The electromagnetic spectrum extends from below frequencies used for modern radio (at the long-wavelength end) through gamma radiation (at the short-wavelength end), covering wavelengths from thousands of kilometers down to a fraction the size of an atom. It is thought that the short wavelength limit is in the vicinity of the Planck length while the long wavelength limit is the size of the universe itself (see physical cosmology), although in principle the spectrum is infinite and continuous.
Electromagnetic Spectrum
The electromagnetic spectrum extends from below frequencies used for modern radio (at the long-wavelength end) through gamma radiation (at the short-wavelength end), covering wavelengths from thousands of kilometers down to a fraction the size of an atom. It is thought that the short wavelength limit is in the vicinity of the Planck length while the long wavelength limit is the size of the universe itself (see physical cosmology), although in principle the spectrum is infinite and continuous.
Salon Electronix PVT
Microwaves
The super high frequency (SHF) and extremely high frequency(EHF) of microwaves come next up the frequency scale. Microwaves are waves which are typically short enough to employ tubular metal wave guides of reasonable diameter. Microwave energy is produced with klystron and magnetron tubes, and with solid state diodes such as Gunn and IMPATT devices. Microwaves are absorbed by molecules that have a dipole moment in liquids. In a microwave oven, this effect is used to heat food. Low-intensity microwave radiation is used in Wi-Fi, although this is at intensity levels unable to cause thermal heating.
Volumetric heating, as used by microwaves, transfer energy through the material electro-magnetically, not as a thermal heat flux. The benefit of this is a more uniform heating and reduced heating time; microwaves can heat material in less than 1% of the time of conventional heating methods.
When active, the average microwave oven is powerful enough to cause interference at close range with poorly shielded electromagnetic fields such as those found in mobile medical devices and cheap consumer electronics.
Radio Frequency
Radio waves can be made to carry information by varying a combination of the amplitude, frequency and phase of the wave within a frequency band and the use of the radio spectrum is regulated by many governments through frequency allocation. When EM radiation impinges upon a conductor, it couples to the conductor, travels along it, and induces an electric current on the surface of that conductor by exciting the electrons of the conducting material. This effect (the skin effect) is used in antennas. EM radiation may also cause certain molecules to absorb energy and thus to heat up, thus causing thermal effects and sometimes burns; this is exploited in microwave ovens.
Electromagnetic spectrum
The electromagnetic spectrum extends from below frequencies used for modern radio (at the long-wavelength end) through gamma radiation (at the short-wavelength end), covering wavelengths from thousands of kilometers down to a fraction the size of an atom. It is thought that the short wavelength limit is in the vicinity of the Planck length while the long wavelength limit is the size of the universe itself (see physical cosmology), although in principle the spectrum is infinite and continuous.
Electronic Devices and Circuits
Analog circuits: Equivalent circuits (large and small-signal) of diodes, BJTs, JFETs, and MOSFETs. Simple diode circuits, clipping, clamping, rectifier. Biasing and bias stability of transistor and FET amplifiers. Amplifiers: single-and multi-stage, differential, operational, feedback and power. Analysis of amplifiers; frequency response of amplifiers. Simple op-amp circuits. Filters. Sinusoidal oscillators; criterion for oscillation; single-transistor and op-amp configurations. Function generators and wave-shaping circuits, Power supplies.
Digital circuits: of Boolean functions; logic gates digital IC families (DTL, TTL, ECL, MOS, CMOS). Combinational circuits: arithmetic circuits, code converters, multiplexers and decoders. Sequential circuits: latches and flip-flops, counters and shift-registers. Sample and hold circuits, ADCs, DACs. Semiconductor memories. Microprocessor(8085): architecture, programming, memory and I/O interfacing.
