## Thursday, 5 December 2019

Faraday's Law Of Electromagnetic Induction :

The meaning Of Electromagnetic Induction is a procedure wherein a conductor is placed in a specific position and attractive field continues changing or attractive field is stationary and a conductor is moving.

This procedure Generates a Voltage or EMF (Electromotive Force) over the electrical conveyor.

The attractive power we took a gander at the power experienced by moving charges in an attractive field.

The power on a current-conveying wire because of the electrons which move inside it when an attractive field is available is a great model.

This Either moving a wire through an attractive field or (equally) changing the quality of the attractive field after some time can make a present stream.

### What is Faraday's Law ?

At that point Faraday's law of electromagnetic acceptance is an essential law of electromagnetism anticipating how an attractive field will associate with an electric circuit to create an electromotive power (EMF).

This wonder is known as electromagnetic acceptance.

The Law Of Faraday expresses that a present will be instigated in a conductor which is presented to a changing attractive field.

The law of Lenz electromagnetic enlistment expresses that the course of this actuated current will be with the end goal that the attractive field made by the instigated current contradicts the underlying changing attractive field which delivered it.

This heading of this present stream can be resolved utilizing Fleming's correct hand rule.

Law of enlistment clarifies the working standard of transformers, engines, generators, and inductors.

This law is named after Michael Faraday, who played out an examination with a magnet and a loop.

When During Faraday's investigation, he found how EMF is prompted in a loop when the motion going through the curl changes.

You Have To Read this : On the off chance that Any adjustment in the attractive field of a curl of wire will make an emf be instigated in the loop.

In this way, emf instigated is called incited emf and if the conductor circuit is shut, the present will likewise circle through the circuit and this current is called initiated current.

You Have To Read this :

Techniques Are change the attractive field:

The moving a magnet towards or away from the loop

• In this way, moving the loop into or out of the attractive field
• By changing the territory of a curl in the attractive field
• By pivoting the loop comparative with the magnet
• You Can Also Read About : Sequential And Combinational Logic Circuits

It communicates that greatness of emf instigated in the loop is identical to the pace of progress of motion that linkages with the curl. The progress linkage of the circle is the consequence of the amount of turns in the twist and movement related with the curl.

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Recipe

How about we Suppose That a magnet is drawing closer towards a curl. Here we consider two moments at time T1 and time T2.

at time T1 Flux linkage with the curl, at time T2 Flux linkage with the curl, In this way, Flux linkage Change,  In this way, the Change in motion linkage Presently, the pace of progress of transition linkage Take The inference on right-hand side we will get The pace of progress of motion linkage As indicated by Faraday's law of electromagnetic enlistment, the pace of progress of motion linkage is equivalent to instigated emf. As Per Lenz's Law.

Where:

Motion Φ in Wb = B.A

B = attractive field quality

A = zone of the curl

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### The best technique to Increase EMF Induced in a Coil

With Expanding the amount of turns on top of it i.e N, from the formulae decided above it is successfully seen that if the amount of turns in a twist is extended, the incited emf also gets extended.

By extending alluring field quality i.e B enveloping the twist Mathematically, if appealing field fabricates, progress increases and if movement grows emf induced will similarly get extended. Speculatively, if the circle is experienced a more grounded appealing field, there will be more lines of intensity for the twist to cut and from now on there will be more emf incited.

By extending the speed of the relative development between the twist and the magnet – If the relative speed between the circle and magnet is extended from its past worth, the twist will cut the lines of movement at a faster rate, so progressively actuated emf would be conveyed.

You May Like To Read : Logic Families

PC hard drives apply the standard of alluring acknowledgment. Recorded data are made on a secured, turning plate.

By and large, scrutinizing these data was made to wear down the standard of selection. Regardless, most information today is passed on in cutting edge rather than straightforward structure—a movement of 0s or 1s are created upon the turning hard drive.

Consequently, most hard drive readout contraptions don't wear down the standard of acknowledgment, anyway use a strategy known as beast magnetoresistance.

Mammoth magnetoresistance is the effect of a colossal contrast in electrical check impelled by an applied alluring field to thin films of subbing ferromagnetic and non attractive layers.

This is one of the essential colossal accomplishments of nanotechnology.

Delineations tablets, or tablet PCs where an uncommonly organized pen is used to draw electronic pictures, moreover applies acknowledgment norms.

The tablets discussed here are set apart as detached tablets, since there are various structures that usage either a battery-worked pen or optical sign to make with.

The idle tablets are not exactly equivalent to the touch tablets and phones immense quantities of us use regularly, anyway may even now be found when denoting your imprint at a business register.

Underneath the screen, are little wires discovering the length and width of the screen.

The pen has an unassuming appealing field beginning from the tip.

As the tip brushes over the screen, a changing appealing field is felt in the wires which changes over into a prompted emf that is changed over into the line you just drew.

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### Power transformers work reliant on Faraday's law

The basic working rule of the electrical generator is Faraday's law of regular selection.

The Induction cooker is the snappiest strategy for cooking. It furthermore tackles the rule of regular selection.

Right when current courses through the twist of copper wire set underneath a cooking compartment, it makes a changing appealing field.

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This subbing or changing alluring field starts an emf and thusly the current in the conductive holder, and we understand that the movement of current reliably conveys heat in it.

Electromagnetic Flow Meter is used to measure the speed of explicit fluids. Right when an appealing field is applied to an electrically ensured channel in which driving fluids are gushing, by then as showed by Faraday's law, an electromotive power is impelled in it.

This started emf is comparative with the speed of fluid gushing.

Structure bases of Electromagnetic speculation, Faraday's idea of lines of intensity is used in comprehended Maxwell's conditions.

