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  • 7/31/2019 Group II Handout

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    ELECTRIC FIELD

    An electric field surrounds electrically charged particles and time-varying magnetic fields. The

    electric field depicts the force exerted on other electrically charged objects by the electricallycharged particle the field is surrounding. The concept of an electric field was introduced

    by Michael Faraday.

    The electric field is a vector field with SI units of newtons per coulomb (N C1

    ) or,equivalently, volts per meter (V m

    1). The SI base units of the electric field are kgms

    3A

    1.

    The strength or magnitude of the field at a given point is defined as the force that would beexerted on a positive test charge of 1 coulomb placed at that point; the direction of the field is

    given by the direction of that force. Electric fields contain electrical energy with energydensity proportional to the square of the field amplitude. The electric field is to charge as

    gravitational acceleration is to mass and force density is to volume.

    A field is associated with a region in space, and we say that a field exists in the region if there is

    a physical phenomenon associated with points in that region. In other words, we can talk of thefields of any physical quantities as being a description of how the quantity varies from one pointto another in the region of the field.

    2 Important Characteristics of Electric Field:Directionthe direction of an electric field at a point is defined as the direction of the force

    upon a positive chargeIntensityforce experienced by a positive charge of 1 Coulomb placed at that point

    http://www.absoluteastronomy.com/topics/Electric_chargehttp://www.absoluteastronomy.com/topics/Magnetic_fieldhttp://www.absoluteastronomy.com/topics/Michael_Faradayhttp://www.absoluteastronomy.com/topics/Vector_fieldhttp://www.absoluteastronomy.com/topics/Sihttp://www.absoluteastronomy.com/topics/Coulombhttp://www.absoluteastronomy.com/topics/Volthttp://www.absoluteastronomy.com/topics/Metrehttp://www.absoluteastronomy.com/topics/Field_strengthhttp://www.absoluteastronomy.com/topics/Energy_densityhttp://www.absoluteastronomy.com/topics/Energy_densityhttp://www.absoluteastronomy.com/topics/Accelerationhttp://www.absoluteastronomy.com/topics/Force_densityhttp://www.absoluteastronomy.com/topics/Force_densityhttp://www.absoluteastronomy.com/topics/Accelerationhttp://www.absoluteastronomy.com/topics/Energy_densityhttp://www.absoluteastronomy.com/topics/Energy_densityhttp://www.absoluteastronomy.com/topics/Energy_densityhttp://www.absoluteastronomy.com/topics/Field_strengthhttp://www.absoluteastronomy.com/topics/Metrehttp://www.absoluteastronomy.com/topics/Volthttp://www.absoluteastronomy.com/topics/Coulombhttp://www.absoluteastronomy.com/topics/Sihttp://www.absoluteastronomy.com/topics/Vector_fieldhttp://www.absoluteastronomy.com/topics/Michael_Faradayhttp://www.absoluteastronomy.com/topics/Magnetic_fieldhttp://www.absoluteastronomy.com/topics/Electric_charge
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    COULOMBS LAW

    Coulomb stated that the force between two very small objects separated in a vacuum or free

    space by a distance which is large compared to the square of the distance between them, or

    where Q1 and Q2 are the positive or negative quantities of charge, R, is the separation, and k isproportionality constant. If the International System of units (SI) is used, Q is measured in

    coulombs (C), R is in meters (m), and the force should ne newtons (N). This will be achieved ifthe constant of proportionality k is written as

    The new constant is called the permittivity of free space and has the magnitude, measured infarads per meter (F/m).

    Coulombs Law is now

    FIGURE 1.If Q1 and Q2 have like signs the vector force F2on Q2 is in

    the same direction as the vector R12.

    Let the vector r1 locate Q1 while r2 locates Q2. Then the vector R12 = r2r2 represents thedirected line segment from Q1 to Q2, as shown in Figure 1. The vector F2 is the force on Q2 and

    is shown for the case where Q1 and Q2have the same sign. The vector form of Coulombs law is

    Where a12 = a unit vector in the direction of R12, or

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    Examples:

    Exercise:

    1.

