What is Self Induction

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Self induction is the property of a coil by virtue of which the coil opposes any change in the strength of current flowing through it by inducing an in itself. For this reason, self induction is also called the inertia of electricity.

If current in a coil L is changed by varying the contact position on a variable resistor, a self induced appears in the coil – while the current is changing.

When current (I) is increasing the self induced appears across the coil in a direction such that it opposes the increase.

When current (I) is decreasing the self induced (e) appears across the coil in a direction such that it opposes the decrease. Therefore it would be in the direction of (I)

Coefficient of self induction

Suppose I= strength of current flowing through a coil at any time.

∅ = amount of magnetic flux linked with all the turns of the coil at that time.

It is found that ∅ ∝I

∅ = LI

Where L is a constant of proportionality and is called coefficient of self induction or self inductance of the coil. The value of L depends on number of turn’s area of cross-section and nature of material of the core on which it he coil is wound.

If I = 1, ∅ = L X 1 or L = ∅

Therefore, coefficient of self induction of a coil is numerically equal to the amount of magnetic flux linked with the coil when unit current flows through the coil.

Now, the induced in the coil is given by

E = d ∅ / dt = – d / dt (LI)

E = – L  dI / dt

If dI / dt = 1, then e = – L X1

Or L = – e 

Hence coefficient of self induction of a coil is equal to the induced in the coil when rate of change of current through the coil is unity.

Inductance is a scalar quantity.

The SI unit of L is Henry. Self inductance of a coil is said to be one Henry (H) when a current change at the rate of 1 ampere/sex through the coil induced an of 1 volt in the coil.

From (12) L = ∅/I

∴ 1 Henry = 1 Weber/ampere

From (13) L = – e / dI / dt

∴ 1 hennery = 1 volt / 1 ampere/ ec 

Thus 1 Henry = 1 Weber/ ampere = 1 volt – sec / ampere 

Smaller units of L are

1 millinery (1 mh) = 10 -3 Henry,

1 micrometry (1μ H) = 10-6 Henry,

Dimensions of inductance L

From (13) L = e dt / dI

As e = work (W) / charge (q) ∴ L = W / q dt / dI

L = [M L2 T-2 ] [ T] / {AT} {AT} = M L T A-2

A wire cannot act as an inductor as magnetic flux linked with wire of negligible area of cross section is zero. The wire has to be in the form of a coil to serve as an inductor. Further, self induced appears only during the time the current is changing.

What will be induced in a 10 H inductor in which current changes from 10 A to 7 A in 9 X 10 -2 s?

Here e = ? L = 10 H, I = 10 A 

I2 + 7 A, dt = 9 X 10 -2s

As e = – L dI / dt

= – L (I2 – I1) / dt = – 10 (7- 10) / 9 X 10-2

∴ e = 333.3 volt.

A coil of inductance 0.5 H is connected to a 13 V battery. Calculate the rate of growth of current?

Here L = 0.5 He = 18 V, dt/dt =?

From e = L dI / dt (leaving neg. sign)

dI / dt = e / L = 18 / 0.5 = 36 As -1

Related Tags: Electromagnetic Induction, Electric Current

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Classification of Algebra Mathematics

Algebra may be divided roughly into following categories:

Elementary algebra, în which properties of operations on real number system are recorded using symbols as “place holders” to denote constants & variables, & rules governing mathematical expressions & equations involving these symbols are studied, This îs usually taught at school under title algebra (or intermediate algebra & college algebra în subsequent years), University-level courses în group theory may also be called elementary algebra,

Abstract algebra, sometimes also called modern algebra, în which algebraic structures such as groups, rings & fields are axiomatically defined & investigated,

Linear algebra, în which specific properties of vector spaces are studied (including matrices);

Universal algebra, în which properties common to all algebraic structures are studied,

Algebraic number theory, în which properties of numbers are studied through algebraic systems, Number theory inspired much of original abstraction în algebra,

Algebraic geometry applies abstract algebra to problems of geometry,

Algebraic combinatory, în which abstract algebraic methods are used to study combinatorial questions,

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Exports Imports Demand

Help with Macroeconomics, Microeconomics, Economics Help

Before we analyse the combined effect of exports and imports on the national income, let us first analysis separately the effect of export and import on the national income. Let us begin by looking at the export and import functions, their determinants and how they affect the aggregate demand and the national income.

Export function and export multiplier

Like C, I and G. exports of goods and services constitute a part of the aggregate demand in an economy and its effect on the economy is also the same. There is however a difference. The demand of consumer goods (C), investment goods (I) and the government purchases (G) originate within the economy and is called domestic demand. By the demand for exports originates outside the economy. It is therefore called the external demand. Let us look at the determinants of exports and the export function.