According to Faraday's law, change in appealing field offers climb to change in electric field and something contrary to this is used in Maxwell's conditions.

It is in like manner used in melodic instruments like an electric guitar, electric violin, etc.

### How We Can Describe Electromagnetic Induction?

Faraday's law, in view of 19ᵗʰ century physicist Michael Faraday. This relates the pace of progress of appealing motion through a hover to the degree of the electro-aim control prompted over it.

The relationship is : The electromotive power or EMF suggests the potential differentiation over the exhausted circle.

For all intents and purposes it is consistently satisfactory to consider EMF voltage since both voltage and EMF are evaluated using a comparative unit, the volt.

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#### Lenz's law :

Lenz's law is a result of assurance of imperativeness applied to electromagnetic selection. It was characterized by Heinrich Lenz in 1833.

While Faraday's law uncovers to us the degree of the EMF conveyed, Lenz's law unveils to us the heading that present will stream.

It communicates that the bearing is for each situation with the ultimate objective that it will negate the modification moving which made it.

This suggests any appealing field made by an impelled current will be the other route to the alteration in the main field.

Lenz's law is regularly merged into Faraday's law with a short sign, the thought of which empowers a comparable orchestrate structure to be used for both the movement and EMF. The result is from time to time called the Faraday-Lenz law, Eventually we routinely oversee appealing acknowledgment in various twists of wire all of which contribute a comparable EMF. Thusly an additional term NNN addressing the amount of turns is normally included.

For instance : ### What is the association between Faraday's law of acceptance and the attractive power?

While the full theoretical supporting of Faraday's law is very stunning, a determined understanding of the prompt relationship with the appealing force on a charged atom is reasonably immediate. Consider an electron which is permitted to move inside a wire. As showed up in figure , the wire is set in a vertical alluring field and moved inverse to the appealing field at steady speed.

The two pieces of the deals are related, forming a circle.

This ensures any work done in making a current in the wire is spread as warmth in the resistance of the wire.

An individual pulls the wire with unfaltering velocity through the appealing field. As they do in that capacity, they have to apply a power.

The predictable appealing field can't do work without any other individual's information (for the most part its quality would need to change), yet it can adjust the course of a power.

For this circumstance a bit of the power that the individual applies is re-facilitated, causing an electromotive power on the electron which goes in the wire,

Working up a current. A part of the work the individual has done pulling the wire finally results in essentialness scattered as warmth inside the hindrance of the wire.

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### Test Of Faraday's : Induction from a magnet traveling through a loop

It might be adequately imitated with insignificant more than family materials.

Faraday used a cardboard chamber with secured wire collapsed over it to outline a circle.

A voltmeter was related over the twist and the impelled EMF read as a magnet was experienced the circle. The plan is showed up in Figure. The observations were according to the accompanying:

Magnet extremely still in or near the twist: No voltage viewed.

Magnet pushing toward the circle: Some voltage evaluated, rising to a top as the magnet moves toward the point of convergence of the twist.

The Magnet experiences the focal point of the twist: Measured voltage rapidly changes sign.

Magnet drops and away from the circle: Voltage assessed the other path to the earlier example of the magnet moving into the twist.

An instance of the EMF evaluated is plotted against magnet position in Figure. These recognitions are unsurprising with Faraday's law. Disregarding the way that the stationary magnet may convey a colossal alluring field, no EMF can be incited because the progress through the twist isn't developing.

Exactly when the magnet moves closer to the circle the movement rapidly augments until the magnet is inside the twist.

As it experiences the circle the alluring progress through the twist begins to decrease. Consequently, the started EMF is convoluted.

### Self Induction

If a long twist of wire of a cross sectional area An and length ℓ with N occupies is related or withdrew from a battery, the changing alluring movement through the circle makes an induced emf.

The induced current makes an alluring field, which limits the alteration in the appealing change. The degree of the started emf can be resolved using Faraday's law.

The appealing field inside the long twist is B = μ0(N/ℓ)I.

The change through the circle is NBA = μ0(N2/ℓ)IA.

The modification experiencing significant change per unit time is μ0(N2/ℓ)A ∆I/∆t = L*∆I/∆t, since I is the primary sum changing with time.

L = μ0(N2/ℓ)A is known as the self inductance of the circle. The units of inductance are Henry (H). 1 H = 1 Vs/A.

The impelled emf can't avoid being emf = - L*∆I/∆t, where the less sign is an aftereffect of Lenz's law.

The impelled emf is comparing to the pace of progress of the current on top of it. It will in general be a couple of times the power supply voltage.

Exactly when a switch in a circuit passing on a huge current is opened, diminishing the current to center in a brief time span interval, this can realize a radiance.

All circuits have self inductance, and we for the most part have emf = - L*∆I/∆t. The self inductance L depends just on the geometry of the circuit.

### Magnet Brake and Eddy Currents

#### Magnet Braking :

1) Suspend the horseshoe magnet by a string over the lab table. Evacuate the magnet's guardian bar, on the off chance that it isn't now expelled.

2) Spin the magnet on its hub on the finish of the string. With some consideration this should be possible so the magnet pivots set up, with small wobbling. Note that the magnet will turn for quite a while ceaselessly, on the other hand winding and loosening up the string. Watch this conduct of the turning magnet for a couple of cycles.

3)Now, bring the aluminum square near shafts of the turning magnet without contacting it by sliding it under the turning magnet. Is there an impact? Record your perceptions.

4) Repeat with the plastic square. Is there any distinction?

5) Hold the suspended magnet very still a centimeter or two over the aluminum hinder that is laying on the table top.

