Electromagnetic induction

The phenomenon of induction of electromotive force due to the relative motion between a magnet and a coil is known as electromagnetic induction.

Faraday discovery:

In 1831, Michael Faraday discovered that when a bar magnet was moved towards or away from a coil connected to a galvanometer as in, the galvanometer shows deflection. The galvanometer shows deflection even if the other pole of the magnet was moved towards or away from the coil. Faraday explained that the deflection is due to the electromotive force induced in the coil due to the motion of the magnet. The emf thus induced in the coil caused current in the galvanometer and the galvanometer needle is deflected.

Later in other experiments, Faraday also observed that the electromotive force is induced in the coil if the magnet is kept fixed and the coil is moved towards or away from the coil. This phenomenon of induction of electromotive force due to the relative motion between a magnet and a coil is known as electromagnetic induction.

Faraday Laws of electromagnetic induction:

First law of faraday’s of electromagnetic induction:

It state that whenever is a change in the magnetic flux lined with a coil an emf is induced in a coil which lasts so long as the change in magnetic flux continues.

Second law of faraday’s of electromagnetic induction:

This law give the measurement of induce emf. a/c to this law the magnitude of emf induce in a is proportional to the rate of change of magnetic flux linked with it i.e.

Induced emf e α dɸ/dt

Therefore e=-k dϕdt$\frac{\mathrm{d}\varphi }{\mathrm{d}\mathrm{t}}$

Where k is called proportionality constant whose value is 1 and

Negative sign show the opposing nature of the induce emf so the above expression may be written as e=-dϕdt$\frac{\mathrm{d}\varphi }{\mathrm{d}\mathrm{t}}$

Let at time t=0 second fluxed with a coil is ɸ1 and after time t=t second flux linked ɸ2 then change in fluxed link is given by ɸ2- ɸ1=dɸ

Therefore induced emf (e)=-dϕt$\frac{\mathrm{d}\varphi }{\mathrm{t}}$

Lenz's law:

According to Lenz's law, the direction of induced current in a coil is such that it always opposes the cause which produces it. This law follows the law of conservation of energy i.e. Energy neither can be created nor can be destroyed but can change from one form to another. In lenz’s law when the magnet is moved towards or away from coil, emf is induced in the coil at the expense of mechanical energy spent by the external agent. Hence in this way mechanical energy is converted to electrical energy. This show the Lenz's law is in accordance with the law of conservation of energy.

Production of E.M.F:

The induced emf can be produced by the following ways

By rotating coil in a magnetic field

By rotating rod in the magnetic field perpendicular to the length and the magnetic field

By passing a alternating current in the coil (self induction)

By mutual induction (passing current in the neighboring coil)

E direction of the emf can be determined by two ways

i. Lenz law: According to Lenz's law, the direction of induced current in a coil is such that it always opposes the cause which produces it. Consider the magnet which is brought towards the coil due to which flux linked with the coil changes .this causes the induced emf in the coil and induced current flows in a circuit as shown in a fig.

thus obtained current produces magnetic field in the coil with the north pole at the left and south pole at the right end .the north pole of the coil repels the north pole of the magnet opposing the motion of the magnet so that work is to be done to move further towards the coil. Similarly if the magnet is taken away from the coil the south pole of the magnet attracts the North Pole and work is to be done against the attractive force .In this way direction of the induced emf can be obtained from the receding motion of the magnet.

ii. Fleming right hand rule: When a conductor such as a wire attached to a circuit moves through a magnetic field, an electric current is induced in the wire due to Faraday's law of induction. The current in the wire can have two possible directions. Fleming's right-hand rule gives which direction the current flows

The right hand is held with the thumb, first finger and second finger mutually perpendicular to each other (at right angles), as shown in the diagram.

The thumb is pointed in the direction of motion of the conductor.

The first finger is pointed in the direction of the magnetic field. (North to south)

Then the second finger represents the direction of the induced or generated current (the direction of the induced current will be the direction of conventional current; from positive to negative).

