Magnetic Field

Magnetic Field

Published by: BhumiRaj Timalsina

Published date: 22 Jun 2021

Magnetic Field Photo

Magnetic Field

Magnet:

A magnet is defined as an object which is capable of producing a magnetic field and attracting unlike poles and rep

Comments 11 Comment in moderation

elling like poles.

Properties of Magnet

Following are the basic properties of magnet:

  • When a magnet is dipped in iron filings, we can observe that the iron filings cling to the end of the magnet as the attraction is maximum at the ends of the magnet. These ends are known as poles of the magnets.
  • Magnetic poles always exist in pairs.
  • Whenever a magnet is suspended freely in mid-air, it always points towards north-south direction. Pole pointing towards geographic north is known as North pole and the pole pointing towards geographic south is known as South pole.
  • Like poles repel while unlike poles attract.
  • The magnetic force between the two magnets is greater when the distance between these magnets are lesser.

Types of Magnets

There are three types of magnets and they are as follows:

  • Permanent magnet
  • Temporary magnet
  • Electromagnets

Magnetic Field

The space in the surrounding of a magnet or any current-carrying conductor in which its magnetic influence can be experienced.

Magnetic Field Lines

Picture

  • They are used to res present the magnetic B field in a region.
  • They are closed continuous curves.
  • Tangent drawn at any point gives the direction of the magnetic field.
  • They cannot interact.
  • Outside a magnet, they are directed from north to south pole and inside a magnet they are directed from south to north.

The magnetic field lines due to a straight current-carrying conductor are concentric circles having a center at the conductor and in a plane perpendicular to the conductor.

Picture

The direction of magnetic field lines can be obtained by Right-Hand Thumb Rule

 Magnetic Flux

Magnetic Flux is defined as the number of magnetic field lines passing through a given closed surface. It gives the measurement of the total magnetic field that passes through a given surface area. Here, the area under consideration can be of any size and under any orientation with respect to the direction of the magnetic field.

  • Magnetic flux is commonly denoted using greek letter Phi or Phi suffix B.
  • Magnetic flux symbol: Φ or ΦB.

Magnetic flux formula is given by:


Picture

ΦB is the magnetic flux.Where,

  • B is the magnetic field
  • A is the area
  • θ the angle at which the field lines pass through the given surface area\

Electrostatic Force and Magnetic Force on a Charged Particle

Recall that in a static, unchanging electric field the force on a particle with charge q will be:

F=qE

Where F is the force vector, q is the charge, and is the electric field vector. Note that the direction of F is identical to E in the case of a positivist charge q, and in the opposite direction in the case of a negatively charged particle. This electric field may be established by a larger charge, Q, acting on the smaller charge q over a distance r so that:

It should be emphasized that the electric force F acts parallel to the electric field E. The curl of the electric force is zero, i.e.:

Equation

A consequence of this is that the electric field may do work and a charge in a pure electric field will follow the tangent of an electric field line.

In contrast, recall that the magnetic force on a charged particle is orthogonal to the magnetic field such that:

F = qv×B = qvBsinθF

where B is the magnetic field vector, v is the velocity of the particle, and θ is the angle between the magnetic field and the particle velocity. The direction of F can be easily determined by the use of the right-hand rule.

Picture

Figure: Right-Hand Rule: Magnetic fields exert forces on moving charges. This force is one of the most basic knowns. The direction of the magnetic force on a moving charge is perpendicular to the plane formed by v and B and follows right-hand rule–1 (RHR-1) as shown. The magnitude of the force is proportional to q, v, B, and the sine of the angle between v and B.

If the particle velocity happens to be aligned parallel to the magnetic field or is zero, the magnetic force will be zero. This differs from the case of an electric field, where the particle velocity has no bearing, on any given instant, on the magnitude or direction of the electric force.

The angle dependence of the magnetic field also causes charged particles to move perpendicular to the magnetic field lines in a circular or helical fashion, while a particle in an electric field will move in a straight line along an electric field line.

A further difference between magnetic and electric forces is that magnetic fields do not net-work, since the particle motion is circular and therefore ends up in the same place. We express this mathematically as:

W=∮B⋅dr=0W=∮B⋅dr=0

Lorentz Force

The Lorentz force is the combined force on a charged particle due to both electric and magnetic fields, which are often considered together for practical applications. If a particle of charge q moves with velocity v in the presence of an electric field E and a magnetic field B, then it will experience a force:

F=q[E+vBsinθ]

 

 

Force on A current-carrying Conductor

When a conductor carrying a current is placed in a magnetic field, the conductor experiences a magnetic force.

