PHY109: ENGINEERING PHYSICS Unit I: Electromagnetic theory

Electromagnetic theory

PHY109: ENGINEERING PHYSICSUnit I: Electromagnetic theory

Deepak Kaushik

Department of Physics

School of Chemical Engineering and Physical Sciences

Lovely Professional University, Punjab.

[email protected]

November 18, 2020 PHY109 (ENGINEERING PHYSICS)


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

Maxwell’s equations

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Outline



Vectors: A physical quantity with direction and magnitude.

In component form:

Example: position vector ( 𝑟 = 𝑥 𝑖 + 𝑦 𝑗+𝑧 𝑘), force, electric field etc.

Product of vectors:a) Scalar (Dot) product:

b) Vector (Cross) product:

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𝐴 = 𝐴 𝐴

𝐴 = 𝐴𝑥 𝑖 + 𝐴𝑦 𝑗+𝐴𝑧 𝑘

o

P (Ax, Ay, Az)

x

Ay

Az

Az

𝐴2

= 𝐴𝑥2 + 𝐴𝑦

2 +𝐴𝑧2

𝐴. 𝐵 = 𝐴 𝐵 𝑐𝑜𝑠𝜃 = 𝐴𝑥𝐵𝑥 + 𝐴𝑦𝐵𝑦+𝐴𝑧𝐵𝑧

𝐴 × 𝐵 = 𝐴 𝐵 𝑠𝑖𝑛𝜃 𝑛 =

𝑖 𝑗 𝑘𝐴𝑥 𝐴𝑦 𝐴𝑧

𝐵𝑥 𝐵𝑦 𝐵𝑧

z

y

Vectors



Field: A function that specifies a particular quantity everywhere in the region.

If the quantity is scalar=> Scalar field.Examples: • Temperature distribution in a building, • sound intensity in a theatre, • electric potential in a region,• refractive index of a stratified medium.

If the quantity is vector=> Vector field.Examples: • Gravitational force on a body in space, • velocity of raindrops in the atmosphere, • magnetic field in a region, • electric field in a region.

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Field concept



• Line integral: Integration along a line/path L.

where, 𝑑𝑙 = 𝑑𝑥 𝑖 + 𝑑𝑦 𝑗+𝑑𝑧 𝑘 is a differential line element of line/path L.

• Circulation: Line integral in a closed path, where a closed path bounds surface.

𝒄𝒊𝒓𝒄𝒖𝒍𝒂𝒕𝒊𝒐𝒏 = 𝐴. 𝑑𝑙

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𝐴. 𝑑𝑙 = 𝐴𝑐𝑜𝑠𝜃 𝑑𝑙

• Surface integral: Integration over a surface S.

• Volume integral: Integration of a scalar field over a volume V

𝐴. 𝑑𝑎 = 𝐴𝑐𝑜𝑠𝜃 𝑑𝑎

∅𝑑𝑣where, 𝑑𝑎 = 𝑑𝑥𝑑𝑦 𝑘

= 𝑑𝑦𝑑𝑧 𝑖= 𝑑𝑧𝑑𝑥 𝑗

is a differential area element of surface S.

• It is a measure of flux through the surface S.

• Integration over closed surface that bounds volume is closed surface integral.

where, 𝑑𝑣 = 𝑑𝑥𝑑𝑦𝑑𝑧is a differential volume element of volume V.

𝐴. 𝑑𝑎

Integral calculus



Derivative: Change of a function f(x) w.r.t. a variable x; 𝑑𝑓 =𝜕𝑓

𝜕𝑥𝑑𝑥

For three variables, change of the function f(x, y, z) can be written as:

𝑑𝑓 =𝜕𝑓

𝜕𝑥𝑑𝑥 +

𝜕𝑓

𝜕𝑦𝑑𝑦 +

𝜕𝑓

𝜕𝑧𝑑𝑧 (1)

where,𝝏𝒇

𝝏𝒙,

𝝏𝒇

𝝏𝒚and

𝝏𝒇

𝝏𝒛are partial derivatives.

