Laws of Electronic/Electrical Engineering: 2013

Sunday, February 17, 2013

Maxwell's equations

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics and electric circuits.

Maxwell's equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents.

They are named after the Scottish physicist and mathematician James Clerk Maxwell who published an early form of those equations between 1861 and 1862.

Conceptually, Maxwell's equations describe how electric charges and electric currents act as sources for the electric and magnetic fields and how they affect each other. Of the four equations, two describe how the fields emanate from charges and the other two equations describe how the fields circulate around their respective sources.

The following equations are the conventional formulation of the Maxwell equations in terms of vector calculus using time dependent vector fields. Symbols in bold represent vector quantities, and symbols in italics represent scalar quantities.

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Ref: Wikipedia

Biot–Savart law

The Biot–Savart law is an equation that describes the magnetic field generated by an electric current.

It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current.

It is named for Jean-Baptiste Biot and Felix Savart who discovered this relationship in 1820.

The Biot–Savart law is used to compute the resultant magnetic field B at position r generated by a steady current I (for example due to a wire)

$\mathbf{B} = \frac{\mu_0}{4\pi} \int_C \frac{I d\mathbf{l} \times \mathbf{r}}{|\mathbf{r}|^3}$

where r is the full displacement vector from the wire element to the point at which the field is being computed, dl is a vector whose magnitude is the length of the differential element of the wire, in the direction ofconventional current, and μ₀ is the magnetic constant

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Ref: Wikipedia

Lorentz force

The Lorentz force is the force on a point charge due to electromagnetic fields.

If a particle of charge q moves with velocity v in the presence of an electric field E and a magnetic field B, then it will experience a force

$\mathbf{F} = q[\mathbf{E} + (\mathbf{v} \times \mathbf{B})]$

( Lorentz force f on a charged particle (of charge q) in motion (instantaneous velocity v). The E field and B field vary in space and time. )

Force on a current carrying wire

When a wire carrying an electrical current is placed in a magnetic field, each of the moving charges, which comprise the current, experiences the Lorentz force, and together they can create a macroscopic force on the wire (sometimes called the Laplace force).

By combining the Lorentz force law above with the definition of electrical current, the following equation results, in the case of a straight, stationary wire:

$\mathbf{F} = I \boldsymbol{\ell} \times \mathbf{B} \,\!$

where ℓ is a vector whose magnitude is the length of wire, and whose direction is along the wire, aligned with the direction of conventional current flow I.

(Right-hand rule for a current-carrying wire in a magnetic field B)

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Ref: Wikipedia

Lenz's law

An induced electromotive force (emf) always gives rise to a current whose magnetic field opposes the original change in magnetic flux.

$\mathcal{E}=-N\frac{\partial \Phi_\mathrm{B}}{\partial t},$

The induced emf ( $\mathcal{E}$ ) and the change in magnetic flux ( $\partial\Phi_\mathrm{B} \,$ ) have opposite signs.

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Ref: Wikipedia

Ampere's law

Ampere's circuital law relates the integrated magnetic field around a closed loop to the electric current passing through the loop.

Using Ampere's law, one can determine the magnetic field associated with a given current or current associated with a given magnetic field, providing there is no time changing electric field present.

The magnetic field in space around an electric current is proportional to the electric current which serves as its source, just as the electric field in space is proportional to the charge which serves as its source.

Ampere's Law states that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability times the electric current enclosed in the loop.

(An electric current produces a magnetic field.)

In terms of total current, which includes both free and bound current, the line integral of the magnetic B-field (in tesla, T) around closed curve C is proportional to the total current I_enc passing through a surface S(enclosed by C):

$\oint_C \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \mu_0 \iint_S \mathbf{J} \cdot \mathrm{d}\mathbf{S} = \mu_0I_\mathrm{enc}$

where J is the total current density (in ampere per square metre, Am⁻²)

Alternatively in terms of free current, the line integral of the magnetic H-field (in ampere per metre, Am⁻¹) around closed curve C equals the free current I_{f, enc} through a surface S:

$\oint_C \mathbf{H} \cdot \mathrm{d}\boldsymbol{\ell} = \iint_S \mathbf{J}_\mathrm{f}\cdot \mathrm{d}\mathbf{S} = I_{\mathrm{f,enc}}$

where J_f is the free current density only.

