Magnetic Effect of Current Part -1

Magnetic Effect of Current

Magnetic Force:

The magnetic force between two moving charges may be described as the effect exerted upon either charge by a magnetic field created by the other.

Magnetic force is a consequence of electromagnetic force and is caused due to the motion of charges. We have learned that a moving charge surrounds itself with a magnetic field.

Magnetic Force on a Current-Carrying Conductor

Let us discuss the force due to the magnetic field in a straight current-carrying rod.

A rod of uniform length l and cross-sectional area A. let the number density of mobile electrons be given by n.

Then the total number of charge carriers can be given by nAI,

where I is the steady current in the rod.

The drift velocity of each mobile carrier is assumed to be given as vd.

When the conducting rod is placed in an external magnetic field of magnitude B, the force applied on the mobile charges or the electrons can be given as:

F= (nAI) qvd×B

Where q is the value of charge on the mobile carrier.

As nqvd is also the current density j and A×|nqvd| is the current I through the conductor, then we can write:

F=[(nqevd)AI]×B=[jAI]×B=Il×B

Where I is the vector of magnitude equal to the length of the conducting rod.

Magnetic Field

Definition -

The region around a magnetic material or a moving electric charge within which the force of magnetism acts.

A pictorial representation of the magnetic field which describes how a magnetic force is distributed within and around a magnetic material

A magnetic field is a vector field in the neighborhood of a magnet, electric current or changing electric field in which magnetic forces are observable.

A magnetic field is produced by moving electric charges and intrinsic magnetic moments of elementary particles associated with a fundamental quantum property known as spin.

 Magnetic field and electric field are both interrelated and are components of the electromagnetic force, one of the four fundamental forces of nature.

Symbol      B or H

Unit  -Tesla

Base Unit- (Newton. Second)/Coulomb

History of Magnetic Field

The research on the magnetic field began in 1269 when French scholar Petrus Peregrinus de Maricourt mapped out the magnetic field on the surface of a spherical magnet using iron needles. He noticed that the resulting field lines crossed at two points. He named these points “poles.” After this observation, he stated that magnets always have North and South poles irrespective of how finely one slices them.

Three centuries later, William Gilbert stated that Earth is a magnet

In 1750 John Mitchell, an English clergyman and philosopher, stated that magnetic poles attract and repel each other.

In 1785, Charles-Augustin de Coulomb experimentally verified Earth’s magnetic field. In the 19th century, French mathematician and geometer Simeon Denis Poisson created the first model of the magnetic field, which he presented in 1824.

By the 19th century, further revelations refined and challenged previously-held notions.

In 1819, Danish physicist and chemist Hans Christian Oersted discovered that an electric current creates a magnetic field around it.

In 1825, André-Marie Ampère proposed a model of magnetism where this force was due to perpetually flowing loops of current, instead of the dipoles of magnetic charge.

In 1831, English scientist Faraday showed that a changing magnetic field generates an electric field. In effect, he discovered electromagnetic induction.

 

Between 1861 and 1865, James Clerk Maxwell published theories on electricity and magnetism. It was known as Maxwell’s equation. These equations describe the relationship between electricity and magnetism.

 

Motion of a Charged Particle in Magnetic Field

The interaction of the electric field and magnetic field and the motion of charged particles in the presence of both the electric and magnetic fields and also have derived the relation of the force acting on the charged particle,

A force acting on a particle is said to perform work when there is a component of the force in the direction of motion of the particles

We have a charged particle carrying a charge q moving in a uniform magnetic field of magnitude B, the magnetic force acts perpendicular to the velocity of the particle.

We say that no work is done by the magnetic force on the particle, and hence, no change in the velocity of the particle can be seen. Mathematically, when the velocity of the particle v is perpendicular to the direction of the magnetic field, we can write,

The magnetic force is directed towards the center of circular motion undergone by the object and acts as a centripetal force. 

Thus, if v and B are perpendicular to each other, the particle describes a circle.

 When a component of velocity is present along the direction of the magnetic field B, then its magnitude remains unchanged throughout the motion, as no effect of a magnetic field is felt upon it, the motion due to the perpendicular component of the velocity is circular in nature.

As the radius of the circular path of the particle is r, the centripetal force acting perpendicular to it towards the center can be given as,

F=mv2r

Also, the magnetic force acts perpendicular to both the velocity and the magnetic field and the magnitude can be given as,

F=qvB

Equating the two, we get

F=mv2r =qvB

Or r=mvqB

Here, r gives the radius of the circle described by the particle. Also, if we write the angular frequency of the particle as ω, then we can write,

v=ωr

So, ω=2πv=qBm

Here, v is the frequency of rotation of the particle. The time for one revolution can be given as,

T=2πω=1v

The distance moved by the particle along the direction of the magnetic field in one rotation is given by its pitch. p=vpT=2πmvpqB

Where vp is the velocity parallel to the magnetic field.

