Other important laws

Fleming Right hand and Left hand rule-
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 as Fleming left and right hand rule.

Fleming Left Hand Rule

It is found that whenever a current carrying conductor is placed inside a magnetic field, a force acts on the conductor, in a direction, perpendicular both to the direction of theelectric 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 of 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.

Right-Hand Rules;

Right-Hand Rules: A Guide to finding the Direction of the Magnetic Force-

F_magnetic - The force a magnetic field exerts on a moving charge

When a charge is placed in a magnetic field, that charge experiences a magnetic force; when two conditions exist:
1) the charge is moving relative to the magnetic field,
2) the charge's velocity has a component perpendicular to the direction of the magnetic field

_{The Right-Hand Rules apply to positive charges or positive (conventional) current}

When using the Right-Hand Rules, it is important to remember that the rules assume charges move in a conventional current (the hypthetical flow of positive charges). In order to apply either Right-Hand Rule to a moving negative charge, the velocity (v) of that charge must be reversed--to represent the analogous conventional current.

Making illustrations of magnetic field and charge interactions in 3D

Because the force exerted on a moving charge by a magnetic field is perpendicular to both the the velocity of the charge and the direction of the field, making illustrations of these interactions involves using the two symbols on the left to denote movement into or out of the plane of the page.

Right-Hand Rule #1 (RHR #1)

Right-Hand Rule #1 determines the directions of magnetic force, conventional current and the magnetic field. Given any two of theses, the third can be found.

Using your right-hand:
point your index finger in the direction of the charge's velocity, v, (recall conventional current).Point your middle finger in the direction of the magnetic field, B.
Your thumb now points in the direction of the magnetic force, F_magnetic.

Right-Hand Rule #2 (RHR #2)

Right-Hand Rule #2 determines the direction of the magnetic field around a current-carrying wire and vice-versa

Using your right-hand:
Curl your fingers into a half-circle around the wire, they point in the direction of the magnetic field, BPoint your thumb in the direction of the conventional current.

Applying the Right-Hand Rules:

The Right-Hand Rules give only the direction of the magnetic field. In order to determine the strength of a magnetic field , some useful mathematical equations can be applied.

For a long, straight wire, the magnetic field, B is:
B = m_oI / 2pr; where,
m_o = 4p x 10^-7 T · m / A and os called the permeability of free space, r is the radial distance from the wire in meters, and I is the current in amperes.

For a single loop of wire, the magnetic field, B through the center of the loop is:
B = m_oI / 2R; where,
m_o is the permeability of free space, and R is the radius of the the circular loop of wire, measured in meters. Both the fields for a coil of wire and a solenoid can be constructed from this equation.

Questions to Consider:

1. A proton is travelling with a speed of 5.0 x 10⁶m / s, when it encounters a magnetic field of magnitude 0.40 T and that is perpendicular to the velocity of the proton. Make a sketch of this situation and indicating the directions of the velocity of the proton, the magnetic field and the magnetic force.

2. Here, a long, straight wire carries a current, I, of 3.0 A. A particle, q with a charge of +6.5 x 10^-6 C, moves parallel to the wire in the direction shown, at a distance of r = 0.050 m and a speed ofv = 280 m / s. Determine the magnitude and direction of the magnetic field experienced by the charge.

Seebeck effect and Seebeck coefficient;

Seebeck Effect

When the two different electrical conductors or semiconductors are kept at different temperatures, the system results in the creation of electrical potential. This was discovered by German physicist Thomas Seebeck (1770-1831). Seebeck discovered this by observing a compass needle which would be deflected when a closed loop was formed between those two different metals or semiconductors. Seebeck initially believed that it was due to the magnetism induced by the temperature difference’s and he called the effect as thermomagnetic effect. However Danish physicist Hans Christian Orsted realized that it’s an electrical current that is induced, which because of Ampere law deflects the magnet.

Explanation of Seebeck Effect: -
The valence electrons in the warmer part of metal are solely responsible for that and the reason behind this is thermal energy. Also because of the kinetic energy of these electrons, these valence electrons igrate more rapidly towards the other (colder) end as compare to the colder part electrons migrate towards warmer part. The concept behind their movement is

- At hot side Fermi distribution is soft i.e. the higher concentration of electrons above the Fermi energy but on cold side the Fermi distribution is sharp i.e. we have fewer electrons above Fermi energy.

- Electrons go where the energy is lower so therefore it will move from warmer end to the colder end which leads to the transporting energy and thus equilibrating temperature eventually

Or in simple words we can come to conclusion that the electrons on a warmer end have a high average momentum as compared to the colder one. Therefore they will take energy with them (more in no.) as ompared to the other one.