History of Electronic Engineering
Electriciansparadise -- Electricians' Guide to Wind Power
Wind is a form of solar energy. Winds are caused by the uneven heating of the atmosphere by the sun, the irregularities of the earth's surface, and rotation of the earth. Wind flow patterns are modified by the earth's terrain, bodies of water, and vegetation. Humans use this wind flow, or motion energy, for many purposes: sailing, flying a kite, and even generating electricity.
The terms wind energy or wind power describe the process by which the wind is used to generate mechanical power or electricity. Wind turbines convert the kinetic energy in the wind into mechanical power. This mechanical power can be used for specific tasks (such as grinding grain or pumping water) or a generator can convert this mechanical power into electricity.
So how do wind turbines make electricity? Simply stated, a wind turbine works the opposite of a fan. Instead of using electricity to make wind, like a fan, wind turbines use wind to make electricity. The wind turns the blades, which spin a shaft, which connects to a generator and makes electricity.
Many wind farms have sprung up in the Midwest in recent years, generating power for utilities. Farmers benefit by receiving land lease payments from wind energy project developers.
Types of Wind Turbines
Modern wind turbines fall into two basic groups: the horizontal-axis variety and the vertical-axis design, like the eggbeater-style Darrieus model, named after its French inventor. Horizontal-axis wind turbines typically either have two or three blades. These three-bladed wind turbines are operated "upwind," with the blades facing into the wind.
GE Wind Energy's 3.6 megawatt wind turbine is one of the largest prototypes ever erected. Larger wind turbines are more efficient and cost effective.
Sizes of Wind TurbinesUtility-scale turbines range in size from 100 kilowatts to as large as several megawatts. Larger turbines are grouped together into wind farms, which provide bulk power to the electrical grid.
Single small turbines, below 100 kilowatts, are used for homes, telecommunications dishes, or water pumping. Small turbines are sometimes used in connection with diesel generators, batteries, and photovoltaic systems. These systems are called hybrid wind systems and are typically used in remote, off-grid locations, where a connection to the utility grid is not available.
Inside the Wind Turbine
Anemometer: Measures the wind speed and transmits wind speed data to the controller.
Blades: Most turbines have either two or three blades. Wind blowing over the blades causes the blades to "lift" and rotate.
Brake: A disc brake, which can be applied mechanically, electrically, or hydraulically to stop the rotor in emergencies.
Controller: The controller starts up the machine at wind speeds of about 8 to 16 miles per hour (mph) and shuts off the machine at about 55 mph. Turbines do not operate at wind speeds above about 55 mph because they might be damaged by the high winds.
Gear box: Gears connect the low-speed shaft to the high-speed shaft and increase the rotational speeds from about 30 to 60 rotations per minute (rpm) to about 1000 to 1800 rpm, the rotational speed required by most generators to produce electricity. The gear box is a costly (and heavy) part of the wind turbine and engineers are exploring "direct-drive" generators that operate at lower rotational speeds and don't need gear boxes.
Generator: Usually an off-the-shelf induction generator that produces 60-cycle AC electricity.
High-speed shaft: Drives the generator.
Low-speed shaft: The rotor turns the low-speed shaft at about 30 to 60 rotations per minute.
Nacelle: The nacelle sits atop the tower and contains the gear box, low- and high-speed shafts, generator, controller, and brake. Some nacelles are large enough for a helicopter to land on.
Pitch: Blades are turned, or pitched, out of the wind to control the rotor speed and keep the rotor from turning in winds that are too high or too low to produce electricity.
Rotor: The blades and the hub together are called the rotor.
Tower: Towers are made from tubular steel, concrete, or steel lattice. Because wind speed increases with height, taller towers enable turbines to capture more energy and generate more electricity.
Wind direction: This is an "upwind" turbine, so-called because it operates facing into the wind. Other turbines are designed to run "downwind," facing away from the wind.
Wind vane: Measures wind direction and communicates with the yaw drive to orient the turbine properly with respect to the wind.
