Summary

• The phenomenon of transferring electric charge from one point in a circuit to another is described by the term electric current. Electric current is defined as the rate of flow of electric charges or electrons through a cross sectional area.

• Current density vector  at any point is defined as the current through a unit normal area at that point.

• Total current flowing through a surface S is given as,

• Electric current is of three types:
1. Convection current
2. Conduction current
3. Displacement current
• The motion of charged particles in free space (vacuum) is said to constitute convection current.
• The motion of the free electrons present in a conductor, by the influence of an electric field, constitutes the conduction current. The relation of conduction current given as,

is known as the point form of Ohmâ€™s law.
• The current flowing in a capacitor is termed as displacement current, given as,

• From the electrical point of view, materials can be classified as conductors (Ïƒ >> 1, Îµr =1), dielectrics (Ïƒ << 1, Îµr â‰¥ 1).
• The resistance of a conductor of uniform cross section is given as,

and in general, for nonuniform cross section, it is given as,

• Jouleâ€™s law states that the rate of heat production by a steady current in any part of an electrical circuit is directly proportional to the resistance and to the square of the current  In differential and integral forms, it is given as,

and
• The electromotive force (emf) in a closed loop is given as,

where  is the emf-producing field, i.e., the field generated by causes other than the static charges.
• Kirchhoffâ€™s Current Law (KCL) in differential and integral forms is given as,

and
• The continuity equation of charges is given as,

• Relaxation time  of a material is the time taken by a charge placed in the interior of a conductor to drop its value to 37% (e-1 = 0.368 â‰ˆ 37%) of its initial value.
• The boundary conditions for the current density for two different conducting media are given as,

and

Combining these two conditions, we get,

where Î¸1 and Î¸2 are the angles with the normal in respective medium.

If the boundary carries a surface charge with density  (C/m2) then by the boundary conditions can be written as,

• Laplace equation for conducting medium is given as âˆ‡2V = 0.
• The dielectric polarisation may be defined as a dynamical response of a system to an externally applied electric field. Polarisation vector is defined as the dipole moment per unit volume of the dielectric, i.e.,

• The dielectric materials which have no free charges and where all electrons are bound and associated with the nearest atoms, are known as nonpolar dielectrics. The dielectric materials in which the molecules or atoms possess a permanent dipole moment which is ordinarily randomly oriented, but which becomes more or less oriented by the application of an external electric field, are known as polar dielectrics.
• The effect of macroscopic polarisation in a given volume of dielectric material is to induce some bound surface and volume charge densities in the dielectric, given as,

and
• Electric displacement is given in terms of electric field and polarisation vector as,

• The relation of polarisation vector with the electric field is given as,

Here, the quantity  is known as the electric susceptibility of a dielectric material which gives a measure of how easily it polarises in response to an electric field.
• Electric permittivity (Îµ) is a physical quantity that describes how an electric field affects and is affected by a dielectric medium. It is determined by the ability of a material to polarise in response to the field, and thereby reduce the total electric field inside the material. Its relations are given as,

Îµ = Îµ0Îµr and Îµr = (1 + Ï‡e)

The dielectric constant or relative permittivity (Îµr) is the ratio of the permittivity of a substance to the permittivity of free space.
• The maximum electric field that a dielectric can withstand without breakdown is known as dielectric strength of the material.
• A dielectric material is said to be homogeneous if the permittivity (Îµ) or conductivity (Ïƒ) does not vary with space in a region.
• A dielectric material is said to be isotropic if the electrical properties of the medium are independent of the direction, i.e.,  and  are in the same direction.
• The Biotâ€“Savart law states that the magnetic field intensity  produced at a point P at a distance r from a differential current element  is

• Three current densities (line current density, ; surface current density  and volume current density, ) are related to each other as,

• The magnetic field intensity  at any point is defined as the force experienced by a north pole of one weber placed at that point. Its unit is newtons per weber (N/Wb) or ampere-turns per metre (AT/m).
• Magnetic flux is defined as the group of magnetic field lines emitted outward from the north pole of a magnet. It is measured in webers and is denoted as Ï•.
• Magnetic flux density () is the amount of magnetic flux per unit area of a section, perpendicular to the direction of magnetic flux; i.e.,

It is a vector quantity (expressed in webers per square metre) that specifies both the strength and direction of the magnetic field.
• The magnetic flux through a surface S is given as,

• The magnetic field intensity is related to the magnetic flux density as,

where Î¼ is a constant, called permeability of the medium. It is given as,

Î¼ = Î¼0 Î¼r
• where  Î¼0 is the permeability of free space, known as absolute permeability  H/m, and Î¼r is the relative permeability.
• As it is not possible to have an isolated magnetic pole, the total magnetic flux through a closed surface must be zero. This is known as Gaussâ€™ law of magnetostatics. It is given as,

and
• Ampereâ€™s circuital law states that the line integral of the magnetic field intensity  around any closed path is equal to the direct current enclosed by the path.

• The magnetic scalar potential (Vm) is defined as,

if
• The magnetic vector potential () is defined as,

• The magnetic field due to a current distribution can be found using the concept of magnetic vector potential and using the relation as,

• The Lorentz force equation relates the force on a moving charged particle in the presence of a magnetic field and is given as,

• The force in an element  of a current-carrying conductor carrying a current I, placed in a magnetic field is given as,

• The force between two current-carrying conductors is given by Ampereâ€™s force law and is written as,

• The magnetic dipole moment is the product of current and area of the loop; its direction is normal to the plane of the loop; its unit is Am2.

• The torque on a current-carrying coil with magnetic moment  placed in a uniform magnetic field  is given as,

• A bar magnet or a small filamentary loop carrying a current is known as a magnetic dipole.
• Magnetisation  is defined as the amount of magnetic moment per unit volume. It is expressed in amperes per metre (A/m).
• The effect of magnetisation in a given volume of magnetic material is to induce some bound surface and volume current densities in the material, given as,

Magnetisation volume current density or bound volume current density (A/m2) =

Magnetisation surface current density or bound surface current density (A/m) =
• The magnetic susceptibility  of a magnetic material is a measure of the degree of magnetisation of a material in response to an applied magnetic field. It relates the magnetic field to the magnetisation vector as,

• Permeability (Î¼) is the degree of magnetisation of a material that responds linearly to an applied magnetic field. It relates the magnetic field to the magnetic flux density as,

• For linear magnetic materials, the relation between different magnetic properties is given as,

• Depending upon the values of the magnetic susceptibility (Ï‡m) or the relative permeability (Î¼r), magnetic materials are broadly classified into three groups as,
1. Paramagnetic (Î¼r > 1, Ï‡m > 0)
2. Diamagnetic (Î¼r < 1, Ï‡m > 0)
3. Ferromagnetic
• The variation of  as a function of the externally applied field  is known as a hysteresis curve or magnetisation curve or Bâ€“H curve.
• The boundary conditions that the magnetic field must satisfy at the interface between two different magnetic media are,

BIn = B2n or Î¼1H1n = Î¼2H2n

And

or  if

Combining these boundary conditions it can be written as,

where Î¸1 and Î¸2 are the angles with the normal in respective medium.
• The ratio of the magnetic flux to the current is called the inductance, or more accurately self-inductance of the circuit.

• Mutual inductance is the ability of one inductor to induce an emf across another inductor placed very close to it.
• The relation between the self inductances of two coils and their mutual inductance is written as,

where, k is the coefficient of coupling.
• Magnetic energy density (energy per unit volume) is given as,

• Total magnetic energy stored can be written in different forms as,