This movement results in the more negative charge at colder part than warmer part, which Leads to the generation of electric potential. If this pair is connected through an electrical circuit. It results in the generation of a DC. However the voltage produced is few microvolt’s (10-6) per Kelvin temperature difference. Now we all are aware of the fact that the voltage increase in series and current increase in parallel. So keeping this fact in mind if we can connect many such devices to increase the voltage (in case of series connection) or to increase the maximum deliverable current (in parallel). Keeping care of only one thing that a large temperature difference is required for this purpose. However one thing must keep in mind that we have to maintain constant, but different temperature and therefore the energy distribution at both the end will be different and hence it leads to the successful mentioned process.

Seebeck Coefficient

The voltage produced between the two points on a conductor when a consistent temperature difference of 1° Kelvin is maintained between them is termed as Seebeck coefficient. One such combination of copper constantan, has a Seebeck cofficient of 41 microvolt per Kelvin at room temperature

Spin Seebeck Effect

However, in the year 2008 it was observed that when the heat is applied to a magnetized metal, its electron rearranges according to its spin. This rearrangement however not responsible for the creation of heat. This effect is K/w as spin Seebeckeffect. This effect used in the development of fast and efficient micro switches.

Applications of Seebeck Effect

1)This Seebeck effect is commonly used in a Thermocouples to measure the temperature differences or to actuate the electronic switches that can turn the system on or off. Commonly used thermocouple metal combinations include constantan/copper, constantan/iron, constantan/chromel and constantan/alumel.

2)The Seebeck effect is used in thermoelectric generator, which function like a heat engine.

3)These also used in some power plants in order to convert waste heat into additional power

4)In automobiles as automotive thermoelectric generators for increasing fuel efficiency.

Biot Savort 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.

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

Therefore final Biot Savart law derivation is,

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.

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.

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,

Now combining these three statements, we can write,

This is the basic form of Biot Savart 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

Here, μ0 used in the expression of contant k is absolute permeability of air or vacuum and it's value is 4π10-7 Wb/ A-m in S.I system of units. μrof 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,

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

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

As per the figure above,

Finally the expression of B comes as,

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,

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,

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

Coulomb's Law;

Coulomb's Law Explanation

It was first observed in 600 BC by Greek philosopher Thales of Miletus, if two bodies are charged with static electricity, they will either repulse or attract each other depending upon the nature of their charge. This was just an observation but he did not establish any mathematical relation for measuring the attraction or repulsion force between charged bodies. After many centuries, in 1785 Charles Augustin de Coulomb, a French physicist, published the actual mathematical relation between two electrically charged bodies and derived an equation for repulsion or attraction force between them. This fundamental relation he referred as Inverse Square Law, which is most popularly known as Coulomb's law. As per this law

Charles Augustin de Coulomb

Where, F is the repulsion or attraction force between two charged bodies.

Q1 and Q2 are the electrical charged of the bodies.

Centers of the bodies are kept at a distance r.

k is a constant depends upon the medium in which charged bodies are situated.

Inverse Square Law or Coulomb's law is very easy to understand if we think the matter in little bit deeply.

Suppose you have two charged bodies one is positively charged and one is negatively charged. As they are oppositely charged they will attract each other if they are kept at a certain distance from each other.

Now if you increase the charge of one body keeping other unchanged, the attraction force is obviously increased. Similarly if you increase the charge of second body keeping first one unchanged, the attraction force between them is again increased.

Hence, force between the charge bodies is proportional to the charge of either bodies or both.

Coulomb's Law

Now, by keeping their charge fixed at Q1 and Q2 if you bring them nearer to each other the force between them increases and if you take them away from each other the force acting between them decreases.

If the distance between their centers is r, it can be proved that the force acting on them is inversely proportional to r2.

It is also obvious that this force is not same for every medium. For example, if the bodies are kept in air, the force is different from the force while they are kept in water. Variation of force according to the medium, is determined by the constant k.

So finally, Coulomb's law comes as

If the charge Q1 and Q2 are similar, the force acting on them would be repulsive but explanation is same as above.

In SI system of unit, force is measured in Newton, distance is measured in meter, charge is measured in coulomb the force acting between the charges is expressed as,

Where, εo is the permittivity or dielectric constant of air and εo = 8.854 X 10 - 12 F / m (farad per meter) and εr is the relative permittivity of the surrounding medium in respect to the permittivity of air.