Yaw drive: Upwind turbines face into the wind; the yaw drive is used to keep the rotor facing into the wind as the wind direction changes. Downwind turbines don't require a yaw drive, the wind blows the rotor downwind.
Yaw motor: Powers the yaw drive.
Advantages and Disadvantages of Wind Energy
Wind energy offers many advantages, which explains why it's the fastest-growing energy source in the world. Research efforts are aimed at addressing the challenges to greater use of wind energy.
Advantages
Wind energy is fueled by the wind, so it's a clean fuel source. Wind energy doesn't pollute the air like power plants that rely on combustion of fossil fuels, such as coal or natural gas. Wind turbines don't produce atmospheric emissions that cause acid rain or greenhouse gasses.
Wind energy is a domestic source of energy, produced in the United States. The nation's wind supply is abundant.
Wind energy relies on the renewable power of the wind, which can't be used up. Wind is actually a form of solar energy; winds are caused by the heating of the atmosphere by the sun, the rotation of the earth, and the earth's surface irregularities.
Wind energy is one of the lowest-priced renewable energy technologies available today, costing between 4 and 6 cents per kilowatt-hour, depending upon the wind resource and project financing of the particular project.
Wind turbines can be built on farms or ranches, thus benefiting the economy in rural areas, where most of the best wind sites are found. Farmers and ranchers can continue to work the land because the wind turbines use only a fraction of the land. Wind power plant owners make rent payments to the farmer or rancher for the use of the land.
Disadvantages Wind power must compete with conventional generation sources on a cost basis. Depending on how energetic a wind site is, the wind farm may or may not be cost competitive. Even though the cost of wind power has decreased dramatically in the past 10 years, the technology requires a higher initial investment than fossil-fueled generators.
The major challenge to using wind as a source of power is that the wind is intermittent and it does not always blow when electricity is needed. Wind energy cannot be stored (unless batteries are used); and not all winds can be harnessed to meet the timing of electricity demands.
Good wind sites are often located in remote locations, far from cities where the electricity is needed.
Wind resource development may compete with other uses for the land and those alternative uses may be more highly valued than electricity generation.
Although wind power plants have relatively little impact on the environment compared to other conventional power plants, there is some concern over the noise produced by the rotor blades, aesthetic (visual) impacts, and sometimes birds have been killed by flying into the rotors. Most of these problems have been resolved or greatly reduced through technological development or by properly siting wind plants.
History of Wind Energy
Since early recorded history, people have been harnessing the energy of the wind. Wind energy propelled boats along the Nile River as early as 5000 B.C. By 200 B.C., simple windmills in China were pumping water, while vertical-axis windmills with woven reed sails were grinding grain in Persia and the Middle East. æ
Early in the twentieth century, windmills were commonly used across the Great Plains to pump water and to generate electricity.
New ways of using the energy of the wind eventually spread around the world. By the 11th century, people in the Middle East were using windmills extensively for food production; returning merchants and crusaders carried this idea back to Europe. The Dutch refined the windmill and adapted it for draining lakes and marshes in the Rhine River Delta. When settlers took this technology to the New World in the late 19th century, they began using windmills to pump water for farms and ranches, and later, to generate electricity for homes and industry.
Industrialization, first in Europe and later in America, led to a gradual decline in the use of windmills. The steam engine replaced European water-pumping windmills. In the 1930s, the Rural Electrification Administration's programs brought inexpensive electric power to most rural areas in the United States.
However, industrialization also sparked the development of larger windmills to generate electricity. Commonly called wind turbines, these machines appeared in Denmark as early as 1890. In the 1940s the largest wind turbine of the time began operating on a Vermont hilltop known as Grandpa's Knob. This turbine, rated at 1.25 megawatts in winds of about 30 mph, fed electric power to the local utility network for several months during World War II.
The popularity of using the energy in the wind has always fluctuated with the price of fossil fuels. When fuel prices fell after World War II, interest in wind turbines waned. But when the price of oil skyrocketed in the 1970s, so did worldwide interest in wind turbine generators.