Export function

Exports of a country are a function of a number of external and internal factors. Some of the important external determinants of exports of a country are: (i) domestic prices fo exports in relation to those importing countries, (ii) income of the importing countries, (iii) importers income-elasticity for imports, (iv)their tariffs and trade policy and (v) their exchange rate policy and foreign exchange restrictions. Some of the important internal determinants of exports of a country include; (i) export policy fo the exporting country, (ii) export duties and subsidies, (iii) availability of exportable surplus, (iv) tirade and tariffs agreements with other countries, and (v) international competitiveness of domestics goods.

Exports and aggregate demand

As noted above, in an open economy exports constitute a part of the aggregate demand. Exports result in inflows of incomes from abroad. A part of this income is consumed and a part saved the increase in consumption due to increase in exports affects the economy in the same manner as the increase in consumption due to increase in income. Since exports constitute a part of the aggregate demand, AD for an open economy is given as

AD = C + I + G + X

Assuming there are no imports, the equilibrium level of the national income with exports can be written as:

Y = C + I + G + X

Where,

C = a + b (Y – T)

The equilibrium level of income in an economy with no imports can be obtained by substituting Eqs. Thus

Y = a + b(Y-T) + I G + X

Y = 1 / 1-b (a – bT + I + G + X)

Export multiplier

Given the Eq, the export multiplier (X-multiplier) can be easily worked out. Assuming an increase in exports,∆X, national income equilibrium equation can be rewritten as:

Y + ∆Y = 1/1-b (a – bT + I + G + X + ∆X)

Subtracting Eqs, we get

∆Y = 1/1-b (∆X)

∆Y/∆X = 1/1-b = X-multiplier

Import function 

Let us now look at the determinants of imports and their effect on the aggregate demand and on the national income. We begin by specifying the import function. Imports are purchases of goods and services from abroad, payments for imports are a leakage from the income stream because payments made for imports make the domestic income flow out of the economy the level of imports determines the level of outflow of domestic income.

Imports and aggregate demand

The aggregate demand as given in Eq. is reduced by the amount of payments fo imports. This negative effect of imports on the aggregate demand is accounted for by including imports (m) as a negative value in the aggregate demand equation. Following the national income accounting conventions, only exports net of imports (X –M) appear in the aggregate demand equation. Thus the aggregate demand equation for an open economy is expressed as:

Y = C + I + G + (X – M)

Equation means that if M > X, the aggregate demand decreases, and if X > M, the aggregate demand increases.

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Ecology îs sub discipline of biology,study of life, The word “ecology” (“Ökologie”) was coined în 1866 byGerman scientist Ernst Haeckel (1834–1919), Ancient philosophers of Greece, including Hippocrates&Aristotle, were amongearliest to record notes&observations onnatural history of plants&animals, Modern ecology branched out of natural history&matured into more rigorous science inlate 19th century, Charles Darwin’s evolutionary treatise includingconcept of adaptation, as it was introduced în 1859, îs pivotal cornerstone în modern ecological theory, Ecology îs not synonymous with environment, environmentalism, natural history environmental science, It îs closely related to physiology, evolutionary biology, geneticsðology, An understanding of how biodiversity affects ecological function îs important focus area în ecological studies, Ecologists seek to explain:

Life processes&adaptations

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Ecology îs human science as well, There are many practical applications of ecology în conservation biology, wetland management, natural resource management (agriculture, forestry, fisheries), city planning (urban ecology), community health, economics, basic&applied science&human social interaction (human ecology), Ecosystems sustain every life supporting function onplanet, including climate regulation, water filtration, soil formation (pedogenesis), food, fibers, medicines, erosion control,&many other natural features of scientific, historical spiritual value,

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Electromagnetic Waves Facts

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Electromagnetic Waves Facts

We know that an electric charge at rest has electric field in the region around it, but no magnetic field. A moving charge produces both the electric and magnetic fields. If a charge is moving with a constant velocity if current is not changing with time), the electric and magnetic fields will not change with time hence no electromagnetic wave can be produce but if the charge is moving with a non- zero acceleration (charge is accelerated) both the magnetic field and electric field will change with space and item it then produces electromagnetic wave. The shows that an accelerated charge emits electromagnetic waves.

L-C circuit is an oscillatory circuit where the charge is oscillating across the capacitor plates. An oscillating charge in L – C circuit has a non- zero acceleration hence it emits electromagnetic waves which have the same frequency as that of the oscillating change.

