Electrostatic fields II
Eng. Narmada Ranaweera
Department of Electrical & Electronics Engineering
University of Sri Jayawardanapura
1
Electrical Flux
• Flux is a measure of the number of field lines passing through an
area. If we define area as a vector, with its direction perpendicular to
the surface, the electric flux is given by,
𝚿 = 𝛆 𝐄. 𝐀 = 𝛆 𝐄 |𝐀| 𝐜𝐨𝐬𝛉
Where θ is the angle between the electric field and the area vector.
• The more field lines pass through the area, the larger the flux.
• In SI units one line of electric flux emanates from +1C and terminates
on -1C.
• Thus, electric flux is measured in Coulombs.
2
Electrical Flux Density
• Electrical flux density D is defined as,
𝑫= 𝜺E
Where ε is called electrical permittivity of the medium.
In vacuum, ε has the value ε0 = 8.85x10-12 F/m
• We represent the electric field by drawing lines that follows the
direction of the field in the region of charge. Because of the definition
of the flux, the density of the lines represents the electric flux density
D. Therefore, these lines are also called ‘electric flux lines’.
3
Electrical flux
𝚿 = 𝐃. 𝐀 = 𝐃 𝐀 𝐜𝐨𝐬𝛉
𝚿 = න 𝑫. 𝒅𝑺
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Example
Electric flux density of a surface is given as,
ഥ = 3𝑥𝑦 𝑎ത 𝑥 + 4𝑥 𝑎ത 𝑧 𝐶/𝑚2
𝐷
What is the net flux through the surface, which lies at z=0 with
0<=x<=5m and 0<=y<=3m?
[Ans. 150C]
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Gauss’s Law
• Gauss's law states that the total electric flux Ψ through any closed
surface is equal to the total charge enclosed by that surface.
6
Gauss’s Law
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Applications of Gauss’s Law
• Gauss's law provides an easy means of finding E or D for symmetrical
charge distributions such as a point charge, an infinite line charge, an
infinite cylindrical surface charge, and a spherical distribution of charge.
• For a symmetric charge distribution, we can construct a mathematical
closed surface (known as a Gaussian surface).
• The surface is chosen such that D is normal or tangential to the Gaussian
surface.
• When D is normal to the surface, D.dS = D dS because D is constant on the
surface. (direction of the dS vector is normal to the surface)
• When D is tangential to the surface, D.dS = 0.
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Point Charge
• Suppose a point charge Q is located at the origin. To determine D at a
point P, it is easy to see that choosing a spherical surface containing P
will satisfy symmetry conditions.
• Thus, a spherical surface centered at the origin is the Gaussian
surface.
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Point Charge
• Since D is everywhere normal to the Gaussian surface, i.e. D = Drar
applying Gauss's law gives,
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Infinite Line Charge
• Suppose the infinite line of uniform charge pL C/m lies along the z-axis
• To determine D at a point P, we choose a cylindrical surface
containing P to satisfy symmetry condition.
• D is constant on and normal to the cylindrical Gaussian surface.
i.e. D = Dρaρ.
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Infinite Line Charge
• Applying Gauss's law to an arbitrary length l of the line,
• Note that ‫𝐷 ׬‬. 𝑑𝑆 evaluated on the top and bottom surfaces of the cylinder
is zero since D has no z-component; that means that D is tangential to
those surfaces. Thus,
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Infinite Sheet of Charge
• Consider the infinite sheet of uniform charge ρs C/m2 lying on the z=0
plane.
• To determine D at point P, we choose a rectangular box that is cut
symmetrically by the sheet of charge and has two of its faces parallel
to the sheet.
• As D is normal to the sheet, D = Dzaz
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Infinite Sheet of Charge
• Applying Gauss's law,
• Note that D.dS evaluated on the sides of the box is zero because D
has no components along ax and ay.
• If the top and bottom area of the box each has area A,
• Thus,
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Uniformly Charged Sphere
• Consider a sphere of radius a with a uniform charge pv C/m3.
• To determine D everywhere, we construct Gaussian surfaces for
cases r < a and r > a separately.
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Uniformly Charged Sphere
• For r < a, the total charge enclosed by the spherical surface of radius
r,
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Uniformly Charged Sphere
• For r > a, the charge enclosed by the surface is the entire charge
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Uniformly Charged Sphere
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Example
Determine electrical flux density D at (4, 0, 3) if there is a point charge
-5π mC at (4, 0, 0) and a line charge 3π mC/m along the y-axis.