The wind turbine technology R&D that followed the oil embargoes of the 1970s refined old ideas and introduced new ways of converting wind energy into useful power. Many of these approaches have been demonstrated in "wind farms" or wind power plants „ groups of turbines that feed electricity into the utility grid „ in the United States and Europe.
Today, the lessons learned from more than a decade of operating wind power plants, along with continuing R&D, have made wind-generated electricity very close in cost to the power from conventional utility generation in some locations. Wind energy is the world's fastest-growing energy source and will power industry, businesses and homes with clean, renewable electricity for many years to come.
Wind Energy Research and Development
The United States faces many challenges as it prepares to meet its energy needs in the twenty-first century. Electricity supply crises, fluctuating natural gas and gasoline prices, heightened concerns about the security of the domestic energy infrastructure and of foreign sources of supply, and uncertainties about the benefits of utility restructuring are all elements of the energy policy challenge. Wind energy is an important part of the diverse energy portfolio that is needed for a stable, reliable energy sector in the United States.
The promise of wind energy is immense; however, reaping the full benefits from this technology rests heavily on sustaining aggressive research, development, and support programs.
In order to expand wind energy's contribution to the nation, the Wind and Hydropower Technology Program's wind energy research focuses on the two elements of its mission:
Increasing the technical viability of wind systems, and
Increasing the use of wind power in the marketplace.
Lenz's Law
An induced electromotive force generates a current that induces a counter magnetic field that opposes the magnetic field generating the current.
Thus, when an external magnetic field approaches a conductor, the current that is produced in the conductor will induce a magnetic field in opposition to the approaching external magnetic field. But when the external magnetic field moves away from the conductor, the induced magnetic field in the conductor reverses direction and opposes the change in the direction of the external magnetic field.
Faraday's Law of Electromagnetic Induction
The amount of induced voltage is determined by:
1. The amount of magnetic flux
The greater the number of magnetic field lines cutting across the conductor, the greater the induced voltage.
2. The rate at which the magnetic field lines cut across the conductor
The faster the field lines cut across a conductor, or the conductor cuts across the field lines, the greater the induced voltage.
Kennelly's Delta-Star Transformation
Similarly, using admittances:YAN = YCA + YAB + (YCAYAB / YBC) = (YABYBC + YBCYCA + YCAYAB) / YBCYBN = YAB + YBC + (YABYBC / YCA) = (YABYBC + YBCYCA + YCAYAB) / YCAYCN = YBC + YCA + (YBCYCA / YAB) = (YABYBC + YBCYCA + YCAYAB) / YAB
In general terms:Zstar = (adjacent Zdelta pair product) / (sum of Zdelta)Ystar = (sum of Ydelta pair products) / (opposite Ydelta)
The above information provided by BOWest Pty Ltd Electrical & Project Engineering
Kennelly's Star-Delta Transformation
Similarly, using admittances:YAB = YANYBN / (YAN + YBN + YCN)YBC = YBNYCN / (YAN + YBN + YCN)YCA = YCNYAN / (YAN + YBN + YCN)
In general terms:Zdelta = (sum of Zstar pair products) / (opposite Zstar)Ydelta = (adjacent Ystar pair product) / (sum of Ystar)
Maximum Power Transfer Theorem
Note that power is zero for an open-circuit (zero current) and for a short-circuit (zero voltage).
Voltage SourceWhen a load resistance RT is connected to a voltage source ES with series resistance RS, maximum power transfer to the load occurs when RT is equal to RS.
Under maximum power transfer conditions, the load resistance RT, load voltage VT, load current IT and load power PT are:RT = RSVT = ES / 2IT = VT / RT = ES / 2RSPT = VT2 / RT = ES2 / 4RS
Current SourceWhen a load conductance GT is connected to a current source IS with shunt conductance GS, maximum power transfer to the load occurs when GT is equal to GS.