• Del operator (𝜵): Derivative vector operator.

𝛻 =𝜕

𝜕𝑥 𝑖 +

𝜕

𝜕𝑦 𝑗 +

𝜕

𝜕𝑧 𝑘 (2)

Using equation (2), equation (1) can be rearranged as:

𝑑𝑓 =𝜕𝑓

𝜕𝑥 𝑖 +

𝜕𝑓

𝜕𝑦 𝑗 +

𝜕𝑓

𝜕𝑧 𝑘 . 𝑑𝑥 𝑖 + 𝑑𝑦 𝑗+𝑑𝑧 𝑘 = 𝛻𝑓. 𝑑𝑙

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Differential calculus


•Gradient: operated on a scalar field.

•Gradient of a scalar field is a vector field.

• It measures maximum increase of the function f.

• Its magnitude gives the slope (rate of increase) along its maximal direction.

• If 𝛻𝑓 = 0, the function f has either a maxima, minima or saddle point.

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𝛻𝑓 =𝜕𝑓

𝜕𝑥 𝑖 +

𝜕𝑓

𝜕𝑦 𝑗 +

𝜕𝑓

𝜕𝑧 𝑘

• Divergence: operated on a vector field.

• Curl: operated on a vector field.

where, 𝐴 =𝐴𝑥 𝑖 + 𝐴𝑦 𝑗 + 𝐴𝑧 𝑘

Divergence of a vector field is a scalar field.

It is a measure of spread of vector 𝐴from a point.

If 𝛻. 𝐴 = 0, 𝐴 is solenoidal.

𝛻. 𝐴 = 𝜕𝐴𝑥

𝜕𝑥+

𝜕𝐴𝑦

𝜕𝑦+

𝜕𝐴𝑧

𝜕𝑧 𝛻 × 𝐴 =

𝑖 𝑗 𝑘𝜕

𝜕𝑥

𝜕

𝜕𝑦

𝜕

𝜕𝑧

𝐴𝑥 𝐴𝑦 𝐴𝑧

where, 𝐴 =𝐴𝑥 𝑖 + 𝐴𝑦 𝑗 + 𝐴𝑧 𝑘

Curl of a vector field is a vector field.

It is a measure of circulation of

vector 𝐴 around a point.

If 𝛻 × 𝐴 = 0, 𝐴 is irrotational. 𝐴 is a conservative vector field.




•Directional derivative: 𝛻𝑓. 𝑛.

•Divergence of curl of a vector is zero.

𝛻. 𝛻 × 𝐴 = 0

•Curl of gradient of a scalar is zero.

𝛻 × 𝛻𝑓 = 0

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Gauss’s theorem (Divergence theorem/Green’s theorem): Total outward flux (closed surface

integral) of a vector field 𝐴 is the volume integral of the divergence of the vector field 𝐴.

𝐴. 𝑑𝑎 = 𝛻. 𝐴 𝒅𝒗

Stokes’ theorem (Curl theorem): The line integral of a vector field 𝐴 around a closed path L

(circulation) is the surface integral of the vector field 𝐴 over the surface bound by the path L.

𝐴. 𝑑𝑙 = 𝛻 × 𝐴 . 𝑑𝑎

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Fundamental theorem


Coulomb’s law:

𝐹 =1

4𝜋𝜀0

𝑞1𝑞2

𝑟2 𝑟

Unit of force: N

Electric field:

𝐸 =1

4𝜋𝜀0

𝑞

𝑟2 𝑟 =

𝐹

𝑄Unit of electric field: N/C or V/m

Electric potential:

𝑉 = − 𝐸. 𝑑𝑙 =𝑊

𝑞

Unit of electric potential: V or J/C

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Electrostatics


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.

𝐸. 𝑑𝑙 = 0

- In differential form we can write, 𝛻 × 𝐸 = 0. This shows 𝐸 is conservative field.

- In differential form, electric field be written as negative of potential gradient i.e.

