Electrical-All u need To know about!!!!: February 2014

Wednesday 26 February 2014

Biot Savart Law

The mathematical expression for magnetic flux density was derived by Jean Baptiste Biot and Felix Savart. Talking the deflection of a compass needle as a measure of the intensity of a current, varying in magnitude and shape, the two scientists concluded that any current element projects into space a magnetic field, the magnetic flux density of which dB, is directly proportional to the length of the element dl, the current I, the sine of the angle and θ between direction of the current and the vector joining a given point of the field and the current element and is inversely proportional to the square of the distance of the given point from the current element, r. this is Biot Savart law statement.

$Hence, \;dB\;\propto \frac{Idlsin\theta}{r^2}\;\;or\;\;dB\;=k\frac{Idlsin\theta}{r^2}$

Where, K is a constant, depends upon the magnetic properties of the medium and system of the units employed. In SI system of unit,

$k\;= \;\frac{\mu_o\mu_r}{4\pi}$

Therefore final Biot Savart law derivation is,

$dB\;= \;\frac{\mu_o\mu_r}{4\pi}\times \frac{Idlsin\theta}{r^2}$

Let us consider a long wire carrying an electric current I and also consider a point p. The wire is presented in the below picture by red color. Let us also consider an infinitely small length of the wire dl at a distance r from the point P as shown. Here r is a distance vector which makes an angle θ with the direction of current in the infinitesimal portion of the wire.

If you try to visualize the condition, you can easily understand the magnetic field density at that point P due to that infinitesimal length dl of wire is directly proportional to current carried by this portion of the wire. That means current through this infinitesimal portion of the wire is increased the magnetic field density due to this infinitesimal length of wire, at point P increases proportionally and if the current through this portion of wire is decreased the magnetic field density at point P due to this infinitesimal length of wire decreases proportionally.
As the electric current through that infinitesimal length of wire is same as the current carried by the wire itself.

$dB\;\propto \;I$

It is also very natural to think that the magnetic field density at that point P due to that infinitesimal length dl of wire is inversely proportional to the square of the straight distance from point P to center of dl. That means distance r of this infinitesimal portion of the wire is increased the magnetic field density due to this infinitesimal length of wire, at point P decreases and if the distance of this portion of wire from point P, is decreased, the magnetic field density at point P due to this infinitesimal length of wire increases accordingly.

$That\; means\;dB\;\propto \;\frac{1}{r^2}$

Lastly, field density at that point P due to that infinitesimal portion of wire is also directly proportional to the actual length of the infinitesimal length dl of wire. As θ be the angle between distance vector r and direction of current through this infinitesimal portion of the wire. The component of dl directly facing perpendicular to the point P is dlsinθ,

$Hence\;dB\;\propto \;dlsin\theta$

Now combining these three statements, we can write,

$dB\;\propto \;\frac{I\cdot dl \cdot sin\theta}{r^2}$

This is the basic form of Biot Savart's Law
Now putting the value of constant k (which we have already introduced at the beginning of this article) in the above expression, we get

$dB\;=\;k\frac{I\cdot dl \cdot sin\theta}{r^2}$ $\Rightarrow dB\;= \;\frac{\mu_o\mu_r}{4\pi}\times \frac{Idlsin\theta}{r^2}$

Here, μ₀ used in the expression of contant k is absolute permeability of air or vacuum and it's value is 4π10^-7 W_b/ A-m in S.I system of units. μ_r of the expression of constant k is relative permeability of the medium.
Now, flux density(B) at the point P due to total length of the current carrying conductor or wire can be represented as,

$B\;=\;\int dB\Rightarrow dB\;= \;\int\frac{\mu_o\mu_r}{4\pi}\times \frac{Idlsin\theta}{r^2}\;= \;\frac{I\mu_o\mu_r}{4\pi}\int \frac{sin\theta}{r^2}dl$

If D is the perpendicular distance of the point P form the wire, then

$rsin\theta\;= \;D\;or\;r\;=\;\frac{D}{sin\theta}$

Now, the expression of flux density B at point P can be rewritten as,

$B\;= \;\frac{I\mu_o\mu_r}{4\pi}\int \frac{sin\theta}{r^2}dl= \;\frac{I\mu_o\mu_r}{4\pi}\int \frac{sin^3\theta}{D^2}dl$ $Again, \;\frac{l}{D}\;=\;cot\theta \Rightarrow l\;=\;Dcot\theta$