Under maximum power transfer conditions, the load conductance GT, load current IT, load voltage VT and load power PT are:GT = GSIT = IS / 2VT = IT / GT = IS / 2GSPT = IT2 / GT = IS2 / 4GS
Complex ImpedancesWhen a load impedance ZT (comprising variable resistance RT and variable reactance XT) is connected to an alternating voltage source ES with series impedance ZS (comprising resistance RS and reactance XS), maximum power transfer to the load occurs when ZT is equal to ZS* (the complex conjugate of ZS) such that RT and RS are equal and XT and XS are equal in magnitude but of opposite sign (one inductive and the other capacitive).
When a load impedance ZT (comprising variable resistance RT and constant reactance XT) is connected to an alternating voltage source ES with series impedance ZS (comprising resistance RS and reactance XS), maximum power transfer to the load occurs when RT is equal to the magnitude of the impedance comprising ZS in series with XT:RT = ZS + XT = (RS2 + (XS + XT)2)1÷2Note that if XT is zero, maximum power transfer occurs when RT is equal to the magnitude of ZS:RT = ZS = (RS2 + XS2)1÷2
When a load impedance ZT with variable magnitude and constant phase angle (constant power factor) is connected to an alternating voltage source ES with series impedance ZS, maximum power transfer to the load occurs when the magnitude of ZT is equal to the magnitude of ZS:(RT2 + XT2)1÷2 = ZT = ZS = (RS2 + XS2)1÷2
Joule's Law
By substitution using Ohm's Law for the corresponding voltage drop V (= IR) across the resistance:P = V2 / R = VI = I2R
Millman's Theorem (Parallel Generator Theorem)
The short-circuit currents available between points P and N due to each of the voltages E1, E2, E3, ... acting through the respective admitances Y1, Y2, Y3, ... are E1Y1, E2Y2, E3Y3, ... so the voltage between points P and N may be expressed as:VPN = SIsc / SY
Reciprocity Theorem
Superposition Theorem
Thevenin and Norton Equivalence
The open circuit, short circuit and load conditions of the Norton model are:Voc = I / YIsc = IVload = I / (Y + Yload)Iload = I - VloadY
Thevenin model from Norton model
Voltage = Current / AdmittanceImpedance = 1 / Admittance
E = I / YZ = Y -1
Norton model from Thevenin model
Current = Voltage / ImpedanceAdmittance = 1 / Impedance
I = E / ZY = Z -1
When performing network reduction for a Thevenin or Norton model, note that:- nodes with zero voltage difference may be short-circuited with no effect on the network current distribution,- branches carrying zero current may be open-circuited with no effect on the network voltage distribution.
Norton's Theorem
Norton's Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single current source and parallel resistance connected to a load. Just as with Thevenin's Theorem, the qualification of “linear” is identical to that found in the Superposition Theorem: all underlying equations must be linear (no exponents or roots).
Thevenin's Theorem
Thevenin's Theorem says you can simplify any linear circuit, regardless of complexity, to an equivalent circuit with a single voltage source and series resistance connected to a load. As in the Superposition Theorem, it must be linear. In other words, passive components such as resistors, inductors and capacitors are okay. Non-linear components such as semiconductors, do not fall under this theorem.
Kirchhoff's Laws
At any instant the sum of all the currents flowing into any circuit node is equal to the sum of all the currents flowing out of that node:
SIin = SIout
Similarly, at any instant the algebraic sum of all the currents at any circuit node is zero:
SI = 0
Kirchhoff's Voltage Law
At any instant the sum of all the voltage sources in any closed circuit is equal to the sum of all the voltage drops in that circuit:
SE = SIZ
Similarly, at any instant the algebraic sum of all the voltages around any closed circuit is zero:
SE - SIZ = 0
In 1847, G. R. Kirchhoff extended the use of Ohm's law by developing a simple concept concerning the voltages contained in a series circuit loop. Kirchhoff's voltage law states:
"The algebraic sum of the voltage drops in any closed path in a circuit and the electromotive forces in that path is equal to zero."