𝐸 = −𝛻𝑉In one dimension,

𝐸 = −𝑑𝑉

𝑑𝑥 𝑥

Thus unit of electric field is V/m.

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Electrostatics


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∅ = 𝐸. 𝑑𝑎 = 𝐸𝑐𝑜𝑠𝜃 𝑑𝑎

Gauss’s law: The total flux through closed surface is the 1/𝜺𝟎 times total charge enclosed in a volume bounded by the surface.

𝐸. 𝑑𝑎 =𝑞𝑒𝑛𝑐

𝜀0

𝐸. 𝑑𝑎 =1

𝜀0 𝜌𝑑𝑣

𝛻. 𝐸 =𝜌

𝜀0,

𝑤ℎ𝑒𝑟𝑒, 𝜌 𝑖𝑠 𝑣𝑜𝑙𝑢𝑚𝑒 𝑐ℎ𝑎𝑟𝑔𝑒 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑎𝑛𝑑 𝑢𝑛𝑖𝑡 𝑖𝑠 𝐶/𝑚3.

Electric flux: Number of electric field lines passes through a surface S.

Maxwell equation in integral form.

Maxwell equation in differential form.

𝜽𝒅𝒂

𝒏𝐸

𝑞 = 𝜌𝑑𝑣


Electrostatics

Unit of electric flux: Nm2/C


Poisson’s equation:

𝑤ℎ𝑒𝑟𝑒, 𝛻2 =𝜕2

𝜕𝑥2+

𝜕2

𝜕𝑦2+

𝜕2

𝜕𝑧2

If volume charge density, 𝜌 = 𝟎,

Continuity equation

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𝛻2𝑉 = −𝜌

𝜀0

𝛻2𝑉 = 0 Laplace equation

𝛻. 𝐽 = −𝜕𝜌

𝜕𝑡


Electrostatics

If volume charge density 𝜌 is constant with time then 𝛻. 𝐽 = 0, which is the steady state condition.


Lorentz force: Force on charge (Q) while moving with velocity ( 𝑣) in electric (𝐸) and magnetic (𝐵) fields:

Magnetic force on current carrying conductor (𝐼𝑑𝑙) can be expressed as:

where, 𝐵 =𝜇0𝐼

4𝜋

𝑑𝑙× 𝑟𝑟2 is the magnetic field due to steady current in line (Biot-Savart’s 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 ( 𝐹𝑚𝑎𝑔) will be

perpendicular to the wires and opposite in direction i.e. attract.

If current between two parallel wires flows in opposite direction, magnetic ( 𝐹𝑚𝑎𝑔) will be

perpendicular to the wires and same in direction i.e. repel.

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𝐹 = 𝑄(𝐸 + 𝑣 × 𝐵)

𝐹𝑚𝑎𝑔 = 𝐼(𝑑𝑙 × 𝐵)


Magnetostatics (when E and B are independent)


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∅ = 𝐵. 𝑑𝑎 = 𝐵𝑐𝑜𝑠𝜃 𝑑𝑎

Magnetic flux: Number of magnetic field lines passes through a surface S.

𝜽𝒅𝒂

𝒏𝐵

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.

𝐵. 𝑑𝑎 = 0

𝛻. 𝐵 = 0



The magnetic field (𝐵) is called solenoidal.


Magnetostatics

Unit of magnetic flux: Wb or Tm2.The unit of magnetic field (B) also written as Wb/m2.


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𝜀 = 𝐸. 𝑑𝑙 = −𝑑∅

𝑑𝑡

𝐸. 𝑑𝑙 = − 𝜕 𝐵

𝜕𝑡. 𝑑𝑎

Faraday’s 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. 𝐵 is constant, last equation reduces to:


Maxwell equation in differential form.𝛻 × 𝐸 = −𝜕𝐵

𝜕𝑡

𝛻 × 𝐸 = 0

which is for electrostatic condition.


For time varying fields

Lenz’s rule


Ampere’s circuital law: Magnetic field (𝐵) in a loop due to a current carrying conductor is 𝜇0

times of current (Ienc) enclosed by the loop.