In an atom an electron while orbiting around the nucleus in a stable orbit thought accelerating does not emit electromagnetic waves electromagnetic waves are emitted only when it falls from higher energy orbit to lower energy orbit.

Electromagnetic waves are also produced when fast moving electors are suddenly stopped by the metal target of high stoic number.

Important facts about the electromagnetic waves

The electromagnetic waves are produces by accelerated or oscillating charge.

These waves do not require any material medium for propagation.

These waves travel in free space with a speed 3 x 10 (seed of light) give by the relation c = 1 √(μ0 ϵ0).

The sinusoidal variation in both electric and magnetic field vectors ( E and B) occurs simultaneously as a result they attain the maxima and minima at the same place and at the same time. The amplitudes of the electric and magnetic fields in free space are electric by E0 / B0 = C.

The directions of variation of electric and magnetic field vectors are perpendicular to each other as well as perpendicular to the direction of propagation of waves. Therefore electromagnetic waves are transverse in nature like light waves.

The velocity of electromagnetic waves depends entirely on the electric and magnetic properties of the medium in which these waves travel and is independent of the amplitude to the field vectors.

The velocity of electromagnetic wave in dielectric is less than c = (3 X 10 ms-1)

The electromagnetic waves carry energy which is divided equally between electric field and magnetic field vectors.

In vacuum the average electric energy density (Ug) and average magnetic energy density (uB) due to static electric field E and magnetic field B which do not vary with time are given by

uE = ½ ϵ0 E2 and uB = 1 / 2 B2 / μ0

Due to both the fields, total average energy density is

u = uE + uB = 1 / 2 ϵ0 E2 + 1 / 2 B2 / μ0

in electromagnetic waves the electric and magnetic fields vary sinusoidal with space (x) and time (t). the above expressions will be valid for electromagnetic waves if E and B are replaced by their rms values therefore,

uE = 1 / 2 ϵ0 E2 and uB = 1 / 2 B2 /μ

here uB = B2 / 2 μ0 = o 1 / 2 μX E2 / c2 = μ0 ϵ0 / 2μ0 E2 = 1 / 2 ϵ0 E2= uE

[ ∴ B = E /c and c = 1 / √(μ0 ϵ0) ]

∴ Total average energy density is u = uE + uB = 2 uE = 2UB = 2 X ½ ϵ0 E2 = ϵ0 E2 = B2 / μ0 = ϵ0 E20 /2 = 1 / μ0B /2.

The electric vector is responsible for the optical effects of an wave and is called the light vector

The electromagnetic waves being uncharged are not deflected by electric and magnetic fields.

Intensity of electromagnetic wave  

Intensity of electromagnetic wave is defined as the energy crossing per second per unit area perpendicular to the direction of propagating of electromagnetic wave.

Consider the propagation of electromagnetic wave with speed c along the X axis take and imaginary cylinder of area of cross section A and length C so that the wave crosses the area A normally let be the average energy density of electromagnetic wave

The energy of electromagnetic wave (U) crossing the area of cross section at P normally in time is the energy of wave contained in a cylinder of length c and area of cross- section A. it is given by U = U (c ∆t) 

The intensity of electromagnetic wave at P is

I = U / A ∆t = u c ∆t A / A ∆t = u c.

In terms of maximum electric fields

U = 1 / 2  ϵ0 E2.

So I = 1 . 2 ϵE2 c = E0 E2 c

In terms of maximum magnetic field

U = 1 /2 B2 / μ0 so I = 1 / 2 B/ μ0 c = 1 / μ0 B2 c

Momentum and radiation pressure of electromagnetic wave. The electromagnetic waves carry energy and momentum. If a portion of electromagnetic wave of energy U is propagating with speed then linear momentum of electromagnetic wave is given by

P = U / C

If the incident electromagnetic wave is completely absorbed by a surface it delivers energy U and momentum U/c to the surface if the electromagnetic waves is totally reflected then the momentum delivered to the surface is 2 U /c as te momentum of electromagnetic wave changes form p to p. this shown that the electromagnetic wave exerts a force on the surface an which it is incident.

This force exerted by electromagnetic wave on unit area of the surface is called radiation pressure. Pressure on the surface it fails. This electromagnetic wave exerts radiation pressures on the surface it falls.

Radiation pressure = force / area = change in momentum/ time/ area = change in momentum are X time

In 1903, the American scientist’s nicol and hull measured radiation pressure of visible light 7 X 10 -6 N/ m2. It means the force due to radiation pressure of visible light on the surface of area 10 cm2 is = (7 X 10-6) X (10 X 10 -4) = 7 X 10 -9 N.