$\scriptstyle \oint_C$ is the closed line integral around the closed curve C,
$\scriptstyle \iint_S$ denotes a 2d surface integral over S enclosed by C
• is the vector dot product,
dℓ is an infinitesimal element (a differential) of the curve C (i.e. a vector with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to the curve C)
dS is the vector area of an infinitesimal element of surface S (that is, a vector with magnitude equal to the area of the infinitesimal surface element, and direction normal to surface S. The direction of the normal must correspond with the orientation of C by the right hand rule),

The B and H fields are related by the constitutive equation

$\mathbf{B} = \mu_0 \mathbf{H} \,\!$

where μ₀ is the magnetic constant.

The Ampere unit:

Ampere is a unit of current.

A current of one ampere is one coulomb of charge going past a given point per second:

$\rm 1\ A=1\tfrac C s.$

In general, charge Q is determined by steady current I flowing for a time t as Q = It.

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Ref: Wikipedia

Saturday, February 16, 2013

Coulomb's law

Coulomb's law or Coulomb's inverse-square law is a law of physics describing the electrostatic interaction between electrically charged particles.

This law states that "The force of attraction or repulsion between two point charges is directly proportional to the product of magnitude of each charge and inversely proportional to the square of distance between them".

A graphical representation of Coulomb's law

If the two charges have the same sign, the electrostatic force between them is repulsive;

if they have different sign, the force between them is attractive.

Mathematical equations:

Scalar form:

$|\boldsymbol{F}|=k_e{|q_1q_2|\over r^2}$

Vector form:

$\boldsymbol{F}=k_e{q_1q_2\boldsymbol{\hat{r}_{21}}\over r_{21}^2}$

where

k e = 1 ⁄ 4π ε 0 (

Coulomb's constant)

The Coulomb unit:

The coulomb is the SI derived unit of electric charge (symbol: Q or q).

It is defined as the charge transported by a steady current of one ampere in one second:

$1 \mathrm{C} = 1 \mathrm{A} \times 1 \mathrm{s}$

One coulomb is also the amount of excess charge on the positive side of a capacitance of one farad charged to a potential difference of one volt:

$1 \mathrm{C} = 1 \mathrm{F} \times 1 \mathrm{V}$

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Ref: Wikipedia

Faraday's law of induction

Faraday's law of induction is a basic law of electromagnetism that predicts how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF).

It is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators and solenoids.

Electromagnetic induction is the production of a potential difference (voltage) across a conductor when it is exposed to a varying magnetic field

The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit.

The Farad unit:

The farad (symbol: F) is the SI derived unit of capacitance. The unit is named after the English physicist Michael Faraday.

A farad is the charge in coulombs which a capacitor will accept for the potential across it to change 1 volt

Example: The voltage across a capacitor with capacitance of 47 nF will increase by 1 volt per second with a 47 nA input current.

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Ref: Wikipedia

Kirchhoff's current law (KCL)

At any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node.

OR

The algebraic sum of currents in a network of conductors meeting at a point is zero.

(The current entering any junction is equal to the current leaving that junction.i₂ + i₃ = i₁ + i₄)

$\sum_{k=1}^n {I}_k = 0$

n is the total number of branches with currents flowing towards or away from the node.

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Ref: Wikipedia

Kirchhoff's voltage law (KVL)

The directed sum of the electrical potential differences(voltage) around any closed network is zero.

OR

The sum of the emfs in any closed loop is equivalent to the sum of the potential drops in that loop.

OR

The algebraic sum of the products of the resistances of the conductors and the currents in them in a closed loop is equal to the total emf available in that loop.

(The sum of all the voltages around the loop is equal to zero. v₁ + v₂ + v₃ - v₄ = 0)

$\sum_{k=1}^n V_k = 0$

Here, n is the total number of voltages measured in a loop.

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Ref: wikipedia

Ohm's Law

Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship:

$I = \frac{V}{R}$

where I is the current through the conductor in units of amperes, V is the potential difference measured across the conductor in units of volts, and R is the resistance of the conductor in units of ohms. More specifically, Ohm's law states that the R in this relation is constant, independent of the current.

In circuit analysis, three equivalent expressions of Ohm's law are used interchangeably:

$I = \frac{V}{R} \quad \text{or}\quad V = IR \quad \text{or} \quad R = \frac{V}{I}.$

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Ref: wikipedia