6) Being mindful so as not to contact the magnet, have one of your lab accomplices delicately pull the aluminum away evenly. Watch and record the impact on the magnet.

7) What happens when you turn around the heading of the development of the aluminum square?

8) Now attempt this with the plastic square and see that the impact can't be because of air flows.

9) The genuine state of the vortex flows in this part is very entangled. Be that as it may, would you be able to own a general expression about powers and the overall movement of magnets and conductors?

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### A Moving Wire in a Magnetic Field

#### Model: A guitar pickup

1) Gently fix around 10 cm length of the copper wire, and fold this length over the wooden dance (you should fold the parts of the bargains over the two tacks on either side of the dance to get the wire rigid enough for this test.) Make the wire section moderately tight, and place the horseshoe magnet into the burden with it opening confronting upward with the wire between its shaft tips.

2) Now associate the parts of the bargains to the parts of the bargains to-banana link utilizing the two short wires with gator cuts on each end. Associate the BNC-to banana link to oscilloscope CH 1 contribution, as in .

A wooden square called a dance holds two parts of the bargains wire over its length. The horseshoe magnet is held with the goal that the wire fits between the shafts of the magnet. The parts of the bargains distend out from the dance. Each end interfaces with a gator cut, which are associated by the BNC-to-banana link to the oscilloscope.

3) coupling [DC]

Alter the oscilloscope show with the goal that the line follow is situated close to the focal point of the screen.

There are uncovered connector tips at the parts of the bargains lead; watch that these don't contact one another and short out your perception.

4) You presently have an attractive pickup as is found in an electric guitar (the wire being the guitar string). Cull the wire, and watch the outcome in the oscilloscope. What is the greatest voltage abundancy you can get along these lines?

5) Try shaking the wire gradually to and fro; is the playfulness the equivalent?

6) If you needed to improve your pickup circuit to acquire bigger sign, which changes may support: a bigger attractive field, a thicker wire, a littler wire opposition? Why?

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## Faraday's Laws of Electromagnetic Induction : First Law And Second Law

### Faraday's Law Of Electromagnetic Induction :

The definition Of Electromagnetic Induction is a process in which a conductor is put in a particular position and magnetic field keeps varying or magnetic field is stationary and a conductor is moving.

This process Generates a Voltage or  EMF (Electromotive Force) across the electrical conductor.

The magnetic force we looked at the force experienced by moving charges in a magnetic field.

The force on a current-carrying wire due to the electrons which move within it when a magnetic field is present is a classic example.

The process also works in reverse.

This Either moving a wire through a magnetic field or (equivalently) changing the strength of the magnetic field over time can cause a current to flow.

You Have To Read this :

#### What is Faraday’s Law ?

Then Faraday’s law of electromagnetic induction  is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF).

This phenomenon is known as electromagnetic induction.

The Law Of Faraday states that a current will be induced in a conductor which is exposed to a changing magnetic field.

The law of Lenz electromagnetic induction states that the direction of this induced current will be such that the magnetic field created by the induced current opposes the initial changing magnetic field which produced it.

This direction of this current flow can be determined using Fleming’s right-hand rule.

Law of induction explains the working principle of transformers, motors, generators, and inductors.

This law is named after Michael Faraday, who performed an experiment with a magnet and a coil.

You Have To Read this :

When During Faraday’s experiment, he discovered how EMF is induced in a coil when the flux passing through the coil changes.

If Any change in the magnetic field of a coil of wire will cause an emf to be induced in the coil.

So, emf induced is called induced emf and if the conductor circuit is closed, the current will also circulate through the circuit and this current is called induced current.

Methods Are change the magnetic field:

1. The moving a magnet towards or away from the coil
2. So, moving the coil into or out of the magnetic field
3. By changing the area of a coil  in the magnetic field
4. By rotating the coil relative to the magnet

It expresses that magnitude of emf induced in the coil is equivalent to the pace of progress of flux that linkages with the coil. The transition linkage of the loop is the result of the quantity of turns in the curl and motion related with the coil.

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#### Formula

Let's Suppose That a magnet is approaching towards a coil. Here we consider two instants at time T1 and time T2.

at time T1 Flux linkage with the coil,

at time T2 Flux linkage with the coil,

This change in flux linkage be,

So, the Change in flux linkage

Now, the rate of change of flux linkage

Take The derivation on right-hand side we will get

The rate of change of flux linkage

According to Faraday’s law of electromagnetic induction, the rate of change of flux linkage is equal to induced emf.

As Per Lenz’s Law.

Where:

Flux Φ in Wb = B.A
B = magnetic field strength
A = area of the coil

#### The most effective method to Increase EMF Induced in a Coil

With Expanding the quantity of turns in the loop i.e N, from the formulae determined above it is effectively observed that if the quantity of turns in a curl is expanded, the actuated emf additionally gets expanded.

By expanding attractive field quality i.e B encompassing the curl Mathematically, if attractive field builds, transition increments and if motion expands emf incited will likewise get expanded. Hypothetically, if the loop is gone through a more grounded attractive field, there will be more lines of power for the curl to cut and henceforth there will be more emf actuated.

By expanding the speed of the relative movement between the curl and the magnet – If the relative speed between the loop and magnet is expanded from its past worth, the curl will cut the lines of motion at a quicker rate, so increasingly instigated emf would be delivered.

You May Like To Read : Logic Families

PC hard drives apply the rule of attractive acceptance. Recorded information are made on a covered, turning plate.

Generally, perusing these information was made to chip away at the standard of enlistment. In any case, most info data today is conveyed in advanced instead of simple structure—a progression of 0s or 1s are composed upon the turning hard drive.