Motion in Combined Electric and Magnetic Fields:

An electric field is created by the electrical charge particle and is perpendicular to the magnetic field.

 The unit of the electric field is Volt/meter and is a vector quantity.

The electric field is conservative in nature.

The magnetic field is the space around the magnet, where the magnetic force is observed. This field is created by the movement of the electric charges.

Magnetic lines represent the direction of the magnetic field. The electric fields are generated around the particles, which are characterised by electric charges.

 The force created by the electric field is much stronger than the force created by the magnetic field. The orbiting motion of charges in a magnetic field is the basis for measuring the mass of an atom. A closed loop is formed by the magnetic field lines, and the electric field lines do not form a loop.

 

Lorentz Force

When a moving charge comes under the influence of magnetic and electric fields, it experiences Lorentz force. Lorentz force is a kind of force that can be calculated as the vector sum of forces created by magnetic and electric fields.

Fnet = FE + FB

Fnet = q(E + v x B)

The below figure shows the representation of the electric field and the magnetic field along with the motion of charge when they are perpendicular to each other.

When the values of E and B are adjusted such that the magnitude of the two forces are equal, the total force acting on the charge is zero, and the charges will move in the field undeflected.

When the strength of electric and magnetic fields are varied to get the forces due to electric and magnetic fields to be equal (FE = FB), then the charge can move in the field without any deflection.

Fnet = 0

FE = FB

qE=Bqv

E=Bv

v=E/B

This case is used when charged particles of a certain velocity (E/B) are used to pass through the crossed fields undeflected. This phenomenon is called a velocity selector. It was applied by J. Thomson to evaluate the charge to the mass ratio in 1897.

Velocity selector:

Velocity selector is also known as Wien filter is a device with a perpendicular arrangement of electric and magnetic fields which is used as a velocity filter.

 

Velocity selector exploits the principle of motion of a charge in a uniform magnetic field. According to this principle, the force experienced by a moving charge with speed v in a uniform magnetic field is given as:

F = Bqv

Where,

F: force experienced by the charge

B: magnetic field

q: moving charge

v: speed of the charge

      The speed of charged particles in velocity selector is

v=EB

Fields of the velocity selector

Uniform electric field: This field is generated by the top plate with negative charges and the bottom plate with the positive charges. These charges result in the formation of the field which is pointing in the upward direction in the figure.

Uniform magnetic field: This field is present uniformly between the two charged plates such that it can be directed either inwards or outwards.

Limitations of the velocity selector

Neither the mass nor the charge of the particles is considered before passing through the filter.

All the uncharged particles pass through the filter.

It is used in mass spectrometers, where charged objects are distinguished as per their charge to mass ratio.

Cyclotron

A cyclotron is a machine used to accelerate charged particles or ions to high energies.

To enhance the energies of charged particles, cyclotron uses magnetic as well as electric fields. It is called crossed fields since the magnetic and electric fields are perpendicular to each other.

 A cyclotron consists of two flat and hollow semicircular metal disc-like containers represented as D1 and D2. These containers are separated by a narrow gap. Disc-like containers connected to a high-frequency oscillator have the capacity to generate an alternating voltage.

Alternating electric field is created between D1 and D2 as well from D2 to D1. The charged particle in the containers is accelerated when it passes through the gap between the metal containers.

When a charged particle moves into a hollow semicircular metal disc-like container, it moves in a circular path and with a constant speed. Due to the shielding effect, the electric field is zero inside the container.

 The frequency of the cyclotron is independent of the speed of the charged particle and the radius of the circular path. These machines are used to bombard nuclei and are used to study nuclear reactions.

Biot-Savart Law

The Biot-Savart law is an equation that gives the magnetic field produced by a current-carrying segment. This segment is  a vector quantity called the current element.

What is the Formula of Biot-Savart’s Law?

What is the Formula of Biot-Savart’s Law?

Consider a wire carrying a current 'i' in a certain direction as shown in the figure above. Take a small thread element of length ds. The direction of this element is along the direction of flow, so it forms a vector ID.

The Biot-Savart law can be applied to know the point magnetic field caused by this small element. Let r be the position vector drawn by the current element from that point, and let θ be the angle between them. Then

 

Where μ0 is the permittivity of free space and is equal to

4π × 10-7 TmA-1.

