitors

## Magnetic Field and Electric Current

F: force [N]
I: current intensity[A]
l: length [m]
B: magnetic flux density

Every magnetic field exerts a force upon a current-carrying conductor. This force is performed by the overlapping of the magnetic fields of the conductor and the magnet. In a magnetic field, a free-moving current-carrying conductor experiences a force perpendicular to the direction of the electric current and to the magnetic field.

Image: “Diagram of the electric (blue) and magnetic (red) fields surrounding a dipole antenna radiating a radio wave.” by Averse. License: CC BY-SA 3.0

The direction of the diversion of a conductor piece in a magnetic field can be determined by the UVW-rule (also called right-hand rule). Spread your thumb, your index finger, and your middle finger perpendicular to one another so that your thumb points in the direction of the technical current, your index finger points in the direction of the magnetic field, and your middle finger points in the direction of the movement of the conductor.

Image: “Right-hand rule” by Canarris. License: CC BY-SA 3.0

If the current direction is perpendicular to the direction of the magnetic field, the force on the straight section of the conductor is equal to the product of electric current, to the length of the conductor section in the magnetic field and to the magnetic field density.

F = I * l * B

The magnetic flux density is given by the magnetic field constant and the magnetic field strength:

B = μ0 * H

## Magnetic Induction and Faraday’s Experiment

Along with the laws of the magnetomotive force, the law of induction represents one of the fundamental laws of electrical engineering. It was discovered by Faraday, who tried to answer the question of whether the reversal of Ampère’s circuital law was possible or not. As it describes the origin of magnetic fields from electric currents, the reversal law would mean that magnetic fields could produce electricity. The following figure shows Faraday’s experiment:

Nearby the switchable coil there is a so-called conductor loop, i.e., a coil with a single turn (N = 1). Its ends are connected to a voltmeter, which penetrates a portion of the conductor with the magnetic flux produced by the closed switch S.

While working with this experimental arrangement, Faraday came to the following conclusions: If the S switch is closed, magnetic flux is formed in the coil with flowing current, as demonstrated in the picture. This current is constant because it is generated by a continuous current. If the switch is open, there is no current, and, as a result, there is no magnetic field. In both cases, the voltmeter does not display any deflections.

The voltage can be detected only during switching the current of the coil on or during switching it off.

Conclusion: When a conductor loop is penetrated by magnetic flux lines, the voltage is created only if the flow comprised by the conductor loop changes over time. If it remains constant, no voltage can be observed. The production of voltage by the time-varying magnetic field is called magnetic induction or induced voltage. It is an electromotive force, or compliance voltage since it makes electricity flow around the circuit of a conductor loop, which supplies the electrical energy—here for example, to the voltmeter.

Note: The current powered by induced voltage is directed in such a way that its own external magnetic field, in conjunction with the (external) magnetic field created by induction, can inhibit the induction process. The induced current counteracts the cause of induction.

## Lorentz Force

When charge carriers move in a magnetic field, they experience a so-called Lorentz force, which can be described in the following vector equation:

F = Q v x B

A particle of charge Q moves in the presence of a magnetic field of density B with velocity v. If only one conductor moves through a magnetic field, its quasi-free electrons experience the Lorentz force.

The Lorentz force acts:

• perpendicular to the particle’s line of flight
• perpendicular to the magnetic force vector

## Alternating Current

ω: magnetic field
Φ0: the largest magnetic flux
t: time

If a conductor loop rotates evenly in a magnetic field, alternating current will be induced. It is characterized by the fact that it periodically changes in size and polarity. Alternating current is represented by a sinusoidal wave on a graph. If the ends of a rotating coil are connected to an external circuit, it creates alternating current whose direction varies sinusoidally with time, and changes once in each period. It is designated as an alternating current.

Suppose that the coil rotates with angular velocity ω in a magnetic field; the magnetic flux passing through the coil is time-dependent and is:

Φ = Φ0 * cos  ω t

Options for describing angular velocity:

• By frequency f: ω = 2 * π * f
• By periodic time T: ω = (2*π)/T

### Alternating Current Circuit with Ohmic Resistance R

Alternating current circuit with Ohmic resistance R

The graphic representation shows that the current is in phase with the voltage. The effective current of the alternating current refers to the current which has to have a direct current in order to be able to produce the same capacity with the same resistance.

### Alternating Current Circuit with Inductive Resistance L

The calculation of the inductive resistance is given by:

XL = Ueff / Ieff = ω * L

The current can also run after the voltage. The phase shift is:

φ = π/2 = 90°

Alternating current circuit with inductive resistance L

### Alternating Current with Capacitive Resistance C

In addition, the current can run ahead of the voltage. The phase shift is:

φ = π/2 = 90°

Alternating current with capacitive resistance C

## Review Questions

Solutions can be found below the references.

1. An electron beam is directed between the plates of a charged parallel-plate capacitor. Before entering the magnetic field of the capacitor, the direction of the electron beam is perpendicular to the drawing layer towards the observer (see figure). In which direction will the electron beam be directed?

1. Direction 1
2. Direction 2
3. Direction 3
4. Direction 4
5. Its direction won’t change; it will keep its own Direction 5.

2. Which of the following statements is false? For our household power outlet, the following alternating voltage applies:

1. The frequency of the alternating current is 50 Hz.
2. The circular frequency of the alternating current is approx. 31 /s.
3. The periodic time of the alternating current is approx. 20 ms.
4. The amplitude of the alternating current is approx. 325 V.
5. The value of the alternating current is approx. 230 V.

3. How is total capacity calculated from the series connection of two capacitors with individual capacitance C1 and C2, respectively?

1. C = 2 * (C1 + C2)
2. C = C1 + C2
3. C = 1/C1 + 1/C2
4. 1/C = 1/(C1+C2)
5. 1/C = (C1 + C2)/(C1*C2)