  • The direction of this force is always right angles to the plane containing both the conductor and the magnetic field and is predicted by Fleming’s Left-Hand Rule.

 

Picture

Referring to the diagram above, F is Force, B is a Magnetic field, ‘I’ is current.

Factors affecting magnetic force on a current-carrying conductor in a magnetic field:

  • Strength of the magnetic field
  • Current flowing through the wire
  • Length of the wire

 

Picture
F = BIlsinθF=BIlsinθ, where

  • F is the force acting on a current-carrying conductor, B is magnetic flux density (magnetic field strength),
  • ‘I’ is the magnitude of a current flowing through the conductor, 
  • l is the length of the conductor,
  • θ is the angle that the conductor makes with the magnetic field.

 

When the conductor is perpendicular to the magnetic field, the force will be maximum. When it is parallel to the magnetic field, the force will be zero.

Torque On Current Loop

Let us consider a rectangular loop such that it carries a current of magnitude I. If we place this loop in a magnetic field, it experiences a torque but no net force, quite similar to what an electric dipole experiences in a uniform electric field.

Picture

Figure: Torque on the current-carrying rectangular loop of wire on a pivot rod when placed in a magnetic field

Let us now consider the case when the magnetic field B is in the plane with the rectangular loop. No force is exerted by the field on the arms of the loop that is parallel to the magnets, but the arms perpendicular to the magnets experience a force given by F1,

1.2

This force is directed into the plane.

Similarly, we can write the expression for a force F2 which is exerted on the arm CD,

1.3

We see that the net force on the loop is zero and the torque on the loop is given by,

1.4

Where ab is the area of the rectangle. Here, the torque tends to rotate the loop in the anti-clockwise direction.

Let us consider the case when the plane of the loop is not along the magnetic field. Let the angle between the field and the normal to the coil be given by θ. We can see that the forces on the arms BC and DA will always act opposite to each other and will be equal in magnitude. Since these forces are the equal opposite and collinear at all points, they cancel out each other’s effect and this results in zero-force or torque. The forces on the arms AB and CD are given by F1 and F2. These forces are equal in magnitude and opposite in direction and can be given by,

1.5

These forces are not collinear and thus act as a couple exerting a torque on the coil. The magnitude of the torque can be given by,

1.6

 

Magnetic Dipole Moment():

The magnetic dipole moment of a coil or loop is the product of current(i) flowing through it and its cross-section area(A) ie

………………….(i)

Also, we have the torque on the coil in a magnetic field is,

τ=BiAsinӨ

τ=μBsinӨ………………..(ii)

τ=B * μ……………(iii)

This is the torque on the coil in vector form. If the coil is rotated or turned through small-angle dӨ, then the small work done is,

dW=τ*dӨ

Hence, the total work done due to the orientation of the coil through angle Ө is

Equaton

Setting this work equal to Ep,

Equaton

This work done is stored in the coil as potential energy and hence P.E. of the coil is,

Equaton

 

Hall effect

Hall effect is defined as the production of a voltage difference across an electrical conductor which is transverse to an electric current and with respect to an applied magnetic field it is perpendicular to the current. Edwin Hall discovered this effect in the year 1879.

Hall field is defined as the field developed across the conductor and Hall voltage is the corresponding potential difference. This principle is observed in the charges involved in the electromagnetic fields.

Hall Effect Derivation

Consider a metal with one type charge carriers that are electrons and is a steady-state condition with no movement of charges in the y-axis direction. Following is the derivation of Hall-effect:

Equaton (at equilibrium, force is downwards due to the magnetic field which is equal to upward electric force)

Where,

  • VH is Hall voltage
  • EH is Hall field
  • v is the drift velocity
  • d is the width of the metal slab
  • B is the magnetic field
  • Bev is a force acting on an electron

I=−nevA

Where,

  • ‘I’ is an electric current
  • n is no.of electrons per unit volume
  • A is the cross-sectional area of the conductor

Equaton

Where,

Equaton Hall coefficient (RH) is defined as the ratio between the induced electric field and the product of the applied magnetic field and current density. In semiconductors, RH is positive for the hole and negative for free electrons.

Equaton

Where,

  • E is an electric field
  • v is the drift velocity
  • RH is the Hall coefficient
  • 𝛍H is the mobility of the hole

Equaton

The ratio between density (x-axis direction) and current density (y-axis direction) is known as Hall angle that measures the average number of radians due to collisions of the particles.

Equaton

Where,

  • R is Hall resistance