As per the figure above,
$Therefore, \; dl\;=\;-\;Dcsc^2\theta d\theta$

Finally the expression of B comes as,
$B\;= \;\frac{I\mu_o\mu_r}{4\pi}\int \frac{sin^3\theta}{D^2}\left[-\;Dcsc^2\theta d\theta\right]\;=\;-\;\frac{I\mu_o\mu_r}{4\pi D}\int sin^3\theta csc^2\theta d\theta\;=\;-\;\frac{I\mu_o\mu_r}{4\pi D}\int sin\theta d\theta$

This angle θ depends upon the length of the wire and the position of the point P. Say for certain limited length of the wire, angle θ as indicated in the figure above varies from θ₁ to θ₂. Hence, flux density at point P due to total length of the conductor is,

$B\;=\;-\;\frac{I\mu_o\mu_r}{4\pi D} \int_{\theta_1}^{\theta_2}sin\theta d\theta\;=\;-\;\frac{I\mu_o\mu_r}{4\pi D} \bigg[-cos\theta\bigg]_{\theta_1}^{\theta_2}\;=\;\frac{I\mu_o\mu_r}{4\pi D} [cos\theta_1 - cos\theta_2]$

Let's imagine the wire is infinitely long, then θ will vary from 0 to π that is θ₁ = 0 to θ₂ = π. Putting these two values in the above final expression of Biot Savart law, we get,

$B\;=\;\frac{I\mu_o\mu_r}{4\pi D} [cos0 - cos\pi]\;=\;\frac{I\mu_o\mu_r}{4\pi D} [1 - (-1)]\;=\;\frac{\mu_o\mu_r}{2\pi D}I$

This is nothing but the expression of Ampere's Law .

Wednesday 19 February 2014

Lenz's Law

Lenz's law is named after the German scientist H. F. E. Lenz in 1834. Lenz's lawobeys Newton's third law of motion (i.e to every action there is always an equal and opposite reaction) and the conservation of energy (i.e energy may neither be created nor destroyed and therefore the sum of all the energies in the system is a constant).
Lenz law is based on Faraday's law of induction so before understanding Lenz's law one should know what Faraday’s law of induction is. When a changing magnetic field is linked with a coil, an emf is induced in it. This change in magnetic field may be caused by changing the magnetic field strength by moving a magnet toward or away from the coil or moving the coil into or out of the magnetic field as desired. Or in simple words we can say that the magnitude of the emf induced in the circuit is proportional to the rate of change of flux.

Heinrich Friedrich Emil Lenz

Lenz's Law

Lenz law states that when an emf is generated by a change in magnetic flux according to Faraday's Law, the polarity of the induced emf is such that it produces an electric current whose magnetic field opposes the change which produces it.

The negative sign is used in Faraday's law of electromagnetic induction, indicates that the induced emf ( ε ) and the change in magnetic flux ( δΦB ) have opposite signs.

Where
ε = Induced emf
δΦB = change in magnetic flux
N = No of turns in coil

Reason for Opposing, Cause of Induced Current in Lenz's Law?

• As stated above Lenz law obeys the law of conservation of energy and if the direction of the magnetic field that creates the current and the magnetic field of the current in a conductor are in same direction, then these two magnetic field would add up and produce the current of twice the magnitude and this would in turn creates more magnetic field, which cause more current and this process continues on and on and thus leads to violation of the law of conservation of energy.

• If the induced current creates a magnetic field which is equal and opposite to the direction of magnetic field that creates it, then only it can resist the change in the magnetic field in the area which is in accordance to the Newton's third law of motion

Explanation of Lenz's Law

For understanding Lenz's law consider two cases :
CASE-I When a magnet is moving towards the coil.

When the north pole of the magnet is approaching towards the coil, the magnetic flux linking the coil increases. According toFaraday's law of electromagnetic induction, when there is change in flux, an emf and hence current is induced in the coil and this current will creates its own magnetic field . Now according to Lenz law, this magnetic field created will oppose its own cause or we can say opposes the increase in flux through the coil and this is possible only if approaching coil side attains north polarity, as we know similar poles repel each other. Once we know the magnetic polarity of the coil side, we can easily determine the direction of the induced current by applying right hand rule. In this case the current flows in anticlockwise direction.