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Divergence of a Vector and Divergence
Theorem
• The divergence of A is the net outward flow of flux per unit volume
over a closed incremental surface.
where ∆v is the volume enclosed by the closed surface S in
which P is located
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Divergence Theorem
• The divergence theorem states that the total outward flux of a vector
field A through the closed surface S is the same as the volume integral
of the divergence of A.
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Example
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Gauss’s Law and Maxwell’s Equations
• Gauss’s Law,
• Applying Divergence Theorem,
.
• This is the first of the four Maxwell's equations to be derived. The equation
states that the volume charge density is the same as the divergence of the
electric flux density.
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Example
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Electric Potential
• Suppose a point charge Q is moved from point A to point B in an
electric field E as shown in Figure.
• From Coulomb's law, the force on Q is F = QE so that the work done in
displacing the charge by dl is
• The negative sign indicates that the work is being done by an external
agent.
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Example
Calculate the work required to move a 10nC charge from the origin to
point (1,1,0) against the static field 𝐸ത = 5𝑎ത𝑥 𝑉/𝑚
1
1
𝑊 = −10 𝑛𝐶 න 5𝑎𝑥 𝑑𝑦𝑎𝑦 += −10 𝑛𝐶 න 5𝑎𝑥 𝑑𝑥𝑎𝑥
0
𝑊 = −50𝑛𝐽
0
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Electric Potential
• The total work done, or the potential energy required, in moving Q
from A to B is
• Dividing W by Q gives the potential energy per unit charge. This
quantity, denoted by VAB, is known as the potential difference
between points A and B.
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Electric Potential
• if the E field in is due to a point charge Q located at the origin, then
• where VB and VA are the potentials (or absolute potentials) at B and A.
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Electric Potential
• The potential at any point is the potential difference between the
point and a chosen point at which the potential is zero.
• In other words, by assuming zero potential at infinity, the potential at
a distance r from the point charge is the work done per unit charge by
an external agent in transferring a test charge from infinity to that
point. Thus
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Kirchoff’s Voltage Law
• For a closed path, the integral returns zero potential difference.
33
Absolute potential
• At some finite radius from a point charge fixed at origin
• If the point charge Q is not located at the origin but at a point whose
position vector is r',
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Electrostatic potential resulting from multiple
point charges
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Example
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Curl of a Vector and Stokes’s theorem
• The circulation of a vector field A around a closed path L
• The curl of A is an axial (or rotational) vector whose magnitude is the
maximum circulation of A per unit area as the area tends to zero and
whose direction is the normal direction of the area when the area is
oriented so as to make the circulation maximum.
• That is,
• where the area ∆S is bounded by the curve L and an is the unit vector
normal to the surface ∆S and is determined using the right-hand rule.
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Stokes’s theorem
• From the definition of the curl of A, we may expect that,
• Stokes's theorem states that the circulation of a vector field A around
a (closed) path L is equal to the surface integral of the curl of A over
the open surface S, bounded by L provided that A and ∇ X A are
continuous on S.
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Maxwell’s equation No.2
• The potential difference between points A and B is independent of
the path taken. Hence,
𝑉𝐵𝐴 = − 𝑉𝐴𝐵
• i.e.
• This shows that the line integral of E along a closed path must be
zero. Physically, this implies that no net work is done in moving a
charge along a closed path in an electrostatic field.
• Applying Stokes's theorem,
or
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Relationship between E and V
• From the way we defined potential, 𝑉 = − ‫𝐸 ׬‬. 𝑑𝑙 , it follows that
• But,
• Comparing the two expressions for dV, we obtain
• Thus,
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THANK YOU..
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DEL Operator
• The Del operator, written ∇, is the vector differential operator.
• In Cartesian coordinates,
• The operator is useful in defining,
1. The gradient of a scalar V, written, as ∇V
2. The divergence of a vector A, written as ∇ • A
3. The curl of a vector A, written as ∇ X A
4. The Laplacian of a scalar V, written as ∇2V
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Gradient of a Scalar field
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Divergence of a Vector field
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Difference Between Gradient and Divergence
46
Curl of a Vector
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Vector Identities involving Curl
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