RRB ALP Physics Trade Notes on Magnetic Effects of Current Electricity

Pole strength magnetic dipole and magnetic dipole moment :

There are two poles in a magnet. North pole and South pole. In a magnet, the unlike charges attract each other whereas the like charges repel each other. 

When two magnets attract each other than action pair is formed. 

These poles are represented n the basis of their respective “POLE STRENGTH” +m and -m respectively. Pole strength is a scalar quantity which represents the strength of the pole.

A magnet can be treated as a dipole since it always has two opposite poles. This arrangement of magnets is called as MAGNET DIPOLE and it has a MAGNETIC DIPOLE MOMENT. It is represented by . It is a vector quantity. Its direction is from -m to +m that means from ‘S’ to ‘N’)

M = m.l_m here l_m = magnetic length of the magnet. l_m is slightly less than l_g (it is the geometrical length of the magnet = end to end distance).

Magnetic field and strength of the magnetic field :

Mathematically, 

Here  = magnetic force on the pole of pole strength m. m may e +ve or -ve and of any value.

S.I. unit of  is Tesla or Weber/m² (abbreviated as T and Wb/m²).

(a)  due to the various source

(i) Due to a single pole :

(similar to the case of a point charge in electrostatics)

in vector form 

here m is with the sign and  = position vector of the test point with respect to the source pole.

(ii) Due to a bar magnet :

(Same as the case of the electric dipole in electrostatics) Independent case never found. Always ‘N’ and ‘S’ exist together as a magnet.

at A (on the axis)  for a << r

at B (on the equatorial)  for a << r

At General point :

Magnetic lines of force of a bar magnet :

Magnet in an external uniform magnetic field :

(same as case of electric dipole)

F_ers = 0 (for any angle)

*here  is angle between  in vector form 

Magnetic effects of current (and moving charge)

 due to a point charge :

 ; here  angle between  -

 with sign

 -and also  -

Direction of  will be found by using the rules of vector product.

Bio-savart’s law ( due to a wire)

here  position vector of the test point w.r.t. 

 angle between  and . The resultant

-due to a straight wire :

Due to a straight wire ‘PQ’ carrying a current ‘i’ the  at A is given by the formula

At points ‘C’ and ‘D’  (think how).

For the case shown in figure

B at 

B due to an infinitely long straight wire is 

Magnetic lines of force by a current carrying straight wire are circular like shown in figure.

 due to a circular loop

(a) At the centre :

N = No. of turns in the loop.

= length of the loop.

N can be fraction or integer.

N can be fraction  or integer.

Semicircular and Quarter of a circle :

(b) On the axis of the loop :

N = No. of turns (integer)

magnetic lines of force due to the current in the ring are like shown in the figure.

The pattern of the magnetic field is comparable with the magnetic field produced by a bar magnet.

The side ‘I’ (the side from which the  emerges out)-of the loop acts as ‘NORTH POLE’ and side II (the side in which the  enters) acts as the ‘SOUTH POLE’.

Solenoid :

the magnetic field at any general point P is given by

where n : number of turns per unit length.

For ‘Ideal Solenoid’ : (l >> R or length is infinite)

The magnetic field inside the solenoid at mid point on its axis is given by

The magnetic field inside the solenoid can be considered same everywhere.

If the material of the solid cylinder has relative permeability  then 

At the ends 

(v) Graph between B and x for ideal solenoid :

AMPERE’s circuital law :

The line integral  on a closed curve of any shape is equal to  (permeability of free space) times the net current I through the area bounded by the curve.

Hollow current carrying infinitely log cylinder :

(I is uniformly distributed on the whole circumference)

(i) for r > R

(ii) r < R, B_in = 0

Graph :

Solid infinite current carrying cylinder :

Assume current is uniformly distributed on the whole cross-section area

current density 

Case (I) : r < R

Case (II) : r > R

Magnetic force on moving charge :

When a charge q moves with velocity -in a magnetic field  then the magnetic force experienced by moving charge is given by following formula :

 Put q with sign

 Instantaneous velocity

Magnetic field at that point.

Note :

 and also 

 power due to magnetic force on a charged particle is zero. (use the formula of power  for its proof).

Since the  so work done by magnetic force is zero in every part of the motion. The magnetic force cannot increase or decrease the speed (or kinetic energy) of a charged particle. Its can only change the direction of velocity.

If  then also magnetic force on charged particle is zero. It moves along a straight line if only magnetic field is acting.

