Electromagnetic theory
PHY109: ENGINEERING PHYSICSUnit I: Electromagnetic theory
Deepak Kaushik
Department of Physics
School of Chemical Engineering and Physical Sciences
Lovely Professional University, Punjab.
November 18, 2020 PHY109 (ENGINEERING PHYSICS)
Electromagnetic theory
Vectors
Scalar and vectors fields
Integral calculus
Differential calculus: Concept of gradient, divergence and curl
Gauss theorem and Stokes theorem
Electrostatics: Gauss law, Poisson and Laplace equations, continuity equation
Magnetostatics and time varying fields
Displacement current, correction in Ampere circuital law, dielectric constant
Maxwellβs equations
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Outline
November 18, 2020 PHY109 (ENGINEERING PHYSICS)
Electromagnetic theory
Vectors: A physical quantity with direction and magnitude.
In component form:
Example: position vector ( π = π₯ π + π¦ π+π§ π), force, electric field etc.
Product of vectors:a) Scalar (Dot) product:
b) Vector (Cross) product:
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π΄ = π΄ π΄
π΄ = π΄π₯ π + π΄π¦ π+π΄π§ π
o
P (Ax, Ay, Az)
x
Ay
Az
Az
π΄2
= π΄π₯2 + π΄π¦
2 +π΄π§2
π΄. π΅ = π΄ π΅ πππ π = π΄π₯π΅π₯ + π΄π¦π΅π¦+π΄π§π΅π§
π΄ Γ π΅ = π΄ π΅ π πππ π =
π π ππ΄π₯ π΄π¦ π΄π§
π΅π₯ π΅π¦ π΅π§
z
y
Vectors
November 18, 2020 PHY109 (ENGINEERING PHYSICS)
Electromagnetic theory
Field: A function that specifies a particular quantity everywhere in the region.
If the quantity is scalar=> Scalar field.Examples: β’ Temperature distribution in a building, β’ sound intensity in a theatre, β’ electric potential in a region,β’ refractive index of a stratified medium.
If the quantity is vector=> Vector field.Examples: β’ Gravitational force on a body in space, β’ velocity of raindrops in the atmosphere, β’ magnetic field in a region, β’ electric field in a region.
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Field concept
November 18, 2020 PHY109 (ENGINEERING PHYSICS)
Electromagnetic theory
β’ Line integral: Integration along a line/path L.
where, ππ = ππ₯ π + ππ¦ π+ππ§ π is a differential line element of line/path L.
β’ Circulation: Line integral in a closed path, where a closed path bounds surface.
πππππππππππ = π΄. ππ
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π΄. ππ = π΄πππ π ππ
β’ Surface integral: Integration over a surface S.
β’ Volume integral: Integration of a scalar field over a volume V
π΄. ππ = π΄πππ π ππ
β ππ£where, ππ = ππ₯ππ¦ π
= ππ¦ππ§ π= ππ§ππ₯ π
is a differential area element of surface S.
β’ It is a measure of flux through the surface S.
β’ Integration over closed surface that bounds volume is closed surface integral.
where, ππ£ = ππ₯ππ¦ππ§is a differential volume element of volume V.
π΄. ππ
Integral calculus
November 18, 2020 PHY109 (ENGINEERING PHYSICS)
Electromagnetic theory
Derivative: Change of a function f(x) w.r.t. a variable x; ππ =ππ
ππ₯ππ₯
For three variables, change of the function f(x, y, z) can be written as:
ππ =ππ
ππ₯ππ₯ +
ππ
ππ¦ππ¦ +
ππ
ππ§ππ§ (1)
where,ππ
ππ,
ππ
ππand
ππ
ππare partial derivatives.
β’ Del operator (π΅): Derivative vector operator.
π» =π
ππ₯ π +
π
ππ¦ π +
π
ππ§ π (2)
Using equation (2), equation (1) can be rearranged as:
ππ =ππ
ππ₯ π +
ππ
ππ¦ π +
ππ
ππ§ π . ππ₯ π + ππ¦ π+ππ§ π = π»π. ππ
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November 18, 2020 PHY109 (ENGINEERING PHYSICS)
Differential calculus
Electromagnetic theory
β’Gradient: operated on a scalar field.
β’Gradient of a scalar field is a vector field.
β’ It measures maximum increase of the function f.
β’ Its magnitude gives the slope (rate of increase) along its maximal direction.
β’ If π»π = 0, the function f has either a maxima, minima or saddle point.
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π»π =ππ
ππ₯ π +
ππ
ππ¦ π +
ππ
ππ§ π
β’ Divergence: operated on a vector field.
β’ Curl: operated on a vector field.
where, π΄ =π΄π₯ π + π΄π¦ π + π΄π§ π
Divergence of a vector field is a scalar field.
It is a measure of spread of vector π΄from a point.
If π». π΄ = 0, π΄ is solenoidal.
