Physics Revision Notes Of Magnetic Effects of Electric Current and Magnetism For NDA
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Electromagnetics

The branch of physics in which we studied the effect of magnetic field produced due to current or moving charge, is known as magnetic effects of electric current.

`text (Magnetic Field)`

It is a region or space around a magnet or current carrying conductor or a moving charge in which its magnetic effect can be felt. Its `Sl` unit is tesla (`T`).

`text (Biot-Savart Law)`

According to this law, "the magnetic field `dB` at the point P due to the small current element of length `dl` is given by

`d B = mu_0/(4 pi) ( i dl sin theta)/r^2 W b // m^2` or tesla

` = mu_0 /(4 pi) (id l xx r)/r^3`

where, `mu_0` is a constant and it is called permeability of free
space

`mu_o = 4 pi xx 10^(-7) Wb//A - m`

Rules to Find the Direction of Magnetic Field

(i) `text(Right hand palm rule)` If we spread our right hand in such a way that thumb is towards the direction of current and fingers are towards that point, where we have to find the direction of field, then the direction of field will he perpendicular to the palm.

(ii) `text(Maxwell's right handed screw rule)` If a right handed cork screw is rotated, so that its tip moves in the direction of flow of current through the conductor, then the rotation of the head of the screw gives the direction of magnetic lines of force.

Applications of Biot-Savart's Law

Ampere's Circuital Law

It states that the line integral of magnetic field around any closed path in vacuum is equal to `mu_0` times the total current enclosed by the loop.

` int B . d l = mu_0 I_text( net)`

The simplified from of it is

`Bi = mu_0 I`

This equation is used in following conditions

(i) At every point of the closed circuit

` B || dl`

( ii) The magnetic field at every point of the closed circuit remains uniform.

`text(Application of Ampere's Circuital Law)`

The main applications of Ampere's circuital law are as follow

( i) The magnetic field due to straight wire of infinite length

` B = mu_o/(2 pi) . I/r`

( ii) The magnetic field due to conducting rod of radius `R` at a point at distance `r` from the rod.

(a) `r < R , B = ((mu_o I)/(2 pi R^2)) . r`

or `B alpha r`

(b) When `r = R` (i.e., on the surface of the conducting rod)

`B = (mu_o I)/(2 pi R)`

(c) `r > R`

`B = mu_o/(2 pi) . I/r`

(iii) The magnetic field due to hollow cylindrical pipe of radius R at a point at distance r from the current carrying hollow pipe.

(a) When `r < R`, then `B = 0`

(b) When `r >= R`, then `B = mu_o/(2 pi) . I/r`


Force between two Parallel Current Carrying Conductor

Two linear parallel conductor carrying currents in the same direction attract each other while in opposite direction they repel each other. Therefore, force between two current carrying parallel conductor per unit length is

`F = mu_o/(4 pi) . (2 I_1 I_2)/r^2`

`text(Magnetic Field of a Moving Point Charge)`

The magnetic field due to a charge q, moving with speed v at a point P at distance r from the charge is given by the following formula.

` vec B = mu_0/( 4 pi) . q/r^2 ( vec v xx vec r)`

` text(Force on a Moving Charge in Magnetic Field)`

Force on a moving charge in the magnetic field is given by

`F_m = q v B sin theta = q(v xx B)`

where, `q =` magnitude of charge,

`v =` velocity of charge,

`B =` intensity of magnetic field

and `theta =` angle between direction of velocity and direction of magnetic field

It is also called Lorentz force.

• Electromagnetic and gravitational forces act on neutral and large bodies.
• If the moving charge is moving perpendicular to the magnetic field, then its kinetic energy remains constant.
• The force acting in the presence of electric and magnetic field simultaneously
`F = q ( E + v xx B)`

`text(Rules to Find the Direction of Force)`

(i) `text(Right hand palm rule)` If a linear conductor is grasped in the palm of the right hand with thumb pointing along the direction of the current, then the finger tips will point in the direction of lines of force.
(ii) `text(Fleming's left hand rule)` If we spread the forefinger, central finger and thumb of our left hand in such a way that these three are perpendicular to each other, then first forefinger is in the direction of magnetic field, second central finger is in the direction of current and thumb will represent the direction of force.

Motion of Charged Particle in a Magnetic Field

When a charged particle enters in a magnetic field perpendicularly, then it moves on a circular path.

Radius of circular path, `r = (mv)/(Bq) = sqrt(2m E_K)/(q B)`.

Time period of partical , ` T = (2 pi m)/(Bq)`

Frequency of- particle, `n = (Bq)/(2 pi m)`.

where, symbols have their usual meanings.
• If a charged particle is moving in the direction of magnetic field, then it experiences no force.
• The velocity of proton moving in a magnetic field changes continuously.

