Electromagnetic Fields - Magnetostatics Complete Study Notes

Biot-Savart’s Law

It states that the differential magnetic field intensity dH produced at a point P by the differential current element Idl is proportional to the product Idl and the sine of the angle a between the element and the line joining P to the element and is inversely proportional to the square of the distance R between P and the element.

Ampere's Law 

Ampere's law allows the calculation of magnetic fields.

Ampere's circuital law states that the line integral of the magnetic field (circulation of H) around a closed path is the net current enclosed by this path.

Ampere's law in differential form:

Ampere's law in integral form:

Ampere’s Circuit Law:

The configurations that can be handled by Ampere's law are:

• Infinite straight lines
• Infinite planes
• Infinite solenoids
• Toroids

Electromagnetic field

A time-varying magnetic field produces an electromotive force (or emf) which may establish a current in a closed circuit,

For transformer,

and for motional emf, 

where u is called energy density

* Displacement Current

where

is known as displacement current density.

Reciprocal of attenuation constant is known as skin depth or penetration depth. It measures the depth at which field intensity reduces to the electron of the original value.

Skin depth 

The Poynting vector P is the power flow vector whose direction is the same as the direction of wave propagation.

P = E × H

Key Points

* Free space

σ = 0, ε = ε₀, μ = μ₀

* Lossless dielectrics

σ = 0, ε = ε_rε₀, μ = μ_rμ₀ or σ << ωε

* Lossy dielectrics

σ ≠ 0, ε = ε_rε₀, μ = μ_rμ₀

* Good conductors

σ = ∞, ε = ε₀, μ = μ_rμ₀ or σ << ωε

Where σ is conductivity, ε is permittivity and μ is the permeability of the medium.

Divergence is a dot product between the gradient operator (the flipped triangle) and the vector field, also the curl is a cross product of the gradient operator and the vector field.

Using the Biot-Savart law for a volume current  we can calculate the divergence and curl of :

The curl of an electric field is zero. The electric field associated with a set of stationary charges has a curl of zero. In this situation, there is no magnetic field. Electric field lines curl up in opposition to a time-varying magnetic field (Faraday's law of induction). Magnetic field lines curl up in response to current flow, ie. moving charges (Ampere's law, first term).

All the Best.

Introduction to Boundary condition for the Magnetic field. 

Lorentz force is the force on an electrically charged particle that moves through a magnetic plus an electric field. The Lorentz force has two vector components, one proportional to the magnetic field and one proportional to the electric field.

The magnetic field B is defined by Lorentz Force Law, and specifically force on a moving charge

F = q v × B

Force in an electrostatic field

F = QE

Force in a magnetostatic field

F = Q (u × B)

So, net force in electromagnetic fields

F = Q (E + u × B)

* Force experienced by a current element ldl in magnetic field B is

dF = ldl × B

* Magnetic flux density can also be defined as the force per unit current element.

* Magnetic moment

m = lSa_n

* Torque on a current loop T = m × B = lS an × B

* For linear materials, magnetization M = χ_mH

* Boundary conditions for magnetic fields

B_1n = B_2n

(H₁ – H₂) ×a_n12 = K

And H_1t = H_2t if K = 0

Magnetic Energy & Inductance relation.

• Induction of an electromotive force occurs in a circuit by varying the magnetic flux linked with the circuit.
• From Faraday's law, the concept of Inductance may be derived from the property of an electric circuit by which an electromotive force is induced in it as the result of a changing magnetic flux.
• Inductance L may be defined in terms of the electromotive force generated to oppose a change in current ΔI in the given time duration Δt.

Emf = -L (ΔI) / (Δt)

• Unit for L is Volt Second / Ampere = Henry.
• 1 henry is 1 volt-second / ampere.
• If the rate of change of current in a circuit ΔI / Δt is one ampere per second and the resulting electromotive force is one volt, then the inductance of the circuit is one henry.
• Since emf results due to the rate of change of magnetic flux , inductance L may also be defined as a measure of the amount of magnetic flux φ produced for a given electric current I as: L = φ/I, where the inductance L is one henry, if current I of one ampere, produces magnetic flux φ of one weber.
• For an inductor of inductance L,
  
• The energy in a magnetostatics field
  
• Inductance is defined as
  
  
  

  Introduction to Magnetomotive Force

  Magnetomotive Force (MMF) performs a similar role in a magnetic circuit to the electromotive force (EMF) of a battery in a basic electrical circuit, thus acting as the 'prime mover' of an electromagnetic system.
                                               
  
  The MMF resulting from passing an electrical current through a coil is given by the electrical current flowing through the coil multiplied by the number of turns, as shown in the following equation:

   

  Equation: MMF, F_m = I × N.

  MMF is measured in amperes (A) rather than ampere-turns, since 'turns' is not an SI unit.

  Magnetomotive force is also used in the derivation of magnetic field strength (H), as shown below. H =  I N / l = F_m / l

  F_m = Hl

  Inductance Introduction

  Magnetic circuits are analogous to resistive electronic circuits if we define the magnetomotive force (MMF) analogous to the voltage (EMF). The flux then plays the same role as the current in electronic circuits so that we define the magnetic analogue to resistance as the reluctance (R).

   

                                       
  MMF = ΦR

  R = MMF/ Φ

  R is proportional to the reciprocal of the inductance.

  The reluctance of a uniform gap, without leakage, is, therefore:

Calculation of the electrical resistance of a wire or bar of uniform cross-section:

where σ is the wire material conductivity, l the length and A is the wire cross-sectional area.

R_eq. = R₁ + R₂ + R₃ +…

Reluctances in Series: The combination of reluctances in series, with a common flux-path, results in:

Reluctances in Parallel: Flux in a magnetic circuit may flow in various leakage paths, as well as the useful ones. It can be shown, by considering two parallel flux-paths, with flux driven by the same MMF, that the equivalent reluctance for two reluctances in parallel is:

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