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    ELECTRIC FIELD INTENSITY

    Electric Field Intensity is the force per unit charge when placed in the electric field. Electric field

    intensity is a vector quantity; it has both magnitude and direction and is denoted by E. Themagnitude of the electric field strength is defined in terms of how it is measured. Suppose that an

    electric charge can be denoted by the symbol Q. This electric charge creates an electric field;

    since Qis the source of the electric field, we will refer to it as the source charge. The strength ofthe source charge's electric field could be measured by any other charge placed somewhere in itssurroundings. The charge that is used to measure the electric field strength is referred to as a test

    charge (denoted by Qt) since it is used to testthe field strength. When placed within the electricfield, the test charge will experience an electric force (F t) - either attractive or repulsive. The

    magnitude of the electric field is simply defined as the force per charge on the test charge.

    Electric Field Intensity due to a Point Charge

    Example:

    Find the E at P(1, 1, 1) caused by four identical 3-nC charges located at P1(1, 1, 0), P2(-1, 1, 0),P3(-1, -1, 0) and P4(1, -1, 0), as shown in Figure 2.

    Figure 2.

    http://www.physicsclassroom.com/Class/1DKin/U1L1b.cfmhttp://www.physicsclassroom.com/Class/1DKin/U1L1b.cfm
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    Exercises:

    Electric Field Intensity due to a Continuous Volume Charge

    1.

    2.

    where: Q = total charge (C)v= volume charge density (C/m

    3)

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    Example:

    Exercise:

    Find the total charge contained in a 2 cm length of the electron beam shown.

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    Electric Field Intensity due to a Line Charge

    Example:Consider an infinite line parallel to thez-axis at x = 6, y=8. Find E at the general field

    point P(x, y, z)

    Solution: replace by the radial distance between the line charge and point, P, R =

    sqrt((x-6)2

    + (y-8)2

    ). Let abe the unit vector of R.

    Exercise:

    where: L= uniform line charge density (C/m) = radial distance between the line charge and point

    1.

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    Referring to Figure 2, the electric flux density is in the radial direction and has a value of

    For Q lines of flux are symmetrically directed outward from the point and pass through an

    imaginary spherical surface of area .

    The radial electric field intensity of a point charge in free space is,

    Therefore,

    Figure 2.

    The electric flux in the region between a pair

    of charged concentric spheres. The direction

    and magnitude ofD are not functions of the

    dielectric between the spheres.

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    Example:

    Exercises:

    1.

    2.

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    DIVERGENCE THEOREM

    The divergence is an operator that measures the magnitude of a vector fields source or sink at agiven point; the divergence of a vector field is a (signed) scalar. For example, for a vectorfield that denotes the velocity of air expanding as it is heated, the divergence of the velocity field

    would have a positive value because the air expands. If the air cools and contracts, the

    divergence is negative. In this specific example the divergence could be thought of as a measureof the change in density.A vector field that has zero divergence everywhere is called solenoidal.

    Letx, y, z be a system ofCartesian coordinates on a 3-dimensional Euclidean space, andlet i,j, k be the corresponding basis of unit vectors.

    The divergence of a continuously differentiable vector field F = Fxi + Fyj + Fzk is defined to bethe scalar-valued function:

    Although expressed in terms of coordinates, the result is invariant under orthogonaltransformations, as the physical interpretation suggests.

    The common notation for the divergence F is a convenient mnemonic, where the dot denotes

    an operation reminiscent of the dot product: take the components of , apply them to thecomponents ofF, and sum the results. As a result, this is considered an abuse of notation.

    Example:

    Calculate the divergence ofF=(y,xy,z).

    Solution: F1x=0,F2y=x,F3z=1divF=0+x+1=x+1.

    The divergence theorem, also known asOstrogradsky's theorem, is a result that relates the

    flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside thesurface.More precisely, the divergence theorem states that the outward flux of a vector field through a

    closed surface is equal to the volume integral of the divergence of the region inside the surface.Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of

    a region.

