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Electric Field Problems Pdf Free


We present two models of the electric field for a canonical problem in electric discharge machining. In particular, an analytical solution based on optimal parameter estimation is discussed, followed by a comparison with numerical solutions based on finite elements and Galerkin boundary elements. The problem is interesting because the structure of the field near the sharp asperity is a critical parameter in realistic models of the electric discharge machining process.




electric field problems pdf free



When a charge is placed in a location with a non-zero electric field, a force will act on it. The magnitude of this force is given by Coulomb's law. If the charge moves, the electric field would be doing work on the electric charge. Thus we can speak of electric potential at a certain point in space, which is equal to the work done by an external agent in carrying a unit of positive charge from an arbitrarily chosen reference point to that point without any acceleration and is typically measured in volts.


The concept of the electric field was introduced by Michael Faraday. An electric field is created by a charged body in the space that surrounds it, and results in a force exerted on any other charges placed within the field. The electric field acts between two charges in a similar manner to the way that the gravitational field acts between two masses, and like it, extends towards infinity and shows an inverse square relationship with distance.[33] However, there is an important difference. Gravity always acts in attraction, drawing two masses together, while the electric field can result in either attraction or repulsion. Since large bodies such as planets generally carry no net charge, the electric field at a distance is usually zero. Thus gravity is the dominant force at distance in the universe, despite being much weaker.[34]


Electric potential is a scalar quantity, that is, it has only magnitude and not direction. It may be viewed as analogous to height: just as a released object will fall through a difference in heights caused by a gravitational field, so a charge will 'fall' across the voltage caused by an electric field.[48] As relief maps show contour lines marking points of equal height, a set of lines marking points of equal potential (known as equipotentials) may be drawn around an electrostatically charged object. The equipotentials cross all lines of force at right angles. They must also lie parallel to a conductor's surface, otherwise this would produce a force that will move the charge carriers to even the potential of the surface.


This relationship between magnetic fields and currents is extremely important, for it led to Michael Faraday's invention of the electric motor in 1821. Faraday's homopolar motor consisted of a permanent magnet sitting in a pool of mercury. A current was allowed through a wire suspended from a pivot above the magnet and dipped into the mercury. The magnet exerted a tangential force on the wire, making it circle around the magnet for as long as the current was maintained.[51]


Experimentation by Faraday in 1831 revealed that a wire moving perpendicular to a magnetic field developed a potential difference between its ends. Further analysis of this process, known as electromagnetic induction, enabled him to state the principle, now known as Faraday's law of induction, that the potential difference induced in a closed circuit is proportional to the rate of change of magnetic flux through the loop. Exploitation of this discovery enabled him to invent the first electrical generator in 1831, in which he converted the mechanical energy of a rotating copper disc to electrical energy.[51] Faraday's disc was inefficient and of no use as a practical generator, but it showed the possibility of generating electric power using magnetism, a possibility that would be taken up by those that followed on from his work.


Electrical power is usually generated by electro-mechanical generators driven by steam produced from fossil fuel combustion, or the heat released from nuclear reactions; or from other sources such as kinetic energy extracted from wind or flowing water. The modern steam turbine invented by Sir Charles Parsons in 1884 today generates about 80 percent of the electric power in the world using a variety of heat sources. Such generators bear no resemblance to Faraday's homopolar disc generator of 1831, but they still rely on his electromagnetic principle that a conductor linking a changing magnetic field induces a potential difference across its ends.[55] The invention in the late nineteenth century of the transformer meant that electrical power could be transmitted more efficiently at a higher voltage but lower current. Efficient electrical transmission meant in turn that electricity could be generated at centralised power stations, where it benefited from economies of scale, and then be despatched relatively long distances to where it was needed.[56][57]


Some organisms, such as sharks, are able to detect and respond to changes in electric fields, an ability known as electroreception,[78] while others, termed electrogenic, are able to generate voltages themselves to serve as a predatory or defensive weapon; these are electric fish in different orders.[3] The order Gymnotiformes, of which the best known example is the electric eel, detect or stun their prey via high voltages generated from modified muscle cells called electrocytes.[3][4] All animals transmit information along their cell membranes with voltage pulses called action potentials, whose functions include communication by the nervous system between neurons and muscles.[79] An electric shock stimulates this system, and causes muscles to contract.[80] Action potentials are also responsible for coordinating activities in certain plants.[79]


Electrostatics is the subfield of electromagnetics describing an electric field caused by static (nonmoving) charges. Starting with free space, assuming a space charge density, , the relationship with the electric field, , is:


This relationship means that, in electrostatics, the space charge density acts like a volume source. The charge-field relationship is not enough, but Maxwell's equations imply the additional requirement that the electric field is irrotational (curl free):


An idealized dielectric material is characterized by the fact that it does not have any free charges but instead has bound charges. At the microscopic level, these bound charges can be displaced by an external electric field and result in induced electric dipoles. These induced electric dipoles are pairs of positive and negative charges that in some way align with the electric field. This results in an electric field inside a dielectric material that is different from that of free space. To get a macroscopic description of this phenomenon, it is convenient to introduce a polarization vector field , and a polarization charge density, . They are related by:


To fully describe electrostatics phenomena, the condition that the electric field is irrotational (Faraday's law) still needs to hold. Since this condition is encoded in the electric potential, the equations of electrostatics can be combined into a single equation:


The field around an object of higher permittivity than its surroundings. The images show a dielectric object, , surrounded by air, , between two capacitor plates (not shown). The top and bottom electrodes have a positive and negative electric potential, respectively. The image to the left shows the magnitude of the electric field, , in color and its direction with arrows. The image to the right shows the magnitude of the electric displacement field, , in color and its direction with arrows. Red and blue represent a high and low magnitude value, respectively.


The field around an object of lower permittivity than its surroundings. The images show an air cavity, , surrounded by a dielectric material, , between two capacitor plates. The top and bottom electrodes have a positive and negative electric potential, respectively. The image to the left shows the magnitude of the electric field, , in color and its direction with arrows. The image to the right shows the magnitude of the electric displacement field, , in color and its direction with arrows. Red and blue represent a high and low magnitude value, respectively.


Gauss's law and Faraday's law can be seen as specifying conditions on the divergence and curl of the electric field, respectively. According to Helmholtz's theorem, this determines the electric field up to a constant. As a side note, this unknown constant is what ultimately makes it necessary to specify a ground level for the electric potential. At material interfaces, the divergence condition implies a condition on the normal component of the field and the curl condition implies a condition on the tangential component of the field. Material interfaces represent discontinuities, and to more easily understand what conditions to impose on boundaries, the corresponding integral forms are usually used. The boundary formulations are then derived by taking the limit of a shrinking closed surface (Gauss's law) and the limit of a shrinking closed contour (Faraday's law), respectively, which enclose a portion of the material interface.


Two charges of opposite signs are enclosed within the green and blue spheres. The electric field is visualized by cyan flux lines. The red sphere has no enclosed charge. Gauss's law states that the flux of the electric displacement field through the green sphere equals the enclosed charge, and similarly for the blue sphere. The red sphere with no enclosed charge has as many incoming field lines as outgoing ones, corresponding to a net zero flux.


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