Figure

Expression for induced e.m.f in a coil rotating in a magnetic field:

Let us consider a rectangular coil having area A is placed in a uniform magnetic field B in a such a way that it is normal to the plane of the coil making the angle δ with the magnetic field B as shown in figure

The component of field B at the right angle to the plane of the coil is Bcosδ . The flux through the coil is ABcosδ

If N be the no.of turns then the flux linkage ɸ is given by NABcosδ ……………1

If the coil turns, about an axis perpendicular to the file direction with the constant angular velocity ω or dδ/dt then the induce emf in the coil is E= −dϕdt$-\frac{\mathrm{d}\varphi }{\mathrm{d}\mathrm{t}}$= −d(NABcosδ)dt=NBAsinδ.dδdt$-\frac{\mathrm{d}\left(\mathrm{N}\mathrm{A}\mathrm{B}\mathrm{c}\mathrm{o}\mathrm{s}\delta \right)}{\mathrm{d}\mathrm{t}}=\mathrm{N}\mathrm{B}\mathrm{A}\mathrm{s}\mathrm{i}\mathrm{n}\delta .\frac{\mathrm{d}\delta }{\mathrm{d}\mathrm{t}}$………….2

We also know ω = dδ/dt …….3

On integrating

Or, ωt=δ…………..4

And ω=2πf…………..5

Now from 2, 3, 4, 5. We have

E=2πf NBAsin2πft$\mathrm{N}\mathrm{B}\mathrm{A}\mathrm{s}\mathrm{i}\mathrm{n}2\pi \mathrm{f}\mathrm{t}$…………6

If sin2πft=1$\mathrm{s}\mathrm{i}\mathrm{n}2\pi \mathrm{f}\mathrm{t}=1$ then emf become maximum the from equation 6 we have

E0=2πf NBA$\mathrm{N}\mathrm{B}\mathrm{A}$……………..7

Form equation 6 and 7 we have

E=E0sin2πft=E$\mathrm{s}\mathrm{i}\mathrm{n}2\pi \mathrm{f}\mathrm{t}=\mathrm{E}$0sinωt$\mathrm{s}\mathrm{i}\mathrm{n}\omega \mathrm{t}$.

Thus a coil rotating with a constant angular velocity in a uniform magnetic field produces a sinosoidally alternating emf. 

Mutual and self inductances:

When a change current (a.c) passing through a coil then the magnetic flux ɸ linked changes. The changes in the magnetic flux linked causes to induce emf in the coil itself. Therefore the phenomenon by which emf is induction in a coil by passing changing current through itself is called self induction.

From faraday’s second law of electromagnetic induction, the self inducted emf (e) is proportional to the rate of change of flux ɸ linked with the coil i.e. e α dϕdt$\frac{\mathrm{d}\varphi }{\mathrm{d}\mathrm{t}}$………1

dϕdt$\frac{\mathrm{d}\varphi }{\mathrm{d}\mathrm{t}}$ is also propotional to the dIdt$\frac{\mathrm{d}\mathrm{I}}{\mathrm{d}\mathrm{t}}$ i.e. dϕdtαdIdt$\frac{\mathrm{d}\varphi }{\mathrm{d}\mathrm{t}}\alpha \frac{\mathrm{d}\mathrm{I}}{\mathrm{d}\mathrm{t}}$…..2

fromequation$\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$1 and 2 we have e α dIdt$\frac{\mathrm{d}\mathrm{I}}{\mathrm{d}\mathrm{t}}$

or,$\mathrm{o}\mathrm{r},$e=-LdIdt$\frac{\mathrm{d}\mathrm{I}}{\mathrm{d}\mathrm{t}}$ where L is proportionality constant called coefficient of self induction.

Or, L=−edIdt$-\frac{\mathbf{e}}{\frac{\mathbf{d}\mathbf{I}}{\mathbf{d}\mathbf{t}}}$

It is also define as the emf induced for unit rate of change of current through itself.

The induction of an emf in the coil due to the flow of current in the neighboring coil is called mutual induction. If P and Q are the two coils place near to each other, the changing current I is set up in the coil P ,then the magnetic field around it also changes due to which flux linked with S is also changing .if ϕ2 is the flux linked with S then, From faraday’s second law of electromagnetic induction, emf (e) is proportional to the rate of change of flux ɸ2 linked with the coil i.e. e α dϕ2dt$\frac{\mathrm{d}{\varphi }_2}{\mathrm{d}\mathrm{t}}$………1

 dϕ2dt$\frac{\mathrm{d}{\varphi }_2}{\mathrm{d}\mathrm{t}}$ is also propotional to the dIdt$\frac{\mathrm{d}\mathrm{I}}{\mathrm{d}\mathrm{t}}$ i.e. dϕ2dt$\frac{\mathrm{d}{\varphi }_2}{\mathrm{d}\mathrm{t}}$αdIdt$\frac{\mathrm{d}\mathrm{I}}{\mathrm{d}\mathrm{t}}$…..2

fromequation$\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$1 and 2 we have e α dIdt$\frac{\mathrm{d}\mathrm{I}}{\mathrm{d}\mathrm{t}}$

or,$\mathrm{o}\mathrm{r},$e = -MdIdt$\frac{\mathrm{d}\mathrm{I}}{\mathrm{d}\mathrm{t}}$ where M is proportionality constant called coefficient of mutual induction.

