Magnetism and Electro-Megnetism Notes

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CHAPTER – 14                                                      MAGNETISM AND ELECTRO-MAGNETISM

MAGNETIC FIELD DUE TO CURRENT

It was discovered by Oersted that when current masses through a conductor, magnetic field is produced. This field is known as “Magnetic Field of Induction” and is denoted bu “B”.

Ampere found that when two current carrying conductors are near each other, they experience force at each other. If the current is in the same direction the force is attractive and if the current is in opposite direction.

When electric charges are at rest they exert electrostatic force of attraction or repulsion on each other. When the charges are in motion they exert electric as well as magnetic force on each other because and isolated moving positive and negative charge create both electric and magnetic field.

MAGNETIC FIELD

Magnetic Field is a space or region around a magnet or current carrying coil of wire where its effect can be felt by small compass needle. Magnetic field of induction can be visualized by magnetic lines of induction.

A line of induction is an endless curve, which can be traced by a compass needle.

MAGNETIC FLUX AND FLUX DENSITY

The total number of magnetic lines of induction passing through a surface is called magnetic flux.

DETERMINING THE CHARGE TO MASS RATIO OF AN ELECTRON

The charge to mass into of an electron was determined by Sir J.J. Thomson by an apparatus which consists of a highly evacuated pear shaped glass pulls into which several metallic electrodes are sealed.

Electrons are produced by heating a tungsten flament F by passing a current through it. The electrons moving sideways are also directed towards the screen by applying negative potential on a hollow cylinder C open on both the sides surrounding the filament. Electrons are accelerated by applying positive potential to discs A and B. If V be the total total P.d between the disc Band the filament F taken then Kinetic Energy.

The beam strikes the screen coated with zinc sulphide after passing through the middle of the two horizontal moetal P’P and a spot of the light produced at O on the screen where the beam strikes and its position is noted.

A magnetic field of induction B is produced in between the plate directed into the paper. The magnetic field is produced by two identical current carrying coils placed on either side of the tube at the position of plates.

The force due to the magnetic field on the moving electron makes them move in a curved path and the light spot shifts from O to O on the screen there from of magnetic field acts as centripetal force

e. V B = mv2 / r

e/m = V/Br ——– I

e/m can be computed if the radius r and the expression of the circular path are in which the beam moves in the field region is determined. The radius r is calculated from the shift of the light spot i.e. r = 3.

A better method of determined V is as under. An electric field E is produced between the plates by applying suitable potential difference to exert a force “Be” on the electron opposite to that due to the magnetic field.

The potential diff. VI is so adjusted that two fields neutralize each other effects and the spot come back to its initial position O. Thus each other effects and the spot come back to its initial position O. Thus

Ee = Be V

Or

V = E/B —– (II)

Where E = V1 / d

d = distance between the plates.

Putting the value of V from eq 2 in 1

e/m = E/B2r

e/m = K75888 x 10(11) e/kg

AMPERE’S LAW

According to this law the sum of the product of the tangential component of the magnetic field of indaction and te length of an element of a closed curve taken in a magnetic field is μo times the current which passes through that area bounded by the curve.

Consider a long straight wire carrying a current 1 in the direction. The lines of force are concentric circles with their common centre on the wire. From these circles consider a circle of radius r. The magnitude of the magnetic field at all points on this circle and inside the circle is same.

Biot and Savart experimentally found that the magnitude of the field depends directly on twice the current and inversely proportional to the distance r from the conductor.

SOLENOIDAL FIELD

A solenoid is a coil of an insulated copper wire wound on a circular cylinder with closed turns. When current passes through it, magnetic field is produced with is uniform and strong inside the solenoid while outside it the field is negligibly weak.

Consider a solenoid through which the current 1 is passing in order to determine the magnetic field of induction B at any point inside the solenoid imagine a closed path “abcda” on the form of a rectangular. The rectangular is divided into four elements of length L1, L2, L3, L4. L1 is along the axis inside the solenoid and L3 is far from the solenoid.

By applying amperes circuital law

B L1 + B. L2 + L2 + B. L3 + B. L4 = μo x current enclosed —– (I)

Since B. L1 is parallel inside the solenoid

B. L1 = BL4 cos 0 = BL4

The field is very weak outside the solenoid is very weak and therefore it can be negnected thus

B. L3 = 0

As B is perpendicular to L2 and L4 inside the solenoid therefore

B. L2 = BL2 cos 90 = 0

B. L4 = BL4 cos 90 = 0

substitute the above values is eq 1

B. L1 + O + 0 + 0 = μo x current closed

B. L1 = μo x current enclosed ——- (II)

If there are n turns per unit length of the solenoid and each turn carries a current I will be “n L1I”

TOROIDAL FIELD

A Toroid or a circular solenoid is a coil of insulated copper wire wound on a circular core with close turn. When the current passes through the toroid, magnetic field is produced which is strong enough inside while outside it is almost zero.

Consider a toroid that consists of N closely packed turns that carry a current I. Imagine a circular curve of concentric the core.

It is evident form of the symmetry at all points of the curve must have the same magnitude an should be tangential to the curve at all points. Divide the circle into small elements each of length ΔL is so small that B and ΔL are parallel to each other.