π». π΄ = ππ΄π₯
ππ₯+
ππ΄π¦
ππ¦+
ππ΄π§
ππ§ π» Γ π΄ =
π π ππ
ππ₯
π
ππ¦
π
ππ§
π΄π₯ π΄π¦ π΄π§
where, π΄ =π΄π₯ π + π΄π¦ π + π΄π§ π
Curl of a vector field is a vector field.
It is a measure of circulation of
vector π΄ around a point.
If π» Γ π΄ = 0, π΄ is irrotational. π΄ is a conservative vector field.
November 18, 2020 PHY109 (ENGINEERING PHYSICS)
Differential calculus
Electromagnetic theory
β’Directional derivative: π»π. π.
β’Divergence of curl of a vector is zero.
π». π» Γ π΄ = 0
β’Curl of gradient of a scalar is zero.
π» Γ π»π = 0
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November 18, 2020 PHY109 (ENGINEERING PHYSICS)
Differential calculus
Electromagnetic theory
Gaussβs theorem (Divergence theorem/Greenβs theorem): Total outward flux (closed surface
integral) of a vector field π΄ is the volume integral of the divergence of the vector field π΄.
π΄. ππ = π». π΄ π π
Stokesβ theorem (Curl theorem): The line integral of a vector field π΄ around a closed path L
(circulation) is the surface integral of the vector field π΄ over the surface bound by the path L.
π΄. ππ = π» Γ π΄ . ππ
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November 18, 2020 PHY109 (ENGINEERING PHYSICS)
Fundamental theorem
Electromagnetic theory
Coulombβs law:
πΉ =1
4ππ0
π1π2
π2 π
Unit of force: N
Electric field:
πΈ =1
4ππ0
π
π2 π =
πΉ
πUnit of electric field: N/C or V/m
Electric potential:
π = β πΈ. ππ =π
π
Unit of electric potential: V or J/C
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November 18, 2020 PHY109 (ENGINEERING PHYSICS)
Electrostatics
Electromagnetic theory
Properties:- Electric potential is path independent i.e. it depends on initial and final positions.
- The line integral of electric field over a closed path is zero i.e.
πΈ. ππ = 0
- In differential form we can write, π» Γ πΈ = 0. This shows πΈ is conservative field.
- In differential form, electric field be written as negative of potential gradient i.e.
πΈ = βπ»πIn one dimension,
πΈ = βππ
ππ₯ π₯
Thus unit of electric field is V/m.
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November 18, 2020 PHY109 (ENGINEERING PHYSICS)
Electrostatics
Electromagnetic theory
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β = πΈ. ππ = πΈπππ π ππ
Gaussβs law: The total flux through closed surface is the 1/πΊπ times total charge enclosed in a volume bounded by the surface.
πΈ. ππ =ππππ
π0
πΈ. ππ =1
π0 πππ£
π». πΈ =π
π0,
π€βπππ, π ππ π£πππ’ππ πβππππ ππππ ππ‘π¦ πππ π’πππ‘ ππ πΆ/π3.
Electric flux: Number of electric field lines passes through a surface S.
Maxwell equation in integral form.
Maxwell equation in differential form.
π½π π
ππΈ
π = πππ£
November 18, 2020 PHY109 (ENGINEERING PHYSICS)
Electrostatics
Unit of electric flux: Nm2/C
Electromagnetic theory
Poissonβs equation:
π€βπππ, π»2 =π2
ππ₯2+
π2
ππ¦2+
π2
ππ§2
If volume charge density, π = π,
Continuity equation
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π»2π = βπ
π0
π»2π = 0 Laplace equation
π». π½ = βππ
ππ‘
November 18, 2020 PHY109 (ENGINEERING PHYSICS)
Electrostatics
If volume charge density π is constant with time then π». π½ = 0, which is the steady state condition.
Electromagnetic theory
Lorentz force: Force on charge (Q) while moving with velocity ( π£) in electric (πΈ) and magnetic (π΅) fields:
Magnetic force on current carrying conductor (πΌππ) can be expressed as:
where, π΅ =π0πΌ
4π
ππΓ ππ2 is the magnetic field due to steady current in line (Biot-Savartβs Law). The
direction of the field is around the circular loop. The unit of magnetic field: T or N(Am)-1.
If current between two parallel wires flows in same direction, magnetic ( πΉπππ) will be
perpendicular to the wires and opposite in direction i.e. attract.
If current between two parallel wires flows in opposite direction, magnetic ( πΉπππ) will be
perpendicular to the wires and same in direction i.e. repel.
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πΉ = π(πΈ + π£ Γ π΅)
πΉπππ = πΌ(ππ Γ π΅)
November 18, 2020 PHY109 (ENGINEERING PHYSICS)
Magnetostatics (when E and B are independent)
Electromagnetic theory
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β = π΅. ππ = π΅πππ π ππ
Magnetic flux: Number of magnetic field lines passes through a surface S.
π½π π
ππ΅
The net magnetic flux through a closed surface is zero i.e. the number of magnetic field lines entering the closed surface is equal to the number of lines coming out of the surface.
π΅. ππ = 0
π». π΅ = 0
Maxwell equation in integral form.
Maxwell equation in differential form.