Subsequently, most hard drive readout gadgets don't chip away at the standard of acceptance, however utilize a procedure known as monster magnetoresistance.

Mammoth magnetoresistance is the impact of a huge difference in electrical obstruction instigated by an applied attractive field to thin movies of substituting ferromagnetic and non magnetic layers.

This is one of the primary huge achievements of nanotechnology.

Illustrations tablets, or tablet PCs where an extraordinarily structured pen is utilized to draw computerized pictures, likewise applies acceptance standards.

The tablets talked about here are marked as aloof tablets, since there are different structures that utilization either a battery-worked pen or optical sign to compose with.

The inactive tablets are not quite the same as the touch tablets and telephones huge numbers of us use normally, however may even now be discovered when marking your mark at a sales register.

Underneath the screen, are small wires stumbling into the length and width of the screen.

The pen has a modest attractive field originating from the tip.

As the tip brushes over the screen, a changing attractive field is felt in the wires which converts into an incited emf that is changed over into the line you just drew.

You May Also Like This : What Is Relationship Between Current And Charge ?

#### Power transformers work dependent on Faraday's law

The essential working guideline of the electrical generator is Faraday's law of common enlistment.

The Induction cooker is the quickest method for cooking. It additionally takes a shot at the guideline of common enlistment.

At the point when current courses through the curl of copper wire set beneath a cooking compartment, it creates a changing attractive field.

You Will Like This : How To Convert Energy One Form To Another

This substituting or changing attractive field initiates an emf and subsequently the current in the conductive holder, and we realize that the progression of current consistently delivers heat in it.

Electromagnetic Flow Meter is utilized to quantify the speed of specific liquids. At the point when an attractive field is applied to an electrically protected pipe in which leading liquids are streaming, at that point as indicated by Faraday's law, an electromotive power is actuated in it.

This initiated emf is relative to the speed of liquid streaming.

Structure bases of Electromagnetic hypothesis, Faraday's concept of lines of power is utilized in understood Maxwell's conditions.

As per Faraday's law, change in attractive field offers ascend to change in electric field and the opposite of this is utilized in Maxwell's conditions.

It is likewise utilized in melodic instruments like an electric guitar, electric violin, and so forth.

#### How We Can Describe Electromagnetic Induction?

Faraday's law, because of 19ᵗʰ century physicist Michael Faraday. This relates the pace of progress of attractive flux  through a circle to the extent of the electro-intention power incited on top of it.

The relationship is :

​
The electromotive power or EMF alludes to the potential contrast over the emptied circle.

Practically speaking it is regularly adequate to consider EMF voltage since both voltage and EMF are estimated utilizing a similar unit, the volt.

You Have To Read : How LED Works ?

Lenz's law :

Lenz's law is an outcome of protection of vitality applied to electromagnetic enlistment. It was defined by Heinrich Lenz in 1833.

While Faraday's law reveals to us the extent of the EMF delivered, Lenz's law discloses to us the heading that present will stream.

It expresses that the bearing is in every case with the end goal that it will contradict the adjustment in motion which created it.

This implies any attractive field created by an actuated current will be the other way to the adjustment in the first field.

Lenz's law is commonly consolidated into Faraday's law with a short sign, the consideration of which enables a similar arrange framework to be utilized for both the motion and EMF. The outcome is now and again called the Faraday-Lenz law,

By and by we regularly manage attractive acceptance in different curls of wire every one of which contribute a similar EMF. Consequently an extra term NNN speaking to the quantity of turns is regularly included.

For example :

#### What is the connection between Faraday's law of induction and the magnetic force?

While the full hypothetical supporting of Faraday's law is very mind boggling, a calculated comprehension of the immediate association with the attractive power on a charged molecule is moderately direct.

Consider an electron which is allowed to move inside a wire. As appeared in figure , the wire is set in a vertical attractive field and moved opposite to the attractive field at consistent speed.

The two parts of the bargains are associated, shaping a circle.

This guarantees any work done in making a current in the wire is disseminated as warmth in the opposition of the wire.

An individual pulls the wire with steady speed through the attractive field. As they do as such, they need to apply a power.

The consistent attractive field can't do work without anyone else's input (generally its quality would need to change), yet it can alter the course of a power.

For this situation a portion of the power that the individual applies is re-coordinated, causing an electromotive power on the electron which goes in the wire,

Building up a current. A portion of the work the individual has done pulling the wire at last outcomes in vitality dispersed as warmth inside the obstruction of the wire.

You May Like To Read About : Skin Effects In Transmission Lines

#### Experiment Of Faraday's : Induction from a magnet moving through a coil

It very well may be effectively imitated with minimal more than family materials.

Faraday utilized a cardboard cylinder with protected wire folded over it to frame a loop.

A voltmeter was associated over the curl and the actuated EMF read as a magnet was gone through the loop. The arrangement is appeared in Figure.

The perceptions were as per the following:

Magnet very still in or close to the curl: No voltage watched.

Magnet pushing toward the loop: Some voltage estimated, ascending to a top as the magnet approaches the focal point of the curl.

The Magnet goes through the center of the curl: Measured voltage quickly changes sign.

Magnet drops and away from the loop: Voltage estimated the other way to the prior instance of the magnet moving into the curl.

A case of the EMF estimated is plotted against magnet position in Figure.

These perceptions are predictable with Faraday's law. In spite of the fact that the stationary magnet may deliver an enormous attractive field, no EMF can be actuated on the grounds that the transition through the curl isn't evolving.

At the point when the magnet draws nearer to the loop the motion quickly increments until the magnet is inside the curl.

