#### The Magnetic Field, Physics tutorial

Introduction:

The first magnetic phenomena to be viewed were those related by natural magnets. These were rough fragments of iron ore found close to the ancient town of Magnesia in Asia. However, the word magnet was derived from the name of that town. Such natural magnets encompass the property of attracting to themselves unmagnetised iron, the consequence being most pronounced at certain areas of the magnet termed as its poles.

It was by means of observing electric currents that the connection between electricity and magnetism was decisively established. Therefore in the year 1820 Haus Christian Oersted (1777 - 1851) at the University of Copenhagen, Demark found that a wire carrying an electric current deflected a close by compass needle.

It is now well acknowledged that electric current generate magnetic fields and that a changing magnetic field generates an electric current. This connection between current and magnetism gave birth to electromagnetism, a subject to which modern civilization is greatly indebted.

Fields Due to Magnets:

The magnetic properties of a magnet come out to originate at certain areas in the magnet which are termed to as the poles. In a bar magnet the poles are the ends.

Here are a few experimental findings regarding magnets:

1) Like poles are of two types.

2) Like poles repel one other and dissimilar pole attract.

3) Poles for all time seem to take place in equivalent and opposite pairs.

4) If no other magnet is close to, a freely suspended magnet sets so that the line connecting its poles (that is, its magnet axis) is roughly parallel to the north-south axis of the earth.

Electric field to define a magnetic field:

The space surrounding a magnet in which a magnetic force is experienced is termed as a magnetic field. The direction of a magnetic field at a point is taken as the direction of the force which acts on the north magnetic pole there.

A magnetic field can be symbolized through magnetic field lines drawn in such a way that:

1) The line or the tangent to it when curved provides the direction of the field at that point

2) The number of lines per unit cross-section area is the sign of the 'strength' of the field.

Field Due to Currents:

Magnetic fields are generated by means of electric currents that can be macroscopic currents in wires or microscopic currents related via electrons in atomic orbits. The magnetic field 'B' is stated in terms of force on moving charge in the Lorentz force law. The interaction of magnetic field having charge leads to numerous practical applications. The magnetic field sources are effectively dipolar in nature, having a north and south magnetic pole. The SI unit for magnetic field is the Tesla that can be view from the magnetic portion of the Lorentz force law Fmagnetic = qvB to be composed of (Newton x second)/(Coulomb x meter). A smaller magnetic field unit is the Gauss (1 Tesla = 10,000 Gauss).

(a) Magnetic field lines of a permanent magnet (b) cylindrical coil (c) iron-core electromagnet, (d) straight current-carrying wire (e) circular current-carrying loop.

Force on a Current in a Magnetic Field:

If a current-carrying conductor lies in a magnetic field, magnetic force are applied on the moving charges in the conductor. Such forces are transmitted to the material of the conductor and the conductor as a whole experienced a force distributed all along its length. The electric motor and the moving coil galvanometer both based on their operation on the magnetic force on conductor - carrying currents.

The force on a current-carrying conductor is:

1) For all time perpendicular to the plane having the conductor and the direction of the field in which it is positioned and

2) Greatest when the conductor is at right angles to the field.

Fleming's left-hand (or motor) rule: The facts regarding the relative directions of current, field and force are summarized by Fleming's left-hand rule which defines that:

Whenever the thumb and first two fingers of the left-hand are held each at right angles to the other, with the first Finger pointing in the direction of the Field and the second finger in the direction of the Current, then the Thumb forecasts the direction of the force or thrust.

Factors Affecting the Force:

The force 'F' on a wire lying at right angles to a magnetic field is directly proportion to the current 'I' in the wire and to the length 'L' of the wire in the field. It as well based on the magnetic field.

Magnetic Flux Density:

The measure of strength of a magnetic field is termed as magnetic flux density or magnetic induction. A magnet is stated to encompass a north and a south pole; two magnets will repel one other whenever like poles face one other and attract one other if opposite poles come close to one other. Electrically charged particles are as well deflected in the magnetic fields.

Magnetic flux density is analogous having characteristics of electric and gravitational fields. Electrical field strength is the force acting on the body per unit charge, and gravitational field strength is the force acting on the body per unit mass. Magnetic fields are made through an electric current. Magnetic flux density (B) is force (F) acting on the conducting material per unit length (l) and per unit of current (I) that can be written as the equation B = F / I x l. The unit of magnetic flux density is termed as the Tesla.

A magnetic field is for all time produced at 90 degrees to a moving electric field. The direction of the electric current, magnetic field and magnetic force are stated by John Ambrose Fleming's left-hand rule. Holding the left hand's first two fingers and thumb at right angles to one other will point out the relative directions the thrust, field and current.

The magnetic field of earth is mainly caused by the present of a molten, rotating iron core at its center. Its flux density is strongest at the South and North poles. Electrically charged particles from the sun are attracted to the poles that cause the aurora borealis, or northern lights, and the aurora australis or the southern lights.