Motion of charged particles under the effect of magnetic force

Particle released if v = 0 then F_M = 0

 particle will remain at rest

 particle will move in a straight line with constant velocity

Initial velocity  and = uniform

 = constant

Now 

The particle moves in a curved path whose radius of curvature is same everywhere, such curve in a plane is only a circle.

 path of the particle is circular.

here p = linear momentum;

k = kinetic energy

Now 

Time period T = 

frequency f = 

Helical path :

If the velocity of the charge is not perpendicular to the magnetic field, the resultant path is a helix.

Radius 

and pitch = 

Charged Particle in 

When a charged particle moves with velocity -in an electric field  and magnetic field  then. The net force experienced by it is given by the following equation.

Combined force is known as Lorentz force.

Case-I:  

In the above situation, particle passes without deviating but its velocity will change due to the electric field. Magnetic force on it = 0.

Case–II:  and 

and charge q is released at the origin.

then its path will be a cyclotron

its velocity in y-direction varies as 

y coordinate at any time t is 

and x coordinate can be given as 

Magnetic force on a current carrying wire :

Suppose a conducting wire, carrying a current i, is placed in a magnetic field.

Magnetic force acting on the wire

Here  vector length of the wire = vector connecting the endpoints of the wire.

Note :

If a current loop of any shape is placed in a uniform  then  on it 

The magnetic moment of a current carrying coil :

M = NiA

N is the number of turns

i is the current in the coil

A is the area of the coil.

Torque on a current loop :

When a current-carrying coil is placed in a uniform magnetic field the net force on it is always zero.

Torque acting on a current carrying coil is

In vector form

where  is the magnetic moment of current carrying coil.

 is the magnetic field.

 is the angle between 

The magnetic force between two parallel current carrying straight wires

Where F is the force on per unit length of each wire.

If i₁ and i₂ are in same direction then fore is attracting and if in opposite direction then force is repulsive.

Terrestrial Magnetism (Earth’s Magnetism) :

(a) Variation or Declination : At a given place the angle between the geographical meridian and the magnetic meridian is called declination, i.e.

(b) Inclination or Angle of Dip: It is the angle which the direction of the resultant intensity of earth’s magnetic field subtends with the horizontal line in magnetic meridian at the given place.

(c) Horizontal Component of Earth’s Magnetic Field B_H : At a given place it is defined as the component of earth’s magnetic field along the horizontal in the magnetic meridian. It is represented by B_H and is measured with the help of a vibration or deflection magnetometer.

If at a place the magnetic field of the earth is B_i and angle of dip, then in accordance with the figure(a).

 and  so that 

and 

Magnetic properties of matter :

Magnetic intensity (H): it is a quantity related to currents in coils and conductors.

• it is a vector quantity
• its dimension is L^–1 A
• its SI unit is Am^–1

Magnetisation (M) : It is equal to the magnetic moment per unit volume.

• it is a vector quantity
• its dimension is L^–1 A
• its SI unit is Am^–1 

Magnetic susceptibility (x) : It is a measure of how a magnetic material responds to an external field.

M = x H

• It is dimensionless quantity

Also, 

Where,  is called relative permeability and it is a dimensionless quality.

Also, 

Where,  is absolute permeability of free space.

Diamagnetism : The individual atoms (or ions or molecules) of a diamagnetic material do not possess a permanent dipole moment of their own. (some diamagnetic materials are Bi, Cu, Pb, Si, nitrogen (at STP), H₂O, NaCl)

Paramagnetism : The individual atoms (or ions or molecules) of a diamagnetic material posses a permanent dipole moment of their own. (some paramagnetic materials are Al, Na, Ca, oxygen (at STP), CuCl₂).

Ferromagnetism : The individual atoms (or ions or molecules) of a diamagnetic material posses a dipole moment of their own. (some ferromagnetic materials are Fe, Co, Ni, Ga)

In terms of susceptibility x, a material is diamagnetic if x is negative. Paramagnetic if x is positive and small and ferromagnetic if x is large and positive.

Hysteresis : The curve between B and H in ferromagnetic materials is complex. It is often not linear and depends on the magnetic history of the sample.

This phenomenon is called hysteresis.

Magnetic hysteresis loops between the intensity of magnetization (I) and H for hard ferromagnetic materials and soft ferromagnetic materials are shown below :

Area of the hysteresis loop is proportional to the thermal energy developed per unit of the volume of the material.

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