As it goes through the loop the attractive transition through the curl starts to diminish. Subsequently, the initiated EMF is turned around.

#### Self Induction

In the event that a long curl of wire of cross sectional territory An and length ℓ with N diverts is associated or disengaged from a battery, the changing attractive motion through the loop creates an instigated emf.

The incited current creates an attractive field, which restricts the adjustment in the attractive transition. The extent of the initiated emf can be determined utilizing Faraday's law.

The attractive field inside the long curl is B = μ0(N/ℓ)I.

The transition through the loop is NBA = μ0(N2/ℓ)IA.

The adjustment in transition per unit time is μ0(N2/ℓ)A ∆I/∆t = L*∆I/∆t, since I is the main amount changing with time.

L = μ0(N2/ℓ)A is known as the self inductance of the loop. The units of inductance are Henry (H). 1 H = 1 Vs/A.

The actuated emf will be emf = - L*∆I/∆t, where the less sign is a result of Lenz's law.

The actuated emf is corresponding to the pace of progress of the current in the loop. It tends to be a few times the power supply voltage.

At the point when a switch in a circuit conveying an enormous current is opened, decreasing the current to focus in a brief timeframe interim, this can bring about a sparkle.

All circuits have self inductance, and we generally have emf = - L*∆I/∆t. The self inductance L depends just on the geometry of the circuit.

### Magnet Brake and Eddy Currents

#### Magnet Braking

1) Suspend the horseshoe magnet by a string over the lab table. Remove the magnet's keeper bar, if it is not already removed.

2) Spin the magnet on its axis on the end of the string. With some care this can be done so that the magnet rotates in place, with little wobbling. Note that the magnet will spin for some time without stopping, alternately winding and unwinding the string. Watch this behavior of the spinning magnet for a few cycles.

3)Now, bring the aluminum block close to poles of the spinning magnet without touching it by sliding it under the spinning magnet. Is there an effect? Record your observations.

4) Repeat with the plastic block. Is there any difference?

#### Eddy Current Propulsion

5) Hold the suspended magnet at rest a centimeter or two above the aluminum block that is resting on the table top.

6) Being careful not to touch the magnet, have one of your lab partners gently pull the aluminum away horizontally. Observe and record the effect on the magnet.

7) What happens when you reverse the direction of the movement of the aluminum block?

8) Now try this with the plastic block and observe that the effect cannot be due to air currents.

9) The actual shape of the eddy currents in this part is quite complicated. However, can you make a general statement about forces and the relative motion of magnets and conductors?

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### A Moving Wire in a Magnetic Field

Example: A guitar pickup

1) Gently straighten about 10 cm length of the copper wire, and wrap this length around the wooden jig (you will need to wrap the ends of this wire around the two tacks on either side of the jig to get the wire taut enough for this experiment.) Make the wire segment relatively taut, and place the horseshoe magnet into the yoke with it opening facing upward with the wire between its pole tips.

2) Now connect the ends of this wire to the ends of the BNC-to-banana cable using the two short wires with alligator clips on each end. Connect the BNC-to banana cable to oscilloscope CH 1 input, as in .

A wooden block called a jig holds two ends of a copper wire across its length.  The horseshoe magnet is held so that the wire fits between the poles of the magnet.  The ends of the wire protrude out from the jig.  Each end connects to an alligator clip, which are connected by the BNC-to-banana cable to the oscilloscope.

3) coupling [DC]
Adjust the oscilloscope display so that the line trace is located near the center of the screen.

Caution:

There are exposed connector tips at the ends of the test lead; watch that these do not touch each other and short out your observation.

4) You now have a magnetic pickup as is found in an electric guitar (the wire being the guitar string). Pluck the wire, and observe the result in the oscilloscope. What is the maximum voltage amplitude you can obtain in this way?

5) Try shaking the wire slowly back and forth; is the amplitude the same?

6) If you wanted to improve your pickup circuit so as to obtain larger signals, which changes might help: a larger magnetic field, a thicker wire, a smaller wire resistance? Why?

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## Gtu Maths - 2(3110015) Papers, Syllabus, Study Material And Example For Examination

Gtu Maths - 2(3110015) :

### Syllabus

#### Chapter 1 : Systems of Linear Equations and Matrices

Weightage Of Chapter : 15%
• Systems of Linear Equations
• Matrices and Elementary Row Operations
• The Inverse of a Square Matrix
• Matrix Equations
• Applications of Systems of Linear Equations

#### Chapter 2 : Linear Combinations and Linear Independence

Weightage Of Chapter : 21%
• Vectors in n R
• Linear Combinations
• Linear Independence
• Vector Spaces
• Definition of a Vector Space
• Subspaces
• Basis and Dimension
• Coordinates and Change of Basis

#### Chapter 3 : Linear Transformations

Weightage Of Chapter : 21%
• Linear Transformations
• The Null Space and Range
• Isomorphisms
• Matrix Representation of Linear Transformations
• Similarity
• Eigenvalues and Eigenvectors
• Diagonalization

#### Chapter 4 : Inner Product Spaces

Weightage Of Chapter : 19%
• The Dot Product on n R and Inner Product Spaces
• Orthonormal Bases
• Orthogonal Complements
• Application: Least Squares Approximation
• Diagonalization of Symmetric Matrices

#### Chapter 5 : Vector Functions

Weightage Of Chapter : 15%
• Vector & Scalar Functions and Fields, Derivatives
• Curve, Arc length, Curvature & Torsion
• Gradient of Scalar Field, Directional Derivative
• Divergence of a Vector Field
• Curl of a Vector Field