Force on an Electron moving in a Magnetic field:

The electric current in a wire is conventionally regarded as the flow of positive charge, however it comprises in fact of a flow of negative electrons in the opposite direction.

Assume that an electron of charge 'e' is moving with the velocity 'v' at right angles to a magnetic field of flux density 'B'.

The electron moves a distance 'l' in a time 't', here t = l/v, and comprises a current I.

Current = flow of charge per second

I = e/t = e/(l/v) = ev/l

Il = ev

But force on a current = BIl

Force on a moving electron = Bev

Torque on a rectangular coil:

Figure above represents a vertical rectangular coil length and breadth 'a' and 'b' correspondingly, carrying a current 'I' with its plane at an angle 'α' to a horizontal magnetic field of magnetic flux density 'B'. Applying Fleming's left-hand rule to the figure, it will be observe that the left-hand vertical side is urged out of the paper, the right-hand vertical side into the paper, and the top and bottom are urged up and down correspondingly. When the coil is free to turn about a vertical axis, only the forces on its vertical sides will encompass a turning effect. The forces on these sides are each BaNI, here 'N' is the number of turns of the coil.

Taking the moment of the forces at 0 we have:

P = 2 BaNI (b/2) cos α

P = BANI cos α Nm

Here A = area of coil = ab. Therefore the torque on the coil is BANI cos α.

Maximum and minimum values of the torque on the coil:

The torque on the coil acquires its maximum value if the plane of the coil is parallel to B and α = 0. The maximum value is BANI. Its minimum value, if the plane of the coil is perpendicular to B and α = 90o, is zero.

The Biot-Savart Law:

The mathematical expression for magnetic flux density was introduced through Jean Baptiste Biot and Felix Savart. Talking the deflection of a compass needle as the measurement of the intensity of a current, varying in shape and magnitude, the two scientists sum up 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 connecting a given point of the field and the  current  element and is inversely proportional to the square of the distance of the specific point from the  current  element, 'r'. This is Biot Savart law statement.

Therefore, dB ∝ (I dl sinθ)/r2 or dB = k (I dl sinθ/r2)

Here, K is the constant, based on the magnetic properties of the medium and system of the units employed.

k = μoμr/4π

Thus final Biot Savart law derivation is:

dB = (μoμr/4π) x (I dl sinθ/r2)

Calculation of the Flux Density:

1) Circular Coil:

Assume that the coil is in air consists of radius 'r', carries a steady current 'I' and is considered to comprise of current elements of length 'δl'. Each and every element is at distance 'r' from the centre 0 of the coil and is at right angles to the line connecting it to 0. That is, θ = 90o. At 0 the net flux density 'B' is the sum of the flux densities 'δB' due to all the elements.

That is,

B = ∑ (μo l dl sinθ)/4π r2 = (μo I sin θ)/4π r2 ∑ dl

However ∑dl = total length of the coil = 2π2 and sin θ = sin 90o = 1

Therefore B = (μo I 2 π r)/4π r2 = μo I/2r

If the coil consists of 'N' turns each of radius 'r'

B = μo NI/2r

2) B at a point on the axis of a circular coil:

B = [(μ NI)/2] sin3 α (r/a = sin α)

3) Very Long Straight Wire:

B = μo I/2π a

The Ampere:

The ampere, often termed to as the amp, is a unit of measurement for the electric current. The word 'electric current', frequently represented by the letter 'I' in computations comprising current, signifies to the rate of flow of electrical current or the amount of electrical charge that passes a specific point in a circuit in a certain amount of time. One ampere is equal to one coulomb per second, and one coulomb is equivalent to 6.24 x 1018 electrons, so one ampere is equal to 6.24 x 1018 electrons passing a given point in a circuit in one second. Coulombs and amperes are closely associated in that the former symbolizes the amount of charge running via a circuit whereas the latter symbolizes the rates at which that charge flows.

The 'ampere' is named after André-Marie Ampère, a French physicist and mathematician who conducted important research in the area of electromagnetism. This belongs to the SI unit system that is the standard international system of units employed in science. This system of units is helpful as most of the people working in science are well-known with them and since they tend to be fairly simple to understand and to change if essential.

To derive an expression for the force, let us assume two long, straight, parallel conductors, distance a part in air, carrying current I1 and I2 correspondingly. The magnetic field at right-hand conductor due to the current I1 in the left-hand one is directed to the paper and its flux density B1 is represented by:

B1 = (μo I1)/2πa

The forces 'F' acting on length 'l' of the right-hand conductor (carrying current I2) is thus:

F = B1 I2 l = (μo I1 I2 l)/2πa

The left-hand conductor experiences an equivalent and opposite force due to being in the field of the right-hand conductor.

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