`text(Force on a Current Carrying Conductor in Magnetic field)`

When a current carrying conductor is placed in magnetic field, then it experiences a force on it, this force is given by
`F_m = B i l sin theta = i l xx B`

where, `B =` intensity of magnetic field,
`i =` current in the conductor,
`l =` length of the conductor
and `theta =` angle between the length of the conductor and direction of magnetic field
When, `theta = 90^o` or `sin theta = 1`, then `F = i l B` (Maximum)
When, `theta = 0` or `sin theta = 0`
`F = i l B xx 0 = 0`
When the current carrying conductor is placed parallel to the field, the acting force will be zero.

`text(Application of Current Carrying Conductor)`

(i) Magnetic force acting on a current carrying conductor is not central force because `F = B I d l sin theta` expression does not depends upon r.
(ii) The force `d F` is always perpendicular to `B` and `i d L`

Magnet

A naturally occurring black coloured substance called lodestone can attract pieces of iron kept nearby. In early days, the Greeks observed this property of loadstone an oxide called magnetite ( `text()_4` ). This type of substance is called magnet.

Magnet is of two types
(i) `text(Natural Magnet)` Natural magnet is a substance found in nature, which has the property of attracting small pieces of iron, this property is called magnetism.
(ii) `text(Artificial Magnets)` The magnet which is made by artificially that is known as artificial magnet. It's shape and size is fixed.
e.g. Horseshoe magnet

` text(Important Facts Related to Magnet)`

(i) Artificial magnet have short life and natural magnet have long life.
(ii) Magnetism of earth have for infinite time.
(iii) Unstable magnet is an induced magnet.
(iv) Magnet are use in electric bell, fan, washing machine etc.

`text(Pole Strength)`

The ability of magnetic pole to attract magnetic material is known as pole strength. It is denoted by `m`. The pole strength of North and South pole is represented by `+ m` and `-m`.

Pole strength `(m) = text( Magnetic force)/text( Magnetic induction) = F/m`

Unit of pole strength is Ampere-meter or Newton/Tesla.

`text(Magnetic Axis)`
The line joining north pole and the south pole of a magnet is called the magnetic axis.

`text(Effective Length of Magnet)`

The distance between the north pole and the south pole of the magnet is called the effective length of the magnet.

` text(Magnetic Dipole)`

Two equal and opposite pole separated by a distance `2l ` are said to constitute a dipole.

`text(Magnetic Moment)`

The magnetic moment of a bar magnet is given by the product of its length and pole strength. It is represented by `M`.

`M = (2l) xx m`
Unit of magnetic moment is ampere - `text(meter)^2`.



Magnetic Field

The area surrounding the magnet in which, another magnet experience a force on it is called magnetic field.

`text(Magnetic Field Lines)`
The imaginary lines which represents the direction of magnetic field is known as magnetic field lines. Some properties of magnetic field lines are given below
• Magnetic line of force always from closed curves.
• They leave the north pole and enter the south pole externally.
• They move from the south pole to the north pole with in the magnet.
• They tend to contract laterally.
• The magnetic field lines never intersect each other.

`text(Some Properties of Magnet)`
(i) `text(Intensity of magnetisation)` The intensity of magnetisation is defined as magnetic moment per unit volume.

i,e ` r = M/V`

where, `M =` magnetic dipole moment,
and `V =` volume of material

(ii) `text(Magnetic permeability)` The magnetic permeability of a material is the measure of degree to which the magnetic field can penetrate or permeate a medium. It is denoted by `mu` is

` mu = B/H`

where, `B =` magnetic induction
and `H =` magnetising field
(iii) `text(Magnetic susceptibility)` The intensity of magnetisation per unit magnetising field is known as magnetic susceptibility

i.e ` phi = I/H`
where, `I =` intensity of magnetisation
and `H =` magnetising field
(iv) `text(Relation between magnetic susceptibility and permeability)` The relative magnetic permeability,
`mu_r = 1 + phi`
where, `mu_r = mu/(mu_0) [ mu_o =` absolute permeability]

`:. mu/mu_c = 1 + phi ` or ` mu = mu_0 (1 + phi)`

Torque on Bar Magnet in Magnetic Field

In figure, a uniform magnetic field `B` is represented by equidistant parallel line. `NS` is a bar magnet of length `2l` and strength of each pole is `m`. The magnet is held at angle `theta` with the direction of `B` Torque

`(tau) = m B l sin theta + m B l sin theta`
`tau =2 mBl sin theta = MB sin theta`

In vector, `tau = vec M xx vec B`

`text(Work Done in Rotating a Dipole in a Magnetic Field)`
The total work done in deflecting the dipole through an angle `theta` from `0°` is
`W = MB (1 - cos theta)`
• If `theta = 0^o`, then `W = MB [1 - cos theta] = 0`
• If `theta = 90°`, then `W = MB`
• If `theta = 18.0°`, then `W = 2 MB`
where, `M` is magnetic dipole moment

`text(Field due to Small Bar Magnet)`

(Magnetic Dipole)
(i) `text(In end on position)` (on axial point)