#### Chapter 6 : Vector Calculus

Weightage Of Chapter : 19%
• Line Integrals
• Path Independence of Line Integrals
• Green`s Theorem in the plane
• Surface Integrals
• Divergence Theorem of Gauss
• Stokes`s Theorem

#### 1. Vector Calculus

Weightage Of Chapter : 33%
• Parametrization of curves. Arc length of curve in space
• Line Integrals, Vector fields and applications as Work, Circulation and Flux
• Path independence
• potential function, piecewise smooth
• connected domain, simply connected domain, fundamental theorem of line integrals
• Conservative fields, component test for conservative fields, exact differential forms, Div, Curl, Green’s theorem in the plane (without proof)
• Parametrization of surfaces
• surface integrals
• Stoke’s theorem (without proof), Divergence Theorem (without proof)

#### 2. Laplace Transform

Weightage Of Chapter : 20%
• Laplace Transform and inverse Laplace transform, Linearity
• First Shifting Theorem (s-Shifting)
• Transforms of Derivatives and Integrals. ODEs
• Unit Step Function (Heaviside Function)
• Second Shifting Theorem (t-Shifting)
• Laplace transform of periodic functions
• Short Impulses, Dime’s Delta Function, Convolution
• Integral Equations, Differentiation and Integration of Transforms
• ODEs with Variable Coefficients, Systems of ODEs

#### 3. Fourier Integral

Weightage Of Chapter : 2%
• Fourier Integral, Fourier Cosine Integral and Fourier Sine Integral.

#### 4. First order ordinary differential equations

Weightage Of Chapter : 12%
• First order ordinary differential equations
• Exact, linear and Bernoulli’s equations, Equations not of first degree: equations solvable for p, equations solvable for y. equations solvable for x and Clairaut’s type

#### 5. Ordinary differential equations

Weightage Of Chapter : 20%
• Ordinary differential equations of higher orders, Homogeneous Linear ODEs of Higher Order
• Homogeneous Linear ODEs with Constant Coefficients
• Euler-Cauchy Equations, Existence and Uniqueness of Solutions
• Linear Dependence and Independence of Solutions
• Wronskian, Nonhomogeneous ODEs, Method of Undetermined Coefficients
• Solution by Variation of Parameters

#### 6. Series Solutions of ODEs

Weightage Of Chapter : 13%
• Series Solutions of ODEs
• Special Functions
• Power Series Method, Legendre’s Equation, Legendre Polynomials
• Frobenius Method, Bessel’s Equation
• Bessel functions of the first kind and their properties

### Reference And Text Books Of Maths 1

Author : Erwin Kreyszig
Publisher : John Wiley and Sons

Author : Peter O'Neill
Publisher : Cengage

Author : Dennis G. Zill
Publisher : Jones and Bartlett Publishers

#### Thomas' Calculus

Author : Maurice D. Weir, Joel Hass
Publisher : Pearson

May 2019
May 2018
Dec 2017
May 2017
Jun 2017

### Question Papers Of New Maths - 2 (3110015)

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### Linear Dependence And Independence

Linear Dependence And Independence :

Let s = {v1,v2,v3…….vr} is a non-empty set of vectors such that

k1v1+k2v2+……..krvr = 0

If the homogeneous system obtained from above equation has only trivial solution

k1 =0 , k2 =0, …. kr = 0

then S is called Linearly independent set.

If the system has non-trivial solution(ex: at least one k is non-zero) then S is called a linearly dependent set. Linearly Dependent if S contains zero vector as 0 = 0v1 + 0v2 +…..+0vr

Linearly Dependent if S contains zero vector as 0 = 0v1 + 0v2 +…..+0vr

#### Example :

Which of the following sets of vectors are linearly dependent ?

(1) (4,-1,2), (-4,10,2), (4,0,1)

(2) (-2,0,1), (3,2,5), (6,-1,1), (7,0,-2)

#### Solution :

Let v1 = (4,-1,2), v2 = (-4,10,2), v3 = (4,0,1)

Consider k1v1+k2v2+k3v3 = 0

k1(4,-1,2) + k2(-4,10,2) + k3(4,0,1) = (0,0,0)

(4k1 -4k2+4k3 , -k1 +10k2,2k1+2k2+ k3 ) = (0,0,0)

Equating corresponding components,

4k1 -4k2+4k3 = 0

-k1 +10k2 = 0

2k1+2k2+ k3 = 0

The augmented matrix of the system is

Reducing the augmented matrix to reduced row echelon form,

Hence,

k1 =0 , k2 =0, k3 = 0

The system has a trivial solution.

Hence,

v1,v2,v3 are linearly independent.

(2) Let v1 = (-2,0,1), v2 = (3,2,5), v3 = (6,-1,1), v4 = (7,0,-2)

Consider k1v1+k2v2+k3v3 +k4v4= 0

k1(-2,0,1) + k2(3,2,5) + k3(6,-1,1) + k4(7,0,-2) = (0,0,0)

(-2k1 + 3k2 + 6k3 + 7k4 , 2k2 – k3 ,k1+5k2+ k3 – 2k4 ) = (0,0,0)

Equating Corresponding Components,

-2k1 + 3k2 + 6k3 + 7k4 = 0

2k2 – k3 = 0

k1+5k2+ k3 – 2k4 = 0

The number of unknowns, r = 4

The number of equations, n = 3

r > n

Hence, v1,v2,v3. v4 are linearly dependent.

Matrices And Elementary Row Operations

### Types of matrix :

#### (1) Row Matrix :

Row matrix or row vector is a matrix having only one row and any number of columns.

[ 2 5 -3 4 ]

#### (2) Column Matrix :

column matrix or column vector is a matrix, having only one column and any number of rows.