    Suppose Vis a subset ofRn (in the case ofn = 3, Vrepresents a volume in 3D space) which

    is compact and has a piecewise smooth boundary. IfF is a continuously differentiable vectorfield defined on a neighborhood ofV, then we have

    The left side is a volume integral over the volume V, the right side is the surface integral over the

    boundary of the volume V. Here Vis quite generally the boundary ofVoriented by outward-pointing normals, and nis the outward pointing unit normal field of the boundary V. (dS maybe used as a shorthand for ndS.) In terms of the intuitive description above, the left-hand side ofthe equation represents the total of the sources in the volume V, and the right-hand side

    represents the total flow across the boundary V.

    http://library.kiwix.org:4201/A/Operator.htmlhttp://library.kiwix.org:4201/A/Vector_field.htmlhttp://library.kiwix.org:4201/A/Vector_field.htmlhttp://library.kiwix.org:4201/A/Vector_field.htmlhttp://library.kiwix.org:4201/A/Vector_field.htmlhttp://library.kiwix.org:4201/A/Velocity.htmlhttp://library.kiwix.org:4201/A/Cartesian_coordinates.htmlhttp://library.kiwix.org:4201/A/Euclidean_space.htmlhttp://library.kiwix.org:4201/A/Basis_linear_algebra_.htmlhttp://library.kiwix.org:4201/A/Vector_field.htmlhttp://library.kiwix.org:4201/A/Orthogonal_matrix.htmlhttp://library.kiwix.org:4201/A/Orthogonal_matrix.htmlhttp://library.kiwix.org:4201/A/Dot_product.htmlhttp://en.wikipedia.org/wiki/Mikhail_Vasilievich_Ostrogradskyhttp://en.wikipedia.org/wiki/Mikhail_Vasilievich_Ostrogradskyhttp://en.wikipedia.org/wiki/Fluxhttp://en.wikipedia.org/wiki/Vector_fieldhttp://en.wikipedia.org/wiki/Surfacehttp://en.wikipedia.org/wiki/Fluxhttp://en.wikipedia.org/wiki/Volume_integralhttp://en.wikipedia.org/wiki/Divergencehttp://library.kiwix.org:4201/A/Compact_space.htmlhttp://library.kiwix.org:4201/A/Boundary_topology_.htmlhttp://library.kiwix.org:4201/A/Boundary_topology_.htmlhttp://library.kiwix.org:4201/A/Compact_space.htmlhttp://en.wikipedia.org/wiki/Divergencehttp://en.wikipedia.org/wiki/Volume_integralhttp://en.wikipedia.org/wiki/Fluxhttp://en.wikipedia.org/wiki/Surfacehttp://en.wikipedia.org/wiki/Vector_fieldhttp://en.wikipedia.org/wiki/Fluxhttp://en.wikipedia.org/wiki/Mikhail_Vasilievich_Ostrogradskyhttp://library.kiwix.org:4201/A/Dot_product.htmlhttp://library.kiwix.org:4201/A/Orthogonal_matrix.htmlhttp://library.kiwix.org:4201/A/Orthogonal_matrix.htmlhttp://library.kiwix.org:4201/A/Orthogonal_matrix.htmlhttp://library.kiwix.org:4201/A/Vector_field.htmlhttp://library.kiwix.org:4201/A/Basis_linear_algebra_.htmlhttp://library.kiwix.org:4201/A/Euclidean_space.htmlhttp://library.kiwix.org:4201/A/Cartesian_coordinates.htmlhttp://library.kiwix.org:4201/A/Velocity.htmlhttp://library.kiwix.org:4201/A/Vector_field.htmlhttp://library.kiwix.org:4201/A/Vector_field.htmlhttp://library.kiwix.org:4201/A/Vector_field.htmlhttp://library.kiwix.org:4201/A/Vector_field.htmlhttp://library.kiwix.org:4201/A/Operator.html
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    Example:

    Compute SFdS whereF=(3x+z77,y2sinx2z,xz+yex5)

    and S is surface of box

    0x1,0y3,0z2.Use outward normal n.Solution: Given the ugly nature of the vector field, it would be hard to compute this integraldirectly. However, the divergence ofF is nice:

    divF=3+2y+x.We use the divergence theorem to convert the surface integral into a triple integral

    SFdS=BdivFdVwhereB is the box

    0x1,0y3,0z2.We compute the triple integral of divF=3+2y+x over the boxB:

    SFdS=103020(3+2y+x)dzdydx=1030(6+4y+2x)dydx=10(18+18+6x)dx=36+3=39.

    Exercise:

    1.