Or, M =−edIdt$-\frac{\mathrm{e}}{\frac{\mathrm{d}\mathrm{I}}{\mathrm{d}\mathrm{t}}}$ negative sign indicates that the direction of emf is such that it opposes the cause of the current.

Construction and working of a transformer:

Transformers is a device used to convert high volt, low current AC into low volt high current AC or vice versa.

The transformer which converts high voltage AC into low voltage is called step down transformers. The transformers which converts low voltage AC into high voltage is called step up transformers. Transformer is based on the principle of mutual induction.

When a magnetic flux is linked with a coil changes emf is induced in the nearby coil. It consists of two coild, primary and secondary of insulated copper wire on the laminated soft iron core. The soft iron core consists of thin strips of iron coated with vanish to insulated from each other and put together to form a single. The primary coil is connected with the source of AC and the secondary is connected with the load.

When the current is passed through the primary coil, magnetic flux linked with the coil changes and emf is induced in the coil.

If θ is the flux linked with the primary coil and Ep is itself induction then the primary voltage. Ep is given by:

Ep = −NPdθdt$-{\mathrm{N}}_{\mathrm{P}}\frac{\mathrm{d}\theta }{\mathrm{d}\mathrm{t}}$…(i)

Where, Np is the number of turns in the primary coil. If there is no loss of flux then the rate of flux linked with the secondary coil will be same.

Then the induced emf is secondary coil,

Or, ES = Ns.dθdt$\frac{\mathrm{d}\theta }{\mathrm{d}\mathrm{t}}$ ….(ii)

Where, NS is the number of turns in secondary coil. Dividing equation (i) and (ii), we get,

Or, EPEs=NpNs$\frac{{\mathrm{E}}_{\mathrm{P}}}{{\mathrm{E}}_{\mathrm{s}}}=\frac{{\mathrm{N}}_{\mathrm{p}}}{{\mathrm{N}}_{\mathrm{s}}}$

Where, NpNs$\frac{{\mathrm{N}}_{\mathrm{p}}}{{\mathrm{N}}_{\mathrm{s}}}$ is called transformer ratio

If NP/NS> 1 then EP> ES and the transformer is step down.

If NP/NS then ES> EP and the transformers in step up.

If Lp and LS is the co – efficient of self – induction of primary and secondary coil respectively.

Then, LpLs=N2PN2s$\frac{{\mathrm{L}}_{\mathrm{p}}}{{\mathrm{L}}_{\mathrm{s}}}=\frac{{\mathrm{N}}_{\mathrm{P}}^2}{{\mathrm{N}}_{\mathrm{s}}^2}$ [L ∝$\propto$ N2]

So, NpNs=(NpLs)−−−−−√$\frac{{\mathrm{N}}_{\mathrm{p}}}{{\mathrm{N}}_{\mathrm{s}}}=\sqrt{\left(\frac{{\mathrm{N}}_{\mathrm{p}}}{{\mathrm{L}}_{\mathrm{s}}}\right)}$

So, EpEs=(CpLs)−−−−−√$\frac{{\mathrm{E}}_{\mathrm{p}}}{{\mathrm{E}}_{\mathrm{s}}}=\sqrt{\left(\frac{{\mathrm{C}}_{\mathrm{p}}}{{\mathrm{L}}_{\mathrm{s}}}\right)}$

If CP is the input current and Isis the output current then, input = IpEp.

Similarly,

Output power = IIES.

If there is no loss of energy then,

Input power = Output power.

Then,

IpEp = ISES

Or, EpEs=IsIp$\frac{{\mathrm{E}}_{\mathrm{p}}}{{\mathrm{E}}_{\mathrm{s}}}=\frac{{\mathrm{I}}_{\mathrm{s}}}{{\mathrm{I}}_{\mathrm{p}}}$.

Hence, the output voltage is increased, the output current will decrease and vice – versa. Therefore, for a long distance transmission line, high voltage is used. If the secondary voltage is high then current will be below and hence the energy loss (I2Kt) due to resistance of wire is low.