To state Kirchhoff's law another way, the voltage drops and voltage sources in a circuit are equal at any given moment in time. If the voltage sources are assumed to have one sign (positive or negative) at that instant and the voltage drops are assumed to have the opposite sign, the result of adding the voltage sources and voltage drops will be zero.
NOTE: The terms electromotive force and emf are used when explaining Kirchhoff's law because Kirchhoff's law is used in alternating current circuits (covered in Module 2). In applying Kirchhoff's law to direct current circuits, the terms electromotive force and emf apply to voltage sources such as batteries or power supplies.
Through the use of Kirchhoff's law, circuit problems can be solved which would be difficult, and often impossible, with knowledge of Ohm's law alone. When Kirchhoff's law is properly applied, an equation can be set up for a closed loop and the unknown circuit values can be calculated.
Morse Code
Originally created for Samuel F. B. Morse's electric telegraph in the early 1840s, Morse code was also extensively used for early radio communication beginning in the 1890s. For the first half of the twentieth century, the majority of high-speed international communication was conducted in Morse code, using telegraph lines, undersea cables, and radio circuits. However, the variable length of the Morse characters made it hard to adapt to automated circuits, so for most electronic communication it has been replaced by machine readable formats, such as Baudot code and ASCII.
The most popular current use of Morse code is by amateur radio operators, although it is no longer a requirement for amateur licensing in many countries. In the professional field, pilots and air traffic controllers are usually familiar with Morse code and require a basic understanding. Navigational aids in the field of aviation, such as VORs and NDBs, constantly transmit their identity in Morse code. Morse code is designed to be read by humans without a decoding device, making it useful for sending automated digital data in voice channels. For emergency signaling, Morse code can be sent by way of improvised sources that can be easily "keyed" on and off, making Morse code one of the most versatile methods of telecommunication in existence.
Designing Digital Logic Circuits
In general the implementation of gate circuits is referred to as Logic Synthesis, which basically can be carried out by hand, but usually some formal method by computer is applied. In this article the design methods for combinational gate circuits are briefly summarized.
The starting point for the design of a logic gate circuit is its desired functionality, having derived from the analysis of the system as a whole, the gate circuit is to make part of. The description can be stated in some algorithmic form or by logic equations, but may be summarized in the form of a table as well. The below example shows a part of such a table for a 7-segment driver that translates the binary code for the values of a decimal digit into the signals that cause the respective segments of the display to light up.
Digit Code Segments A-G
0 0000 1 1 1 1 1 1 0 -A-
1 0001 0 1 1 0 0 0 0 F B
2 0010 1 1 0 1 1 0 1 |-G-|
3 0011 1 1 1 1 0 0 1 E C
. .... . . . . . . . -D-
The implementation process starts with a logic minimization phase, to be described below, in order to simplify the function table by combining the separate terms into larger ones containing fewer variables.
Next the minimized result may be split up in smaller parts by a factorization procedure and is eventually mapped onto the available basic logic cells of the target technology. This operation is commonly referred to as Logic Optimization.[3]
Circuit design
Formal circuit design usually involves the following stages:
sometimes, writing the requirement specification after liaising with the customer
writing a technical proposal to meet the requirements of the customer specification
synthesising on paper a schematic circuit diagram, an abstract electrical or electronic circuit that will meet the specifications
calculating the component values to meet the operating specifications under specified conditions
performing simulations to verify the correctness of the design
building a breadboard or other prototype version of the design and testing against specification
making any alterations to the circuit to achieve compliance
choosing a method of construction as well as all the parts and materials to be used
presenting component and layout information to draughtspersons, and layout and mechanical engineers, for prototype production
testing or type-testing a number of prototypes to ensure compliance with customer requirements
signing and approving the final manufacturing drawings
post-design services (obsolescence of components etc.)