CASE-II When a magnet is moving away from the coil

When the north pole of the magnet is moving away from the coil, the magnetic flux linking the coil decreases. According to Faraday's law of electromagnetic induction, an emf and hence current is induced in the coil and this current will creates its own magnetic field . Now according to Lenz's law, this magnetic field created will oppose its own cause or we can say opposes the decrease in flux through the coil and this is possible only if approaching coil side attains south polarity, as we know dissimilar poles attract each other. Once we know the magnetic polarity of the coil side, we can easily determine the direction of the induced current by applying right hand rule. In this case the current flows in clockwise direction.

NOTE : For finding the directions of magnetic field or electric current use Right hand thumb rule i.e if the fingers of the right hand are placed around the wire so that the thumb points in the direction of current flow, then the curling of fingers will show the direction of the magnetic field produced by the wire.

Right hand thumb rule

The Lenz law can be summarized as under:

• If the magnetic flux Ф linking a coil increases, the direction of current in the coil will be such that it oppose the increase in flux and hence the induced current will produce its flux in a direction as shown below (using right hand thumb rule).

• If magnetic flux Ф linking a coil is decreasing, the flux produced by the current in the coil is such that it aid the main flux and hence the direction of current is as shown below

Application of Lenz's Law

• Lenz law can be used to understand the concept of stored magnetic energy in an inductor. When a source of emf is connected across an inductor, a current starts flowing through it. The back emf will oppose this increase in current through the inductor. In order to establish the flow of current, the external source of emf has to do some work to overcome this opposition. This work done by the emf is stored in the inductor and it can be recovered after removing the external source of emf from the circuit

• This law indicates that the induced emf and the change in flux have opposite signs which provide a physical interpretation of the choice of sign in Faraday's law of induction.

• Lenz's law is also applied to electric generators. When an electric current is induced in a generator, the direction of this induced current is such that it opposes its cause i.e rotation of generator (as in accordance to Lenz's law) and hence the generator requires more mechanical energy. It also provides back emf in case of electric motors.

• Lenz’s law is also used in electromagnetic braking and induction cook tops.

Kirchhoff Voltage Law

ohms law

Faradays law of electromagnetic induction

Faraday's First Law

Any change in the magnetic field of a coil of wire will cause an emf to be induced in the coil. This 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.
Method to change magnetic field:
1. by moving a magnet toward or away from the coil
2. by moving the coil into or out of the magnetic field.
3. by changing area of a coil placed in the magnetic field
4. by rotating the coil relative to the magnet.

Faraday's Second Law

It states that the magnitude of emf induced in the coil is equal to the rate of change of flux linkages with the coil. The flux linkages of the coil is the product of number of turns in the coil and flux associated with the coil.

Faraday Law Formula

Faraday's law

Consider a magnet approaching towards a coil. Here we consider two instants at time T1 and time T2.
Flux linkage with the coil at time, T1 = NΦ1 Wb
Flux linkage with the coil at time, T2 = NΦ2 wb
Change in flux linkage = N(Φ2 - Φ1)
Let this change in flux linkage be, Φ = Φ2 - Φ1
So, the Change in flux linkage = NΦ
Now the rate of change of flux linkage = NΦ / t
Take derivative on right hand side we will get
The rate of change of flux linkage = NdΦ/dt
But according to Faraday's law of electromagnetic induction the rate of change of flux linkage is equal to induced emf

Considering Lenz's Law

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

HOW TO INCREASE EMF INDUCED IN A COIL

• By increasing the number of turns in the coil i.e N- From the formulae derived above it is easily seen that if number of turns of coil is increased, the induced emf is also increased.

• By increasing magnetic field strength i.e B surrounding the coil- Mathematically if magnetic field increases, flux increases and if flux increases emf induced will also increased. Theoretically if the coil is passed through a stronger magnetic field, there will be more lines of force for coil to cut and hence there will be more emf induced.

• By increasing the speed of the relative motion between the coil and the magnet - If the relative speed between the coil and magnet is increased from its previous value, the coil will cut the lines of flux at a faster rate so more induced emf would be produced.