Theory and working of an ac generator:

A.C. generator is an electronic mechanic which converts mechanical energy in to electrical energy. It work on the principle of electromagnetic induction i.e. when the coil is rotated in the uniform magnetic field then the emf is developed on the rotating coil.

Construction of A.C. generator:

Figure

Armature: the rectangular coil (PQRS) consist large no. of turn which wounded the soft iron core is called armature.

Strong magnetic field: the rectangular coil PQRS is rotated in the strong magnetic field produce due to magnet N and S.

Slip ring: the rectangular coil PQRS are connected with two ring R1 and R2seperatly. The ring R1and R2rotated with the rectangular coil PQRS.

Brush: the two metallic brush B1andB2 are fixed and connected with load through which output obtained

Working

When the rectangular coil is roated in the strong magnetic field then the magnetic line of force is cut and flux link changes then a/c to faraday’s law of electromagnetic emf is induced in the coil .the current flow out through the brush B1 in one direction while in B2 in other direction i.e. the direction of flow of current are in opposite to each half cycle. Similar process also occurring other half cycle. This phenomenon is repeated continuously.

Theory: Let the axis of the coil is perpendicular to the magnetic field and coil PQRS are rotated with constant angular velocity ‘ω’ as shown in figure

Let ɵ be the angle normal to the coil and magnetic field B at any instant‘t’ then we have

ɵ=ωt…………..1

the component of magnetic field to the plane of the coil is B cosɵ then from equation 1 we have B cosωt let A be the area of the coil then the magnetic flux

ɸ=BAcosω

since the rectangular coil consisting large no. of turn say N then the magnetic flux due to N no. of coil is ɸ=BNAcosωt…………..2

Differentiating the equation 2 with respect time we get

E=dɸ/dt = -BNAωsinωt………….3

We also have from faraday’s law of electromagnetic induction

E= - dɸ/dt………….4

Now from eq. 3 and 4 we have

E=BNAωsinωt……..5

Let E0=BNAω

Then from equation 5 we have

E=E0sinωt………6.

Dividing the eq. 6 by R we get I=I0sinωt.

Now the variation of induce emf of the coil with respect of the magnetic field are given below

ɵ0.	E=E0sinωt	Result
0	0	Plane of coil is B
90	E0	Plane of coil is along B
180	0	Plane of coil is B
270	-E0	Plane of coil is along B
360	0	Plane of coil is B

This table show that the E increase from zero to maximum and decrease from maximum to minimum.

Theory and working of an Dc generator:

A DC generator converts the mechanical energy in to the electrical energy .It is based on the principle of the electromagnetic induction.

Construction

It consists of the split rings called commutator C1 and C2${\mathrm{C}}_2$ , the rectangular coil (PQRS)consist large no. of turn which wounded the soft iron core is called armature, field magnets and the brushes

Working

When the armature rotates, commutator also rotates with it and alternatively come in contact with the brushes b1 and B2. When the coil is horizontal as in figure no .9(a),thecommutator C1 connects while the arm RS of the armature to the end A and the commutator C2 connects w the arm PQ of the armature to the end A and the commutator C2 connects the arm PQ of the armature to the end b of the load. When the armature move anticlockwise ,the arm PQ moves upward and the arm RS moves downward .An e.m.f. is induced in the armature and current flows in the load (RL) from A to B in accordance with Flemings’s Right Hand Rule.

When a half revolution is completed, the position of the armature is reversed. The side PQ of the coil is close to the N pole of the magnet and side RS is close e to the S-pole of the magnet. As the armature is rotating, the current in it flow in opposite direction. However at this time the commutator C1 and C2 have also changed the position as they rotate with the armature. As a result, the commutator C2 connects the arm PQ to the end A of the load while the commutator C1 connects the arm SR to the end B of the load. Then, current passes through the load again from A to B.

Transformer in high power transmission:

The power at the power generating station is converted into high voltage at low current before transmitting it to long distances: If P is the power that is transmitted at voltage V from a power station to a city through a power line whose resistance is R then the current set up in the power line is I =P/V.

The current I=P/V flows through the power line during in transmission. Therefore the energy dissipated across the power line in the form of heat is given by H= I 2R= (P/V)2 R =P 2R/V2.this heat is the energy that gets lost on the way and is not available for the consumption in the city. For an efficient power transmission, H should be as small as possible for which V should be as high as possible. If V is high , I is low .it is therefore the power is generating station is converted in to a high voltage at low current before transmitting it to long distances .the power is converted to high voltage at low current by using step up transformer.