`B = mu_o/(4 pi) (2M)/r^3`

where, `r` is distance of the required point from the centre of dipole
(ii) `text(In broad side on position)` (on perpendicular bisector)
` B = mu_0/(4 pi) M/r^3`
(iii) At any general point `(r, theta)` relative to centre of dipole

` B = mu_0/(4 pi) M/r^3 sqrt( 1 + 3 cos^2 theta)`

Earth's Magnetism

The earth is a natural source of magnetic field. One magnetic field present everywhere near the surface of the earth. A freely suspended magnet always points in the north-south direction even in the absence of any other magnet. This suggests that the earth itself behaves as a magnet which causes a freely suspended magnet (or magnetic needle) to point always in a particular direction i.e. north and south. The
shape of earth's magnetic field resembles that of a bar magnet of length one-fifth of earth's diameter buried at its centre. The south pole of a earth's magnet is towards the earth's north pole (geographical north), while the north pole of earth's magnet is towards the earth's south pole (geographical south). Thus, there is a magnetic S-pole near the geographical north and a magnetic' N -pole near the geographical south. The positions of the earth's magnetic poles are not well defined on the globe,
they are spread over an area.

Magnetic Elements

To have a complete knowledge of the earth's magnetism at a place, the following three elements must be known
(i) `text(Angle of declination)` The angle between the magnetic meridian and geographical meridian at a place is called the angle of declination (or simply the declination) at that place.
(ii) `text(Angle of dip or inclination)` The angle which the axis of needle makes with the horizontal, is called angle of dip (`theta`).
(iii) `text(Horizontal component of the earth's field)` The direction of the earth's field at the magnetic poles is normal to the earth's surface (i.e., in vertical direction) and at magnetic equator it is parallel to the earth's surface, (i.e., in horizontal direction).
Thus, the resultant earth's field can be resolved in two components.

The horizontal component `B_H` along `AB`
`B_H = B_e, cos theta` ..........(i)
The vertical component `B_v` along `AD`
`B_V = B_e sin theta` ..........(ii)
From Eqs. (i) and (ii), we get

`:. = (B_V)/(B_H) = (B_e sin theta)/(B_e cos theta) = tan theta`

or `B_V = B_H tan theta`

Again from Eqs. (i) and (ii), we get

`B_H^2 + B_V^2 = B_e^2 (cos^2 theta + sin^2 theta)`

or ` B_e = sqrt( B_H^2 + B_V^2 )`

Classification of Magnetic Substances

On the basis of magnetic properties, different materials have been classified into three categories
(i) `text(Diamagnetic substance)` These substances when placed in an external magnetic field, then acquire feeble magnetism opposite to the direction of the magnetic field. e.g. `Bi, Zn, Au, NaCI, H_2O`, etc.
(ii) `text(Paramagnetic substance)` These substances when placed in an external magnetic field, then acquire feeble magnetism in the direction of the magnetic field.
e.g. `AI, Na, Pt, Mn, CuCl_2 , O_2`, etc.
(iii) `text (Ferromagnetic substance)` These substances when placed in an external magnetic field are strongly magnetised in the direction of the field.
e.g. `Fe, Ni, Co, Fe_2O_ 3` , etc.

`text(Curie's law)`

According to curie's Law, the magnetic susceptibility of paramagnetic substance is inversely proportional to the absolute temperature. i.e.

` chi alpha 1/T`

`text(Curie Temperature)`
Above a certain temperature the substance loses its ferromagnetic character and begins to behave as a paramagnetic substance. This particular temperature is called the curie temperature of the substance.

`text(Magnetic Flux)`

It is defined as the total number of magnetic lines of force passing normally through any surface. If surface of area A is placed perpendicular to uniform magnetic field B. Then, magnetic flux `phi = BA`. Its unit is `Wb` (weber) or `kg-m^2 // s^2 A`. If a plane is parallel to the magnetic field, then no flux link will pass through it and magnetic flux link with coil will be zero. If the coil is rotated through `90°` in the magnetic field, then magnetic flux linked with the coil is zero.

Electromagnetic Induction

Electromagnetic Induction is the phenomenon due to which an induced emf is set up in a conductor or in an electric circuit, on changing the magnetic flux linked with it.

`text(Faraday's Laws of Electromagnetic Induction)`

Faraday had introduced two laws of electromagnetic induction which are as follows

(i) `text(First Law) ` Whenever the magnetic flux linked with a coil changes, an induced emf induced in the circuit.

(ii) `text(Second Law)` Induced emf is equal to the negative rate of change of magnetic flux
`e = - (Delta phi)/(Delta t)` .

If circuit has a coil of `N` turns, then

` e = - ( Delta (N phi))/(Delta t)`

` text(Induced Current and Induced Charge)`

If a circuit have a resistance `(R)` then
(i) Induced current, `i = N/R (Delta phi)/(Delta t)`.
(ii) Induced charge, `q = N/R Delta phi`.
where, `R =` resistance of the circuit


 
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