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𝐵 . 𝑑𝑙 = 𝜇0Ienc

𝛻 × 𝐵 = 𝜇0 𝐽, 𝑤ℎ𝑒𝑟𝑒, I = 𝐽 . 𝑑𝑎

Ampere’s 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 𝐽𝐷 = 𝜀0𝜕𝐸

𝜕𝑡:

𝐵 . 𝑑𝑙 = 𝜇0Ienc + 𝜇0𝜀0

𝑑

𝑑𝑡 𝐸 . 𝑑𝑎

𝛻 × 𝐵 = 𝜇0 𝐽 + 𝐽𝐷




For time varying fields


For parallel plate capacitor, capacitance can be expressed as:

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𝑐 = 𝜀0

𝐴

𝑑, 𝑤ℎ𝑒𝑟𝑒, 𝜀0 𝑖𝑠 𝑝𝑒𝑟𝑚𝑖𝑡𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 𝑓𝑟𝑒𝑒 𝑠𝑝𝑎𝑐𝑒.

Introducing a dielectric material the capacitance is:

The displacement current for dielectrics can be written as:

𝐽𝐷 = 𝜀𝜕𝐸

𝜕𝑡

𝑐 = 𝜀𝐴

𝑑, 𝑤ℎ𝑒𝑟𝑒, 𝜀 𝑖𝑠 𝑝𝑒𝑟𝑚𝑖𝑡𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 𝑑𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙.

𝑎𝑛𝑑, 𝜀 = 𝜀𝑟𝜀0

𝜀𝑟 is diectric constant of the material.


Dielectric constant


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where 𝐽𝐷 = 𝜀0𝜕𝐸

𝜕𝑡is displacement current density

𝛻 × 𝐵 = 𝜇0 𝐽 + 𝐽𝐷

where 𝐽 is current density, I = 𝐽 . 𝑑𝑎

𝛻. 𝐸 =𝜌

𝜀0.

where 𝜌 is volume charge density, 𝑞 = 𝜌𝑑𝑣

𝛻. 𝐵 = 0

𝛻 × 𝐸 = −𝑑𝐵

𝑑𝑡

Maxwell’s equation

𝐵 . 𝑑𝑙 = 𝜇0Ienc + 𝜇0𝜀0

𝑑

𝑑𝑡 𝐸 . 𝑑𝑎

𝐸. 𝑑𝑙 = − 𝜕 𝐵

𝜕𝑡. 𝑑𝑎

𝐵. 𝑑𝑎 = 0

𝐸. 𝑑𝑎 =1

𝜀0 𝜌𝑑𝑣

In differential form In integral form


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𝛻 × 𝐵 = 𝜇0𝜀0

𝜕𝐸

𝜕𝑡

𝛻. 𝐸 = 0 𝛻. 𝐵 = 0

𝛻 × 𝐸 = −𝑑𝐵

𝑑𝑡

Maxwell’s equations in free space where free charge density and current is zero; ρ = 0 & J = 0.

By decoupling the above linear differential equations for 𝐸 and 𝐵 we get a second order differential

wave equation:

𝛻2𝐸 − 𝜇0𝜀0

𝜕2𝐸

𝜕𝑡2= 0 𝛻2𝐵 − 𝜇0𝜀0

𝜕2𝐵

𝜕𝑡2= 0

𝑐 =1

𝜇0𝜀0≈ 3 × 108 𝑚/𝑠 is speed of electromagnetic waves.

𝐸 = 𝐸0𝑒𝑖(𝑘𝑧−𝜔𝑡+𝜑1) 𝑛

The solution of the wave equations is:

𝐵 = 𝐵0𝑒𝑖(𝑘𝑧−𝜔𝑡+𝜑2) 𝑚

where k, w, 𝜑 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.

Maxwell’s equation in free space


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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Bibliography

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PHY109: ENGINEERING PHYSICS Unit I: Electromagnetic theory