class 12 Notes for Electric potential

An electric potential (also called the electric field potential or the electrostatic potential) is the amount of electric potential energy that a unitary point electric charge would have if located at any point in space, and is equal to the work done by an external agent in carrying a unit of positive charge from infinity to that point without any acceleration.

According to theoretical electromagnetics, electric potential is a scalar quantity denoted by V, equal to the electric potential energy of any charged particle at any location (measured in joules) divided by the charge of that particle (measured in coulombs). By dividing out the charge on the particle a remainder is obtained that is a property of the electric field itself.

This value can be calculated in either a static (time-invariant) or a dynamic (varying with time)electric field at a specific time in units of joules per coulomb (J C⁻¹), or volts (V). The electric potential at infinity is assumed to be zero.

A generalized electric scalar potential is also used in electrodynamics when time-varying electromagnetic fields are present, but this can not be so simply calculated. The electric potential and the magnetic vector potential together form a four vector, so that the two kinds of potential are mixed under Lorentz transformations.

Introduction[edit]

Classical mechanics explores concepts such as force, energy, potential etc. Force and potential energy are directly related. A net force acting on any object will cause it to accelerate. As an object moves in the direction in which the force accelerates it, its potential energy decreases: the gravitational potential energy of a cannonball at the top of a hill is greater than at the base of the hill. As it rolls downhill its potential energy decreases, being translated to motion, inertial (kinetic) energy.

It is possible to define the potential of certain force fields so that the potential energy of an object in that field depends only on the position of the object with respect to the field. Two such force fields are the gravitational field and an electric field (in the absence of time-varying magnetic fields). Such fields must affect objects due to the intrinsic properties of the object (e.g., mass or charge) and the position of the object.

Objects may possess a property known as electric charge and an electric field exerts a force on charged objects. If the charged object has a positive charge the force will be in the direction of the electric field vector at that point while if the charge is negative the force will be in the opposite direction. The magnitude of the force is given by the quantity of the charge multiplied by the magnitude of the electric field vector.

Electrostatics[edit]

Main article: Electrostatics

The electric potential at a point r in a static electric field E is given by the line integral

where C is an arbitrary path connecting the point with zero potential to r. When the curl ∇ × E is zero, the line integral above does not depend on the specific path C chosen but only on its endpoints. In this case, the electric field is conservative and determined by the gradient of the potential:

Then, by Gauss's law, the potential satisfies Poisson's equation:

where ρ is the total charge density (including bound charge) and ∇· denotes the divergence.

The concept of electric potential is closely linked with potential energy. A test charge q has an electric potential energy U_E given by

The potential energy and hence also the electric potential is only defined up to an additive constant: one must arbitrarily choose a position where the potential energy and the electric potential are zero.

These equations cannot be used if the curl ∇ × E ≠ 0, i.e., in the case of a nonconservative electric field (caused by a changing magnetic field; see Maxwell's equations). The generalization of electric potential to this case is described below.

Electric potential due to a point charge[edit]

The electric potential created by a charge Q isV=Q/(4πε_or). Different values of Q will make different values of electric potential V (shown in the image).

The electric potential created by a point charge Q, at a distance r from the charge (relative to the potential at infinity), can be shown to be

where ε₀ is the electric constant (permittivity of vacuum). This is known as the Coulomb potential.

The electric potential due to a system of point charges is equal to the sum of the point charges' individual potentials. This fact simplifies calculations significantly, since addition of potential (scalar) fields is much easier than addition of the electric (vector) fields.

The equation given above for the electric potential (and all the equations used here) are in the forms required by SI units. In some other (less common) systems of units, such as CGS-Gaussian, many of these equations would be altered.

Generalization to electrodynamics[edit]

When time-varying magnetic fields are present (which is true whenever there are time-varying electric fields and vice versa), it is not possible to describe the electric field simply in terms of a scalar potential V because the electric field is no longer conservative:  is path-dependent because  (Faraday's law of induction).

Instead, one can still define a scalar potential by also including the magnetic vector potential A. In particular, A is defined to satisfy:

where B is the magnetic field. Because the divergence of the magnetic field is always zero due to the absence of magnetic monopoles, such anA can always be found. Given this, the quantity

is a conservative field by Faraday's law and one can therefore write

where V is the scalar potential defined by the conservative field F.