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    CONDUCTORS

    The electrons of different types of atoms have different degrees of freedom to move around.

    Because these virtually unbound electrons are free to leave their respective atoms and floataround in the space between adjacent atoms, they are often calledfree electrons. This relative

    mobility of electrons within a material is known as electric conductivity. Conductivity () isdetermined by the types of atoms in a material (the number of protons in each atom's nucleus,

    determining its chemical identity) and how the atoms are linked together with one another.

    Materials with high electron mobility (many free electrons) are called conductors, whilematerials with low electron mobility (few or no free electrons) are called insulators.

    Conductance (G) is the ability of a material to pass electrons. Conductance is the reciprocal of

    resistance R. The factors that affect the magnitude of resistance are exactly the same forconductance, but they affect conductance in the opposite manner. Therefore, conductance is

    directly proportional to area, and inversely proportional to the length of the material. Whereas

    the symbol used to represent resistance (R) is the Greek letter omega ( ). In terms of resistanceand conductance:

    R =

    1

    , G=

    1

    G R

    Metallic Conductors

    Metallic conductors are those which allow the electricity to pass through them without

    undergoing any chemical change. For example, copper, silver etc. In metallic conductors, theconductance is due to the movement of electrons under the influence of applied electrical

    potential. The stream of electrons constitutes the current.

    In metals, the conductivity strongly depends on the number of valence electrons available peratom. In general, the electrical conductance of solids depends upon the energy gap between the

    filled valence band and next higher vacant energy band. The outermost filled energy band iscalled valencebandand the next empty band in which electrons can move is called conduction

    band. The spaces between valence band and conduction band represent energies forbidden toelectrons and are called energy gaps or forbidden zone.

    Energy Bands for Solids

    a) The insulator shows alarge energy gap

    b) The semiconductor hasonly a small energy gap

    c) The conductor exhibits noenergy gap between the valence

    and conduction bands

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    **With a field E, an electron having a charge Q = -e will experience a force

    In free space, the electron would accelerate and continuously increase its velocity. This velocity

    vd is termed as the drift velocity, and it is linearly related to the electric field intensity(E) by themobility of the electron (e in m2/V.s) in a given material so that,

    Current density (J) is linearly related to the charge density as well as to the velocity. Therefore,

    where e is the free electron charge density, a negative value. The relationship between J and Efor metallic conductor, however, is also specified by the conductivity

    where is measured in Siemens per meter (S/m). One siemens is (1S) is the basic unit ofconductance in the SI system and is equal to one ampere per volt (A/V). Formerly, the unit of

    conductance was called the mho and was symbolized by . The conductivity then may be

    expressed in terms of charge density and the electron mobility.

    **when J and E are uniform, then

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    Example:

    1.

    2.

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    Semiconductors

    Asemiconductor is a material with electrical conductivity intermediate in magnitude betweenthat of a conductor and an insulator.

    CHARGE CARRIERS IN SEMICONDUCTORS

    There are two types of current carriers present in a semiconductor material, electrons and holes.At high temperature, electrons move from valance band to conduction band and as a result a

    vacancy is created in the valence band at a place where an electron was present before shifting toconduction band. The vacancy is a hole and is seat of positive charge having the same value of

    electron. Therefore the electrical conduction in semiconductors is due to motion of electrons inconduction band and also due to motion of holes in valence band.

    Both carrier move in an electric field, and they move in opposite directions; hence eachcontributes a component of the total current which is in the same direction as that provided by

    the other. The conductivity is therefore a function of both hole and electron concentration andmobilities,

    Mobility constants for semiconductors at 300K:

    Silicon (Si) e = 0.12 m2/Vs

    h = 0.025 m2/Vs

    Germanium (Ge) e = 0.36 m

    2

    /Vsh = 0.17 m2/Vs

    Exercise:

    1.

    http://en.wikipedia.org/wiki/Electrical_conductivityhttp://en.wikipedia.org/wiki/Electrical_Conductorhttp://en.wikipedia.org/wiki/Insulator_(electrical)http://en.wikipedia.org/wiki/Insulator_(electrical)http://en.wikipedia.org/wiki/Electrical_Conductorhttp://en.wikipedia.org/wiki/Electrical_conductivity
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    DIELECTRICS

    Dielectrics are non-conducting substances, they have no charge carriers or no free electrons.Rather, they are bound in place by atomic and molecular forces and can only shift positions

    slightly in response to external fields. They are called bound charges, in contrast to the free

    charges that determine conductivity. The bound charges can be treated as any other sources ofthe electrostatic field.