Fleming Rules

Whenever a current carrying conductor comes under a magnetic field, there will be a force acting on the conductor and on the other hand, if a conductor is forcefully brought under a magnetic field there will be an induced current in that conductor. In both of the phenomenon there is an relation between magnetic field, electric current and force. This relation directionally determined by Fleming Left Hand rule and Fleming Right Hand rule respectively. 'Directionally' means these rules do not show the magnitude but show the direction of any of the three parameters (magnetic field, electric current, force) if the direction of other two are known. Fleming Left Hand rule is mainly applicable for electric motor and Fleming Right Hand rule is mainly applicable for electric generator. In late 19th century, John Ambrose Fleming, introduced these both rules and as per his name the rules are well known asFleming left and right hand rule.

Fleming Left Hand Rule

It is found that whenever an electric current carrying conductor is placed inside a magnetic field, a force acts on the conductor, in a direction, perpendicular both to the direction of the electric current and the magnetic field. In the figure it shown that a portion of a conductor of length L placed vertically in a uniform horizontal magnetic field strength H, produced by two magnetic pole N and S. If i is the electric current flowing through this conductor , the magnitude of the force acts on the conductor is,

F = BiL

F = Bil

Hold out your left hand with forefinger, second finger and thumb at right angle to one another. If the fore finger represents the direction of the field and the second finger that of the current, then thumb gives the direction of the force.

While electric current flows through a conductor one magnetic field is induced around it. This can be imagined by considering numbers of closed magnetic lines of force around the conductor. The direction of magnetic lines of force can be determiner by Maxwell's corkscrew rule or right-hand grip rule. As per these rules the direction of the magnetic lines of force (or flux lines) is clockwise if the current is flowing away from the viewer that is if the direction of current through the conductor is inward from the reference plane as shown in the figure.

Field Around a Current Carrying Conductor

Now if a horizontal magnetic field is applied externally to the conductor, these two magnetic fields i.e. field around the conductor due to current through it and the externally applied field will interact each other. We observe in the picture, the magnetic lines of force of external magnetic field are form N to S pole that is from left to right. The magnetic lines of force of external magnetic field and magnetic lines of force due to current in the conductor are in same direction, above the conductor and they are in opposite direction below the conductor. Hence there will be larger numbers of co-directional magnetic lines of force above the conductor than that of below the conductor. Consequently, there will be a larger concentration of magnetic lines of force in a small space above the conductor. As magnetic lines of force are no longer straight lines, they are under tension like stretched rubber bands. As a result there will be a force which tends to move the conductor from more concentrated magnetic field to less concentrated magnetic field that is from present position to downwards. Now if you observe the direction of current, force and magnetic field in the above explanation, you will find that the directions are according to Fleming left hand rule.

Interaction of magnetic fields and current-carrying conductors

Fleming Right Hand Rule

As per Faraday's law of electromagnetic induction, whenever a conductor moves inside a magnetic field, there will be an induced current in it. If this conductor is forcefully moved inside the magnetic field, there will be a relation between the direction of applied force, magnetic field and the electric current. This relation among these three directions, is determined by by Fleming Right Hand rule

This rule states "Hold out the right hand with the first finger, second finger and thumb at right angles to each other. If forefinger represents the direction of the line of force, the thumb points in the direction of motion or applied force, then second finger points in the direction of the induced current.

Monday 17 February 2014

Kirchhoff's Laws

There are some simple relationship between currents and voltages of different branches of an electrical circuit. These relationship are determined by some basic laws which are known as Kirchhoff laws or more specifically Kirchhoff Current and Voltage laws. These laws are very helpful in determining the equivalent electrical resistance or impedance (in case of AC) of a complex network and the currents flowing in the various branches of the network. These laws are first derived by Guatov Robert Kirchhoff and hence these laws are also referred as Kirchhoff Laws.