The electrostatic potential is simply the special case of this definition where A is time-invariant. On the other hand, for time-varying fields,

unlike electrostatics.

Units[edit]

The SI unit of electric potential is the volt (in honor of Alessandro Volta), which is why a difference in electric potential between two points is known as voltage. Older units are rarely used today. Variants of the centimeter gram second system of units included a number of different units for electric potential, including the abvolt and the statvolt.

Galvani potential versus electrochemical potential[edit]

Main articles: Galvani potential, Electrochemical potential and Fermi level

Inside metals (and other solids and liquids), the energy of an electron is affected not only by the electric potential, but also by the specific atomic environment that it is in. When a voltmeter is connected between two different types of metal, it measures not the electric potential difference, but instead the potential difference corrected for the different atomic environments.^[1] The quantity measured by a voltmeter is calledelectrochemical potential or fermi level, while the pure unadjusted electric potential is sometimes called Galvani potential. The terms "voltage" and "electric potential" are a bit ambiguous in that, in practice, they can refer to either of these in different contexts.

thanks for reading 

class 12 Notes for Electrostatics (Part 5)

'Static' electricity[edit]

Main article: Static electricity

Lightning over Oradea in Romania

Before the year 1832, when Michael Faraday published the results of his experiment on the identity of electricities, physicists thought "static electricity" was somehow different from other electrical charges. Michael Faraday proved that the electricity induced from the magnet, voltaic electricity produced by a battery, and static electricity are all the same.

Static electricity is usually caused when certain materials are rubbed against each other, like wool on plastic or the soles of shoes on carpet. The process causes electrons to be pulled from the surface of one material and relocated on the surface of the other material.

A static shock occurs when the surface of the second material, negatively charged with electrons, touches a positively charged conductor, or vice versa.

Static electricity is commonly used in xerography, air filters, and some automotive paints. Static electricity is a buildup of electric charges on two objects that have become separated from each other. Small electrical components can easily be damaged by static electricity. Component manufacturers use a number of antistatic devices to avoid this.

Static electricity and chemical industry[edit]

When different materials are brought together and then separated, an accumulation of electric charge can occur which leaves one material positively charged while the other becomes negatively charged. The mild shock that you receive when touching a grounded object after walking on carpet is an example of excess electrical charge accumulating in your body from frictional charging between your shoes and the carpet. The resulting charge build-up upon your body can generate a strong electrical discharge. Although experimenting with static electricity may be fun, similar sparks create severe hazards in those industries dealing with flammable substances, where a small electrical spark may ignite explosive mixtures with devastating consequences.

A similar charging mechanism can occur within low conductivity fluids flowing through pipelines—a process called flow electrification. Fluids which have low electrical conductivity (below 50 picosiemens per meter), are called accumulators. Fluids having conductivities above 50 pS/m are called non-accumulators. In non-accumulators, charges recombine as fast as they are separated and hence electrostatic charge generation is not significant. In the petrochemical industry, 50 pS/m is the recommended minimum value of electrical conductivity for adequate removal of charge from a fluid.

An important concept for insulating fluids is the static relaxation time. This is similar to the time constant (tau) within an RC circuit. For insulating materials, it is the ratio of the static dielectric constant divided by the electrical conductivity of the material. For hydrocarbon fluids, this is sometimes approximated by dividing the number 18 by the electrical conductivity of the fluid. Thus a fluid that has an electrical conductivity of 1 pS/cm (100 pS/m) will have an estimated relaxation time of about 18 seconds. The excess charge within a fluid will be almost completely dissipated after 4 to 5 times the relaxation time, or 90 seconds for the fluid in the above example.

Charge generation increases at higher fluid velocities and larger pipe diameters, becoming quite significant in pipes 8 inches (200 mm) or larger. Static charge generation in these systems is best controlled by limiting fluid velocity. The British standard BS PD CLC/TR 50404:2003 (formerly BS-5958-Part 2) Code of Practice for Control of Undesirable Static Electricity prescribes velocity limits. Because of its large impact on dielectric constant, the recommended velocity for hydrocarbon fluids containing water should be limited to 1 m/s.

Bonding and earthing are the usual ways by which charge buildup can be prevented. For fluids with electrical conductivity below 10 pS/m, bonding and earthing are not adequate for charge dissipation, and anti-static additives may be required.