    If an external field is applied, it turns out that charges are induced on the surface which in turnproduces a field and opposes the external field. The opposing field does not exactly cancel theexternal field but only reduces it.

    The characteristic which all dielectric materials have in common is their ability to store electric

    energy. This storage takes place by means of a shift in the relative positions of the internal,bound positive and negative charges against the normal molecular and atomic forces.

    Dipole Moment

    The electric dipole moment (p) of anything be it an atom stretched in an external electricfield, a polar molecule, or two oppositely charged metal spheres is defined as the product ofcharge and separation.

    where Q is the positive one of the two bound charges composing the dipole, and d is the vector

    from the negative to the positive charge. The units ofp are coulombs-meters.

    Polarization of Dielectrics

    Dielectric polarization arises from the electrical response of individual molecules of a mediumand may be classified as electronic, atomic, orientation, and space-charge or interfacial

    polarization, according to the mechanism involved.

    The polarization (P) is the dipole moment per unit volume, with the SI unit of coulomb per

    square meter, or

    Different materials polarize to different degrees which is represented by the quantity e(chi sub e) known as the electric susceptibilitybut for most every material, the stronger thefield (E), the greater the polarization (P). Add a constant of proportionality and we're all set.

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    The electric susceptibility is a dimensionless parameter that varies with material. The electric

    susceptibility e of a dielectric material is a measure of how easily it polarizes in response to anelectric field. This, in turn, determines the electric permittivity of the material and thus

    influences many other phenomena in that medium, from the capacitance of capacitors to

    the speed of light. Its value ranges from 0 for empty space to whatever. The constant ofproportionality (epsilon nought) is known as the permittivity of free space.

    The bound charge within the closed surface is taken through the integral:

    And the total enclosed charge, bound charges plus free charges:

    Theelectric flux densityD is related to the polarization density P by

    The relative permittivity or dielectric constant of the material is another dimensionless

    quantity and is equal to (1 +Xe). And, , is thepermittivity.

    Example:

    A slab of dielectric material has a relative dielectric constant of 3.8 and contains a uniform

    electric flux density of 8nC/m2. If the material is lossless, find: (a) E; (b) P

    http://en.wikipedia.org/wiki/Polarization_densityhttp://en.wikipedia.org/wiki/Permittivityhttp://en.wikipedia.org/wiki/Capacitorshttp://en.wikipedia.org/wiki/Speed_of_lighthttp://en.wikipedia.org/wiki/Electric_displacementhttp://en.wikipedia.org/wiki/Electric_displacementhttp://en.wikipedia.org/wiki/Electric_displacementhttp://en.wikipedia.org/wiki/Electric_displacementhttp://en.wikipedia.org/wiki/Speed_of_lighthttp://en.wikipedia.org/wiki/Capacitorshttp://en.wikipedia.org/wiki/Permittivityhttp://en.wikipedia.org/wiki/Polarization_density
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    CAPACITANCE

    The capacitance (C) of an electrostatic system is the ratio of the quantity of charge separated (Q)

    to the potential difference applied (V).

    where d is the separation, S is the area, is the charge density and is the permittivity.

    Energy Stored in a Capacitor (WE)

    Example:

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    Exercise:

    Total Capacitance in Parallel

    Total Capacitance in Series

    Charge Distribution and Voltage Distribution when Capacitors are in Parallel

    Charge Distribution and Voltage Distribution when Capacitors are in Series

    1.

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    Example:

    1.The total capacitance of two capacitors is 0.03F when joined in series and 0.16F when

    connected in parallel. Find the capacitance of each capacitor.

    2. Three capacitors are connected in series across a 135-volts supply, the voltages across them

    are 30, 45, & 60 and the charge in each is 4500C. Find the capacitance of each capacitor and

    that of the combination.

    TABLE 1. Conductivity for a Number of Metallic Conductors

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    TABLE 2. Relative Permittivity for Common Insulating and Dielectric Materials