The magnetic field (π΅) is called solenoidal.
November 18, 2020 PHY109 (ENGINEERING PHYSICS)
Magnetostatics
Unit of magnetic flux: Wb or Tm2.The unit of magnetic field (B) also written as Wb/m2.
Electromagnetic theory
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π = πΈ. ππ = βπβ
ππ‘
πΈ. ππ = β π π΅
ππ‘. ππ
Faradayβs law: work done in moving a test charge around a closed loop equals the rate of decrease of the magnetic flux through the enclosed surface.
For steady state condition i.e. π΅ is constant, last equation reduces to:
Maxwell equation in integral form.
Maxwell equation in differential form.π» Γ πΈ = βππ΅
ππ‘
π» Γ πΈ = 0
which is for electrostatic condition.
November 18, 2020 PHY109 (ENGINEERING PHYSICS)
For time varying fields
Lenzβs rule
Electromagnetic theory
Ampereβs circuital law: Magnetic field (π΅) in a loop due to a current carrying conductor is π0
times of current (Ienc) enclosed by the loop.
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π΅ . ππ = π0Ienc
π» Γ π΅ = π0 π½, π€βπππ, I = π½ . ππ
Ampereβs circuital law fails to explain in time varying fields. For example charging/discharging of capacitor connected in a circuit.
The equation is modified by Maxwell by introducing displacement current π½π· = π0ππΈ
ππ‘:
π΅ . ππ = π0Ienc + π0π0
π
ππ‘ πΈ . ππ
π» Γ π΅ = π0 π½ + π½π·
Maxwell equation in integral form.
Maxwell equation in differential form.
November 18, 2020 PHY109 (ENGINEERING PHYSICS)
For time varying fields
Electromagnetic theory
For parallel plate capacitor, capacitance can be expressed as:
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π = π0
π΄
π, π€βπππ, π0 ππ ππππππ‘ππ£ππ‘π¦ ππ ππππ π ππππ.
Introducing a dielectric material the capacitance is:
The displacement current for dielectrics can be written as:
π½π· = πππΈ
ππ‘
π = ππ΄
π, π€βπππ, π ππ ππππππ‘ππ£ππ‘π¦ ππ πππππππ‘πππ πππ‘πππππ.
πππ, π = πππ0
ππ is diectric constant of the material.
November 18, 2020 PHY109 (ENGINEERING PHYSICS)
Dielectric constant
Electromagnetic theory
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where π½π· = π0ππΈ
ππ‘is displacement current density
π» Γ π΅ = π0 π½ + π½π·
where π½ is current density, I = π½ . ππ
π». πΈ =π
π0.
where π is volume charge density, π = πππ£
π». π΅ = 0
π» Γ πΈ = βππ΅
ππ‘
Maxwellβs equation
π΅ . ππ = π0Ienc + π0π0
π
ππ‘ πΈ . ππ
πΈ. ππ = β π π΅
ππ‘. ππ
π΅. ππ = 0
πΈ. ππ =1
π0 πππ£
In differential form In integral form
Electromagnetic theory
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π» Γ π΅ = π0π0
ππΈ
ππ‘
π». πΈ = 0 π». π΅ = 0
π» Γ πΈ = βππ΅
ππ‘
Maxwellβs equations in free space where free charge density and current is zero; Ο = 0 & J = 0.
By decoupling the above linear differential equations for πΈ and π΅ we get a second order differential
wave equation:
π»2πΈ β π0π0
π2πΈ
ππ‘2= 0 π»2π΅ β π0π0
π2π΅
ππ‘2= 0
π =1
π0π0β 3 Γ 108 π/π is speed of electromagnetic waves.
πΈ = πΈ0ππ(ππ§βππ‘+π1) π
The solution of the wave equations is:
π΅ = π΅0ππ(ππ§βππ‘+π2) π
where k, w, π is wave vector, angular frequency and phase of the waves. z is the direction of propagation of
the em wave. Unit vectors n and m are the polarization direction of electric and magnetic field.
E0 and B0 are the amplitude of the wave; E0 = cB0; c = w/k.
Maxwellβs equation in free space
Electromagnetic theory
Text Books:
ENGINEERING PHYSICS by HITENDRA K MALIK AND A K SINGH, MCGRAW HILL EDUCATION, 1st Edition, (2009).
References:
ENGINEERING PHYSICS by B K PANDEY AND S CHATURVEDI, CENGAGE LEARNING, 1st Edition, (2009).
ENGINEERING PHYSICS by D K BHATTACHARYA, POONAM TONDON OXFORD UNIVERSITY PRESS.
Further Readings:
INTRODUCTION TO ELECTRODYNAMICS, DAVID J. GRIFFITHS, PHI LEARNING PRIVATE LIMITED, 3rd Edition, (2009).
PRINCIPLES OF ELECTROMAGNETICS, MATHEW N. O. SADIKU, OXFORD UNIVERSITY PRESS, 4th Edition, (2009).
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November 18, 2020 PHY109 (ENGINEERING PHYSICS)
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