#### (3) Zero Or Null Matrix :

Zero matrix is a Matrix in which all the elements are zero or null matrix and is denote by 0.

#### (4) Square Matrix :

A square matrix is in which the number of rows is equal to the number of columns.

#### (5) Diagonal Matrix :

A square matrix, all of whose non-diagonal elements are zero and at least one diagonal element is non-zero, is called a diagonal matrix.

#### (6) Unit Or Identity Matrix :

Unit or identity matrix is a diagonal matrix, all of whose diagonal elements are unity and is denoted by I.

#### (7) Scalar Matrix :

Scalar matrix is a diagonal matrix, all of whose diagonal elements are equal.

#### (8) Upper Triangular Matrix :

Upper triangular matrix is a square matrix, in which all the element below the diagonal are zero.

#### (9) Lower Triangular Matrix :

Lower triangular matrix is a square matrix, in which all the elements above the diagonal are zero.

#### (10) Trace of a Matrix :

Trace of a matrix is the sum of all the diagonal elements of a square matrix .

And
Trace of A = 2 + 6 + 3 = 11

#### (11) Transpose of a Matrix :

Transpose of a matrix is a matrix obtained by interchanging rows and columns of a matrix and is denoted by AT.

#### (12) Determinant of a Matrix :

If A is a square matrix, then determinant of A is represented as |A| or det(A).

#### (13) Singular and Non-singular Matrices :

A square matrix A called singular if det(A) = 0 and non-singular if det(A) ≠ 0.

### Elementary Row Operations :

#### Echelon Form of a Matrix :

A matrix A said to be in row echelon form if it satisfies the following properties :

(1) Every zero row of the matrix A occurs below a non-zero row.

(2) The first non-zero number from the left of a non-zero row is 1. This is called a leading 1.

(3) For each non-zero row, the leading 1 appears to the right and below any leading 1 in the preceding rows.

#### Types Of Echelon Form :

(1) Row Echelon Form

(2) Reduced Row Echelon Form

### First Shifting Theorem :

First shifting theorem is used when we have to find out laplace transform of function which have eat or e-at multiply with function.

It is simple if we have eat multiply with the function then replace s with (s – a ) in the equation.

It is simple if we have e-at multiply with the function then replace s with (s + a) in the equation.

If L{f(t)} = F(s) then,

#### First Shifting Theorem

Above both equations shows when we have to use F(s+a) and when we have to use F(s-a).

There are some basic laplace transform of first shifting.

So, let’s see basic equations when function is multiply with eat or e-at.

#### Example :

First we have to forget e-3t and find laplace of the remaining function of the equation.

Then, we have to replace s with (s+3) because we have e-3t and according to the equation if we have e-at then replace it with (s+a).

#### Solution :

As we learn in above example first we have to find laplace of the function except e-3t.

After that replace s with (s+3) according to the equation.

At the end simplify the solution and get the answer.

In this article we learn how to find laplace of the function when eat or e-at is multiplied with the function.

### Linear Transformations

Let V and W be two vector spaces. A linear transformation ( T : V -> W) is a function T from V to W such that

(a) T(u+v) = T(u) + T(v)

(b) T(ku) = kT(u)

for all vectors u and v in V and all scalars k.

If V = W, the linear transformation T : V -> V is called a linear operator.

#### Example :

Show that the following functions are linear transformations.

(1) T : R2 -> R2 where T(x,y) = (x+2y , 3x-y)

(2) T : R3 -> R2 where T(x,y,z) = (2x-y+z , y-4z)

#### Solution :

Let u = (x1,y1) and v = (x2,y2) be the vectors in R2 and k be any scalar.

T(u) = (x1+2y1 , 3x1 – y1)

T(v) = (x2+2y2 , 3x2-y2)

(1) u + v = (x1,y1) + (x2,y2)

= (x1+ x2 , y1 + y2 )

T( u + v ) = (x1 + x2 + 2y1 + 2y2 , 3×1 + 3×2 – y1 – y2)

= (x1 + 2y1 , 3×1 – y1) + (x2 + 2y2 , 3×2 – y2)

= (x1 + 2y1 + x2 + 2y2 , 3×1 – y1 + 3×2 – y2 )

thus, = T(u) + T(v)

(2) ku = k(x1,y1) = (kx1,ky1)

T(ku) = (kx1 + 2ky1 , 3kx1 – ky1 )

= k (x1 + 2y1 , 3x1 – y1 )

= kT(u)

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### Basis And Dimension Of Homogeneous system:

Basis And Dimension For solution space of the Homogeneous systems.

The Basis and Dimensions for the solution space of this system can be found as follows :

(1) Solve system using Gauss Elimination Method.

If the system has only trivial solution space is {0}.

Which has no basis and hence the dimension of the solution space is zero.

(2) If Solution has arbitrary constants t1,t2…tn and express x as a linear combination of vectors x1,x2…xn with t1,t2…tn as coefficients.

Ex : x = t1x1 + t2x2 +…tnxn

(3) The set of vectors x1,x2…xn from a basis for the solution space of Ax = 0 and hence the dimension of the solution space is n.

#### Example :

Determine the dimension and a basis for the solution space of the system

x1  + x2 -2x3 = 0

-2x1  -2x2 + 4x3 = 0

– x1  – x2 + 2x3 = 0

#### Solution :

The Matrix form of the system is

The augmented matrix of the system is

Reducing The augmented matrix to row echelon form,

The corresponding system of equation is

x1  + x2 – 2x3 = 0

x1   = – x2 + 2x3

Assigning the free variables x2 and x3 arbitrary values t1  and t2 respectively,

x1   = – t1 + 2t2

x2 = t1

and x3 = t2 is the solution of the system.