Kirchhoff's Current Law

In an electrical circuit the electric current flows rationally as electrical quantity. As the flow of current is considered as flow of quantity, at any point in the circuit the total current enters is exactly equal to the total current leaves the point. The point may be considered any where in the circuit.
kirchhoff current law

Suppose the point is on the conductor through which the current is flowing, then the same current crosses the point which can alternatively said that the current enters at the point and same will leave the point. As we said the point may be any where on the circuit, so it can also be a junction point in the circuit. So total quantity of current enters at the junction point must be exactly equal to total quantity of current leave the junction. This is very basic thing about flowing of electric current and fortunately Kirchhoff Current law says the same. The law is also known as Kirchhoff First Law and this law stated that at any junction point in the electrical circuit, the summation of all the branch currents is zero. If we consider all the currents enter in the junction are considered as positive current then convention of all the branch currents leaving the junction are negative. Now if we add all these positive and negative signed currents obviously we will get result of zero.

Mathematical Form

We have a junction where n number of beaches meet together.

Let's I1, I2, I3, ...................... Im are the current of branches 1, 2, 3, ......m and

Im + 1, Im + 2, Im + 3, ...................... In are the current of branches m + 1, m + 2, m + 3, ......n respectively.

The currents in branches 1, 2, 3 ....m are entering to the junction.

Whereas currents in branches m + 1, m + 2, m + 3 ....n are leaving from the junction.

So the currents in the branches 1, 2, 3 ....m may be considered as positive as per general convention and similarly the currents in the branches m + 1, m + 2, m + 3 ....n may be considered as negative.

Hence all the branch currents in respect of the said junction are -

+ I1, + I2, + I3,................+ Im, − Im + 1, − Im + 2, − Im + 3, .................. and − In.

Now, the summation of all currents at the junction is-

I1 + I2 + I3 + ................+ Im − Im + 1 − Im + 2 − Im + 3..................− In.

This is equal to zero according to Kirchhoff Current Law.

Therefore, I1 + I2 + I3 + ................+ Im − Im + 1 − Im + 2 − Im + 3..................− In = 0.

The mathematical form of Kirchhoff First Law is ∑ I = 0 at any junction of electrical network.

Kirchhoff's Voltage Law

Kirchhoff Voltage Law

This law deals with the voltage drops at various branches in an electrical circuits. Think about one point on an closed loop in an electrical circuit. If some one goes to any other point on the same loop, he or she will find that the potential at that second point may be different from first point. If he or she continues to go to some different point in the loop, he or she may find some different potential at that new location. If he or she goes on further along that closed loop ultimately he or she reaches the initial point from where the journey was started. That means he or she comes back to the same potential point after crossing through different voltage levels. It can be alternatively said that net voltage gain and net voltage drops along a closed loop are equal. That is what Kirchhoff Voltage law states. This law is alternatively known as Kirchhoff Second Law.

If we consider a closed loop, conventionally if we consider all the voltage gains along the loop are positive then all the voltage drops along the loop should be considered as negative. The summation of all these voltages in a closed loop is equal to zero. Suppose n numbers of back to back connected elements form a closed loop. Among these circuit element m number elements are voltage source and n - m number of elements drop voltage such as resistors.

The voltages of sources are V1, V2, V3,................... Vm.

And voltage drops across the resistors respectively, Vm + 1, Vm + 2, Vm + 3,..................... Vn.

As it said that the voltage gain conventionally considered as positive, and voltage drops are considered as negative, the voltages along the closed loop are -

+ V1, + V2, + V3,................... + Vm, − Vm + 1, − Vm + 2, − Vm + 3,.....................− Vn.

Now according to Kirchhoff Voltage law the summation of all these voltages results to zero.

That means, V1 + V2 + V3 + ................... + Vm − Vm + 1 − Vm + 2 − Vm + 3 + .....................− Vn = 0.

So accordingly Kirchhoff Second Law, ∑V = 0.

Application of Kirchhoff's Laws to Circuits

The current distribution in various branches of a circuit can easily be found out by applying Kirchhoff Current law at different nodes or junction points in the circuit. After that Kirchhoff Voltage law is applied each possible loops in the circuit and generate algebraic equation for every loop. By solving these all equations, one can easily find out different unknown currents, voltages and resistances in the circuits.

Some Popular Conventions We Generally use During Applying KVL

1) The resistive drops in a loop due to current flowing in clockwise direction must be taken as positive drops.

2) The resistive drops in a loop due to current flowing in anti-clockwise direction must be taken as negative drops.

3) The battery emf causing current to flow in clockwise direction in a loop is considered as positive.

4) The battery emf causing current to flow in anti-clockwise direction is referred as negative.