Applicable standards[edit]

1.BS PD CLC/TR 50404:2003 Code of Practice for Control of Undesirable Static Electricity

2.NFPA 77 (2007) Recommended Practice on Static Electricity

3.API RP 2003 (1998) Protection Against Ignitions Arising Out of Static, Lightning, and Stray Currents

Electrostatic induction in commercial applications[edit]

Electrostatic induction was used in the past to build high-voltage generators known as Influence machines. The main component that emerged in theses times is the capacitor. Electrostatic induction is also used for electro-mechanic precipitation or projection.In such technologies, charged particles of small sizes are collected or deposited intentionally on surfaces. Applications range from Electrostatic precipitator to Spray painting or Inkjet printing. Recently a new Wireless power Transfer Technology has been based on electrostatic induction between oscillating distant dipoles.

this is my last part for the topic "Electrostatics"

 thanks for reading

class 12 Notes for Electrostatics (Part 4)

Triboelectric series[edit]

Main article: Triboelectric effect

The triboelectric effect is a type of contact electrification in which certain materials become electrically charged when they are brought into contact with a different material and then separated. One of the materials acquires a positive charge, and the other acquires an equal negative charge. The polarity and strength of the charges produced differ according to the materials, surface roughness, temperature, strain, and other properties. Amber, for example, can acquire an electric charge by friction with a material like wool. This property, first recorded by Thales of Miletus, was the first electrical phenomenon investigated by man. Other examples of materials that can acquire a significant charge when rubbed together include glass rubbed with silk, and hard rubber rubbed with fur.

Electrostatic generators[edit]

Main article: Electrostatic generator

The presence of surface charge imbalance means that the objects will exhibit attractive or repulsive forces. This surface charge imbalance, which yields static electricity, can be generated by touching two differing surfaces together and then separating them due to the phenomena ofcontact electrification and the triboelectric effect. Rubbing two nonconductive objects generates a great amount of static electricity. This is not just the result of friction; two nonconductive surfaces can become charged by just being placed one on top of the other. Since most surfaces have a rough texture, it takes longer to achieve charging through contact than through rubbing. Rubbing objects together increases amount of adhesive contact between the two surfaces. Usually insulators, e.g., substances that do not conduct electricity, are good at both generating, and holding, a surface charge. Some examples of these substances are rubber, plastic, glass, and pith. Conductive objects only rarely generate charge imbalance except, for example, when a metal surface is impacted by solid or liquid nonconductors. The charge that is transferred during contact electrification is stored on the surface of each object. Static electric generators, devices which produce very high voltage at very low current and used for classroom physics demonstrations, rely on this effect.

Note that the presence of electric current does not detract from the electrostatic forces nor from the sparking, from the corona discharge, or other phenomena. Both phenomena can exist simultaneously in the same system.

Charge neutralization[edit]

Natural electrostatic phenomena are most familiar as an occasional annoyance in seasons of low humidity, but can be destructive and harmful in some situations (e.g. electronics manufacturing). When working in direct contact with integrated circuit electronics (especially delicateMOSFETs), or in the presence of flammable gas, care must be taken to avoid accumulating and suddenly discharging a static charge (seeelectrostatic discharge).

Charge induction[edit]

Main article: Electrostatic induction

Charge induction occurs when a negatively charged object repels (the negatively charged) electrons from the surface of a second object. This creates a region in the second object that is more positively charged. An attractive force is then exerted between the objects. For example, when a balloon is rubbed, the balloon will stick to the wall as an attractive force is exerted by two oppositely charged surfaces (the surface of the wall gains an electric charge due to charge induction, as the free electrons at the surface of the wall are repelled by the negative balloon, creating a positive wall surface, which is subsequently attracted to the surface of the balloon). You can explore the effect with a simulation of theballoon and static electricity.

now we will meet on part 5.

class 12 Notes for Electrostatics (Part 3)

Electrostatic approximation[edit]

The validity of the electrostatic approximation rests on the assumption that the electric field is irrotational:

From Faraday's law, this assumption implies the absence or near-absence of time-varying magnetic fields:

In other words, electrostatics does not require the absence of magnetic fields or electric currents. Rather, if magnetic fields or electric currentsdo exist, they must not change with time, or in the worst-case, they must change with time only very slowly. In some problems, both electrostatics and magnetostatics may be required for accurate predictions, but the coupling between the two can still be ignored. Electrostatics and magnetostatics can both be seen as Galinean limits for electromagnetism.^[2]

Electrostatic potential[edit]

Because the electric field is irrotational, it is possible to express the electric field as the gradient of a scalar function,, called the electrostatic potential (also known as the voltage). An electric field, , points from regions of high electric potential to regions of low electric potential, expressed mathematically as

The Gradient Theorem can be used to establish that the electrostatic potential is the amount of work per unit charge required to move a charge from point  to point  is the following line integral:

From these equations, we see that the electric potential is constant in any region for which the electric field vanishes (such as occurs inside a conducting object).