Computer aided design (CAD)
Electronics Theory
Circuit analysis is the study of methods of solving generally linear systems for unknown variables such as the voltage at a certain node or the current though a certain branch of a network. A common analytical tool for this is the SPICE circuit simulator.
Also important to electronics is the study and understanding of electromagnetic field theory.
Strain (mechanical) effects
Strain (mechanical) effects
Temperature effects
where T is its temperature, T0 is a reference temperature (usually room temperature), R0 is the resistance at T0, and α is the percentage change in resistivity per unit temperature. The constant α depends only on the material being considered. The relationship stated is actually only an approximate one, the true physics being somewhat non-linear, or looking at it another way, α itself varies with temperature. For this reason it is usual to specify the temperature that α was measured at with a suffix, such as α15 and the relationship only holds in a range of temperatures around the reference.[10]
Intrinsic semiconductors exhibit the opposite temperature behavior, becoming better conductors as the temperature increases. This occurs because the electrons are bumped to the conduction energy band by the thermal energy, where they can flow freely and in doing so they leave behind holes in the valence band which can also flow freely.
Extrinsic semiconductors have much more complex temperature behaviour. First the electrons (or holes) leave the donors (or acceptors) giving a decreasing resistance. Then there is a fairly flat phase in which the semiconductor is normally operated where almost all of the donors (or acceptors) have lost their electrons (or holes) but the number of electrons that have jumped right over the energy gap is negligible compared to the number of electrons (or holes) from the donors (or acceptors). Finally as the temperature increases further the carriers that jump the energy gap becomes the dominant figure and the material starts behaving like an intrinsic semiconductor
Magnetic effects
See Lorentz force for more on this and Hall effect for some other implications of a magnetic field. This equation is not a modification to Ohm's law. Rather, it is analogous in circuit analysis terms to taking into account inductance as well as resistance.
Physics of Electronics
where J is the current density (current per unit area, unlike the simpler I, units of amperes, of Ohm's law), σ is the conductivity (which can be a tensor in anisotropic materials) and E is the electric field (units of volts per meter, unlike the simpler V, units of volts, of Ohm's law). While the notation above does not explicitly depict the variables, each are vectors and each are functions of three position variables. That is, in the case of J, using cartesian coordinates, there are actually three separate equations, one for each component of the vector, each equation having three independent position variables. For example, the components of J in the x, y and z directions would be Jx(x,y,z), Jy(x,y,z) and Jz(x,y,z).
Current flowing through a uniform conductor with a uniform field appliedThe advantage of this form is that it describes electrical behaviour at a single point and does not depend on the geometry of the conductor being considered. It only depends on the material of the conductor which determines the conductivity. That this is a form of Ohm's law can be shown by considering a conductor of length l, uniform cross-sectional area a and a uniform field applied along its length.
The potential difference between two points is defined as
with the element of path along the integration of electric field vector E. For a uniform applied field and defining the voltage in the usual convention of opposite direction to the field;
Substituting current per unit area, J, for I / a (a being the cross section of the conductor), the continuum form becomes:
The electrical resistance of a uniform conductor is given, in terms of conductivity, by:
After substitution Ohm's law takes on the more familiar, yet macroscopic and averaged version:
A perfect crystal lattice, with low enough thermal motion and no deviations from periodic structure, would have no resistivity,[3] but a real metal has crystallographic defects, impurities, multiple isotopes, and thermal motion of the atoms. Electrons scatter from all of these, resulting in resistance to their flow.
Ohm's law
A voltage source, V, drives an electric current, I, through resistor, R, the three quantities obeying Ohm's law: I = V/R.Ohm's law applies to electrical circuits; it states that the current through a conductor between two points is directly proportional to the potential difference or voltage across the two points, and inversely proportional to the resistance between them.
The mathematical equation that describes this relationship is:
where I is the current