By amperes law

Σ B : ΔL = μo x current enclosed

ΣB ΔL Cos 0 = μo x current enclosed

ΣB ΔL = μo x current enclosed

BΣ ΔL = μo x current enclosed

Σ ΔL = 2 π r

B 2 π r = μo x current enclosed ——– (I)

Cases

If the circular path 1 is outside the core on the inner side of the toroid if enclose no current. Thus eq 1 become

B 2 π r = μo x 0 = 0

B = 0

If the circular path 2 is outside the core on the outer side of the toroid each turn of the winding passes twice through the area bounded by this path carrying equal currents in opposite directions thus the net current through the area is zero hence eq 1 becomes

B 2 π r = μo x 0 = 0

B = 0

If the circular path 3 is within the core the area bounded by the curve will be threaded by N turns each carrying 1. Thus Current enclosed = NI

Therefore eq 1 becomes

B 2 π r = μo NI

B = μo NI / 2 π r

ELECTROMAGNETIC INDUCTION

The phenomenon in which an Emf is set up in a coil placed in a magnetic field whenever the flux through it is changing is called ELECTROMAGNETIC INDUCTION. If the coil forms a part of a closed circuit the induced Emf cases a current to flow in the circuit. This current is called INDUCED CURRENCY.

The magnitude of induced emf depends upon the rate at which the flux through the coil charges. It also depends on the number of turns on the coil.

The magnetic flux through a circuit can be changed in a number of different ways. By changing the relative position of the coil w.r.t to a magnetic field or current bearing solenoid.

By changing current in the neighbouring coil or by changing current in the coil itself.

By moving a straight conductor in the magnetic field in such a way that it cut the magnetic lines of force.

FLUX LINKAGE

The product of number of turns N and the flux ф through each turn of the coil is called flux linkage i.e.

Flux Linkage = N ф

FARADAY’S LAW OF ELECTROMAGNETIC INDUCTION

A Emf is induced in a coil through which the magnetic flux is changing. The Emf lasts so long as the change of flux is in progress and becomes zero as soon as the flux through the coil becomes constant or zero.

SELF INDUCTION

Consider a coil through which an electric current is flowing. Due to this current magnetic field will be produced which links with the coil itself. If for any reason the current changes the magnetic flux also changes and hence an Emf is induced in the coil this phenomenon is known as SELF INDUCTANCE. In accordance with Lenz Law, the emf posses the change that has induced it and it is therefore known as back emf.

If the current is increasing the back emf opposes the increase. If the current decreasing it opposes the decrease.

The back emf is directly proportional to the rate of change of current. If ΔL change in current Δ t then back emf E is given.

e = L Δl / Δt ——- (I)

Where L = self inductance of the coil.

The measure of the ability of a coil to give rise to a back emf is called the Self inductance. Its value depends on the dimensions of the coil, the number of turns and the permeability of the core material. Its unit is henry.

Henry

The self inductance of a coil is 1 Henry if the current varying through is at the rate of 1 amp/sec, induces a back emf of 1 volt.

If N be the number of turns in the coil and Δ φ be the change of flux in time Δ t then by Faraday’s Law.

Є = -N Δφ / Δt —– (II)

-N Δφ / Δt = – Δl / Δt

N Δφ = L Δl

Δ (Nφ) = Δ (Ll)

Nφ = L1

MUTUAL INDUCTION

Consider two coils close to each other. One coil is connected to a source of emf and the other with a galvanometer. The coil which is connected to the emf is called the primary coil and the other is called secondary coil. Some of the magnetic flux produced by the current in the primary coil is changed the magnetic flux in the secondary coil also changes and hence an emf is induced in the secondary this phenomenon is called mutual induction.

The back emf “ξ” induced in the secondary coil is directly proportional to the rate of change of current Δ1 / Δt in primary coil and is given by

Є2 = -M ΔI / Δt ——– (I)

Where M is the mutual inductance of the pair of coils. Its value depends upon the number of turns of the coil, their cross-sectional area, their closeness and core material. Its unit is Henry.

If N2 be the number of turns in the secondary and Δф / Δt be the rate of change of flux in it then by faraday’s law.

Є2 = -N2 Δφ2 / Δt —— (II)

Comparing 1 and 2

-N2 Δφ2 / Δt = – M Δ1 / Δt

N2 Δφ2 = M Δ1

Δ(N2 φ2) = Δ(M 1)

N2 φ2 = M 1

Non-Inductive Winding

In bridge circuits such as used for resistance measurements self inductance is a nuisance.

When the galvanometer key of bridge is closed the current in the arms of bridge are re-distributed unless the bridge happens to the balanced. When the currents are being re-distributed these are changing and self induction delays the reading of new equilibrium. Thus the galvanometer key thus not corresponds to steady state which the bridge will eventually reach. Its me therefore be misleading.

To minimize their self inductance coils of the bridge and re-resistance boxes are so wound as to setup extremely small magnetic field.

The wire is doubled back on itself before being coiled.

In this type of winding current flows in opposite direction in the double wires and consequently the magnetic field and hence the magnetic flux setup by one wire in neutralized by that due the other wire. Hence self induced emf will not be produced when the current through the circuit changes.