Electrostatic energy[edit]

Main articles: Electric potential energy and Energy density

A single test particle's potential energy, , can be calculated from a line integral of the work, . We integrate from a point at infinity, and assume a collection of  particles of charge , are already situated at the points . This potential energy (in Joules) is:

where  is the distance of each charge  from the test charge , which situated at the point , and  is the electric potential that would be at  if the test charge were not present. If only two charges are present, the potential energy is . The total electric potential energy due a collection of N charges is calculating by assembling these particles one at a time:

where the following sum from, j = 1 to N, excludes i = j:

This electric potential,  is what would be measured at  if the charge  were missing. This formula obviously excludes the (infinite) energy that would be required to assemble each point charge from a disperse cloud of charge. The sum over charges can be converted into an integral over charge density using the prescription :

,

This second expression for electrostatic energy uses the fact that the electric field is the negative gradient of the electric potential, as well asvector calculus identities in a way that resembles integration by parts. These two integrals for electric field energy seem to indicate two mutually exclusive formulas for electrostatic energy density, namely  and ; they yield equal values for the total electrostatic energy only if both are integrated over all space.

Electrostatic pressure[edit]

On a conductor, a surface charge will experience a force in the presence of an electric field. This force is the average of the discontinuous electric field at the surface charge. This average in terms of the field just outside the surface amounts to:

,

This pressure tends to draw the conductor into the field, regardless of the sign of the surface charge.

rest continue in part 4.

class 12 Notes for Electrostatics (Part 2)

Electric field

Electric field lines are useful for visualizing the electric field. Field lines begin on positive charge and terminate on negative charge. Electric field lines are parallel to the direction of the electric field, and the density of these field lines is a measure of the magnitude of the electric field at any given point. The electric field, , (in units of voltsper meter) is a vector field that can be defined everywhere, except at the location of point charges (where it diverges to infinity). It is convenient to place a hypothetical test charge at a point (where no charges are present). By Coulomb's Law, this test charge will experience a force that can be used to define the electric field as follow

(See the Lorentz equation if the charge is not stationary.)

Consider a collection of  particles of charge , located at points  (called source points), the electric field at  (called the field point) is:

where  is the displacement vector from a source point to the field point , and  is a unit vector that indicates the direction of the field. For a single point charge at the origin, the magnitude of this electric field is  and points away from that charge is positive. That fact that the force (and hence the field) can be calculated by summing over all the contributions due to individual source particles is an example of the superposition principle. The electric field produced by a distribution of charges is given by the volume charge density  and can be obtained by converting this sum into a triple integral:

The electrostatic field (lines with arrows) of a nearby positive charge (+)causes the mobile charges in conductive objects to separate due to electrostatic induction. Negative charges (blue) are attracted and move to the surface of the object facing the external charge. Positive charges (red) are repelled and move to the surface facing away. These induced surface charges are exactly the right size and shape so their opposing electric field cancels the electric field of the external charge throughout the interior of the metal. Therefore the electrostatic field everywhere inside a conductive object is zero, and the electrostatic potential is constant.

Gauss's law[edit]

Gauss's law states that " the total electric flux through any closed surface in free space of any shape drawn in an electric field is proportional to the total electric charge enclosed by the surface." Mathematically, Gauss's law takes the form of an integral equation:

where  is a volume element. If the charge is distributed over a surface or along a line, replace  by  or . The Divergence Theorem allows Gauss's Law to be written in differential form:

where  is the divergence operator.

Poisson and Laplace equations[edit]

The definition of electrostatic potential, combined with the differential form of Gauss's law (above), provides a relationship between the potential Φ and the charge density ρ:

This relationship is a form of Poisson's equation. In the absence of unpaired electric charge, the equation becomes Laplace's equation:

now rest continue in part 3.