The force is a vector quantity, which means it has direction in addition to its magnitude. Here, the force acts along the straight line joining the centers of charged bodies.
The expression (1) can be rewritten as

Unit of charge determined from Coulomb's law

In equation if we put Q1 = Q2 = 1 unit, r = 1 m, εr = 1, we get,

Hence, unit of charge can be defined as that charge which when placed in air at a distance of one meter from an equal and similar charge, is repelled with a force of 9 X 109 Newton. This unit of charge is named as coulomb after name of Charles Augustin de Coulomb.

Definition of relative permittivity
As per Coulomb's law the expression of force acting between two charged bodies kept in a medium, is given as

The expression of force acting between two charged bodies kept in air or vacuum, is given as

Therefore, relative permittivity of a medium is the ratio of forces experienced between two charged bodies placed same distance apart in air and medium.

Joule's Law;

We know about the heating effect of electric current, when it flows through a circuit due to collision between electrons and atoms of wire. But precisely how much heat is generated during electric current flow through a wire, on what conditions and parameters does it depend ?
How we know about this ? To solve this problem Joule coined a formula which explains this phenomenon accurately. This is known as Joule’s law. This law is explained in detail afterwards.

Joule’s Law of Heating

The heat which is produced due to the flow of current within an electric wire is expressed in Joules. Now the mathematical representation or explanation ofJoule’s law is given in the following manner.

i) The amount heat produced in current conducting wire is proportional to the square of the amount of current that is flowing through the circuit, when the electrical resistance of the wire and the time of current flow is constant.

i.e. H ∝ i2 (When R & t are constant)

ii) The amount of heat produced is proportional to the electrical resistance of the wire when the current in the circuit and the time of current flow is constant.

i.e. H ∝ R (when i & t are constant)

iii) Heat generated due to flow of current is proportional to the time of current flow, when the resistance and amount of current flow is constant.

i.e. H ∝ t (when i & R are constant)
When these three conditions are merged, the resulting formula is like this -

Here ‘H’ is the heat generated in Joules, ‘i’ is the current flowing through the circuit in ampere and ‘t’ is in seconds. When any three of these are known the other one can be equated out.

Here, 'J' is a constant, known as Joule's mechanical equivalent of heat. Mechanical equivalent of heat may be defined as the number of work units which, when completely converted into heat, furnish one unit of heat.

Obviously the value of J will depend on the choice of units for work and heat.
It has been found that
J = 4.2 joules/cal (1 joule = 107 ergs) = 1400 ft. lbs./CHU = 778 ft. lbs/B Th U

It should be noted that the above values are not very accurate but are good enough for general work.
Now according to Joule's law I2Rt = work done in joules electrically when I ampere of current are maintained through a resistor of R ohms for t second.

By eliminating I and R in turn, in the above expression with the help of Ohm's law, we get alternative forms as

Gauss's Law ;

We know that there is always an electric field around a positive or negative electrical charge and that electric field there is flow of energy tubes or flux. Actually this flux is radiated/emanated from the electric charge. Now amount of this flow of flux depends upon the quantity of charge it is emanating from. To find out this relation the Gauss's theorem was introduced. This theorem can be considered as one of the most powerful and most useful theorem in the field of electrical science. We can find out the amount of flux radiated through the surface area surrounding the charge from this theorem.

This theorem states that the total electric flux through any closed surface surrounding a charge is equal to the net positive charge enclosed by that surface.

Suppose the charges Q1, Q2_ _ _ _Qi, _ _ _ Qn are enclosed by a surface then the theorem may be expressed mathematically by surface integral as

Where D is the flux density in coulombs/m2 and dS is the outwardly directed vector.

Explanation of Gauss's Theorem

For explaining the Gauss's theorem, it is better to go through an example for proper understanding.

Let Q be the charge at the center of a sphere and the flux emanated from the charge is normal to the surface. Now, this theorem states that the total flux emanated from the charge will be equal to Q coulombs and this can be proved mathematically also. But what about when the charge is not placed at the center but at any other point other than the center (as shown in the figure)

Gauss's Theorem 1) Charge at center 2) Charge at a point other than center

At that time the flux line are not normal to the surface surrounding the charge, then this flux is resolved into two components which are perpendicular to each other, the horizontal one is the sinθ component and the vertical one is the cosθ component. Now when the sum of these components is taken for all the charges then the net result is equal to the total charge of the system which proves Gauss's theorem.

Proof of Gauss’s Theorem

Let us consider a point charge Q located in a homogeneous isotropic medium of permittivity ε.

The electric field intensity at any point at a distance r from the charge is

The flex density is given as

Now from the figure the flux through area dS

Where θ is the angle between D and the normal to dS

Now, dScosθ is the projection of dS is normal to the radius vector. By definition of a solid angle