The solution vector is

Hence,

Dimension = 2

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### Linear Combination Of Vectors

Linear Combination Of Vectors :

A Vector v is called a linear combination of vectors v1,v2,…..vr, it is expressed as

v=k1v1+k2v2+……..krvr

where k1,k2,….kr are scalars.

Note : If r =1 then v =k1v1 , This shows that a vector v is a linear combination of a single vector v1 , if it is a scalar multiple of v1 .

Vector Expressed as a Linear Combination Of Given Vectors:

Method For check if a vector v is a linear combination of given vectors
v1,v2,…..vr is as follows:

(1) Express v as linear combination of v1,v2,…..vr

v=k1v1+k2v2+……..krvr

(2) If the system of equations in (1) is consistent then v is a linear combination of v1,v2,…..vr .

If it is inconsistent, then v is not a linear combination of v1,v2,…..vr .

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#### Example :

Which of the following are linear combinations of v1 = (0,-2,2) and v2 = (1,3,-1) ?

(1) (3,1,5)

#### Solution :

Let v = k1v1+k2v2

(1) (3,1,5) = k1(0,-2,2) + k2(1,3,-1)

= (0,-2k1,2k1) + (k2,3k2,-k2)

= (k2 , -2k1+3k2 , 2k1-k2)

Equating corresponding components,

k2 = 3

-2k1+3k2 = 1

2k1-k2 = 5

The augmented matrix of the system is

Reducing the augmented matrix to row echelon form,

Interchange Row 1 and Row 2.

multiply Row 1 with -1/2.

Subtract two times row 1 from row 3.

subtract two times row 2 from row 3.

The system of equation is consistent.

Hence, v is a linear combination of v1 and v2.

The corresponding system of equation is

k1  – (3/2)k2 = -(1/2)

k2 = 3

Solving these equations,

k1 = 4 ,
k2 = 3

Hence, v = 4v1+3v2

### Subspaces In Vector Space

Subspaces In Vector Space :

A non-empty subset W of a vector space V is called a subspace of V if W is itself a vector space under the operations defined on V.

Note :

Every vector space has at least two subspaces, itself and the subspace {0}.

The Subspace {0} is called the zero subspace consisting only of the zero vector.

#### Theorem :

If W is a non-empty subset of vector space V, then W is a subspace of V if and only if the following axioms hold.

Axiom 1 : If u and v are vectors in W then u + v is in W.

Axiom 2 : If k is any scalar and u is a vector in W, then ku is in W.

#### Example :

Show that W = { (x,y) | x = 3y } is a subspace of R2. State all possible subspaces of R2.

#### Solution :

Let u = { (x1,y1) | x1 = 3y1 } and v = { (x2,y2) | x2 = 3y2 } are in W and k is any scalar.

Axiom 1 :

u + v = (x1,y1) + (x2,y2)

= (x1 + x2 , y1 + y2)

But , x1 = 3y1 and x2 = 3y2

Therefore, x1 + x2 = 3 (y1 + y2)

u + v = { (x1 + x2 , y1 + y2) | x1 + x2 = 3 (y1 + y2) }

Thus, u + v is in W.

Axiom 2 :

ku = k (x1,y1)

= (kx1,ky1)

But, x1 = 3y1

Therefor, kx1 = 3(ky1)

ku = { (kx1,ky1) | kx1 = 3(ky1) }

Thus, ku is in W.

Hence, W is a subspace of R2.

All possible subspaces of R2 are

(1) {0} (2) R2 (3) Lines passing through the origin.

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### Vector Spaces

Vector Spaces  :

If the following axioms are satisfied with by all objects u,v,w in V and all scalars k1,k2 then V is called a vector space.

The objects in V are vectors,

(1) If u and v are objects in V then u + v in V.

(2) u + v = v + u

(3) u + (v + w) = (u + v) + w

(4) u + 0 = 0 + u = u

(5) u + (-u) = 0

(6) If k1 is any scalar and u is an object in V, then k1u is in V.

(7) k1(k2u) = (k1k2)u

(8) k1(u+v) = k1u + k1v

(9) (k1+k2)u = k1u + k2u

(10) 1u = u

#### Example :

Determine whether the set V of all pairs of real numbers (x,y) with the operations (x1,y1) + (x2,y2) = (x1+x2+1 , y1+y2+1) and k(x,y) = (kx,ky) is a vector space.

#### Solution :

(1) u + v = (x1,y1) + (x2,y2) = (x1+x2+1 , y1+y2+1)

Since x1,y1,x2,y2 are real numbers x1+x2+1 and y1+y2+1 are also real numbers.

Therefore, u+v is also an object in V.

(2) u + v = (x1+x2+1 , y1+y2+1)

= (x2+x1+1 , y2+y1+1)

= v + u

(3) u + (v + w) = (x1,y1) + [ (x2,y2) + (x3,y3) ]

(4) Let (a,b) be an object in V such that

Also, u + (a,b) = u

Hence, (-1,-1) is the zero vector in V.

(5) Let (a,b) be an object in V such that

Also, (a,b) + u = (-1,-1)

Hence, (-x,-2,-y,-2) is the negative of u in V.

(6) k1u = k1(x1,y1)

= (k1x1,k1y1)

Since k1x1 and k1y1 are real numbers, k1u is an object in V.

Hence, V is closed under scalar multiplication.

(7) k1 (u+v) = k1 (x1+x2+1 , y1+y2+1)

V is not distributive under scalar multiplication.

Hence, V is not a vector space.

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