The direction of the magnetic field is always in a plane perpendicular to the line and position vector of the element. This is given by the rule of the right hand, where the thumb points in the direction of the normal current and the other fingers in the direction of the magnetic field.

Applications of the law of Biot-Savart

The Biot-Savart law to calculate magnetic response even at the atomic or molecular level.

It is also used in aerodynamic theory to calculate the speed induced by vortex lines.

Importance of Biot-Savart's Law:

The law of Biot-Savart is similar to the law of Coulomb of electrostatics.

 The law also applies to very small wires that carry current. The law applies to symmetrical current distribution.

 

Magnetic Field Intensity Due to a Straight Current-Carrying Conductor of Finite Length

 

As we know, current-carrying conductors experience magnetic fields. Each mass creates a gravitational field and can also interact with that field. A charge creates an electric field and also interacts with it. Since a moving charge interacts with a magnetic field, we can expect it to create this field as well.

Consider a straight wire AB carrying a current (I) and determine the magnetic field strength at P.

Look at the picture above. According to the Biot-Savart law, the magnetic field at point P is given by 

 

Let AB be a conductor carrying a current I. Consider a point P at a certain distance from the center of the conductor. Consider a small current carrying element "dl" placed at point c. It is at a distance 'r' from point p. "l" is the distance between the center of the coil and dl is the length. Using the Biot-Savart law, the magnetic field of the current carrying element "dl" at P,

 Magnetic field due to a current carrying conductor

Current is usually defined as the charging rate. We already know that static charges produce an electric field that is proportional to the magnitude of charge. The same principle can be applied here. Moving charges create magnetic fields that are proportional to current, and thus a current-carrying conductor creates a magnetic effect around itself. This magnetic field is usually caused by subatomic particles in the conductor, such as moving electrons in atomic orbits.       

A magnetic field has both magnitude and direction. Therefore, it is a vector quantity labeled B (in the diagram below). The magnetic field produced by a current-carrying conductor depends on the current in the conductor and the distance from the point. The direction of the magnetic field is perpendicular to the wire. If you wrap the fingers of your right hand around the wire with the thumb pointing in the direction of the current, the direction of the curl of the fingers gives the direction of the magnetic field. This becomes clearer in the diagram below, where the red lines represent the magnetic field lines.

Magnetic Field Intensity Due To A Circular Loop Of A Current-Carrying Conductor

The magnetic field strength of a circular loop is considered in its axial position. For this let "a" be the radius of the circular loop. The current through the loop is "i". The intensity of the magnetic field is determined at point "P". This point is at a distance x from the center of the circular loop. Those two elements are used to determine the intensity of the field at point "P". The figure below shows a circular circuit carrying.

 

 

The magnetic field intensity due to the current- carrying circular loop along the axial position is given as:

Since the axial position is considered, the intensity of the magnetic field will be along the axis.


Therefore, the above equation is the magnetic field intensity of a circular current-carrying loop.

If there are N numbers of turns in the loop, then the magnetic field is given as:

Magnetic field at the center of Circular Coil Carrying Current:

 

 

Consider a circular coil of radius a and carrying current I in the direction shown in Figure. Suppose the loop lies in the plane of paper. It is desired to find the magnetic field at the centre O of the coil. Suppose the entire circular coil is divided into a large number of current elements, each of length dl. According to Biot-Savart law, the magnetic field  

dB  at the centre O of the coil due to current element I

dl is given by,

 

where  

r is the position vector of point O from the current element. The magnitude of dB at the centre O is

 

The direction of dB is perpendicular to the plane of the coil and is directed inwards. Since each current element contributes to the magnetic field in the same direction, the total magnetic field B at the center O can be found by integrating the above equation around the loop i.e.

 

For each current element, angle between   dl  and   r  is 90°. Also distance of each current element from the center O is a.

Magnetic Field on the Axis of a Circular Current Loop – Derivation:

Before we learn more about the magnetic field on the axis of a circular circuit, we need to understand the basic law of magnetism. We know that there is an equation that describes the magnetic field produced by a constant current. The law of Biot-Savart correlates the strength of the  field with the length, proximity and direction of the electric current. The Biot-Savart law is an equation that gives the  field produced by a current-carrying segment. A flux-carrying element is considered a vector quantity. The Biot-Savart law is obtained by the equation:

Where

μ0 is the permeability of free space and is equal to 4π × 10-7 TmA-1.

Now we will learn how to control the magnetic field along the axis of a circular circuit. Estimate the magnetic field due to the circular coil on its axis. The evaluation requires the sum of the "idl" effects of the smallest flow elements. "i" is direct current and this  evaluation in  vacuum or free space.

 See the image below. From the picture, we can see that the "i" represents a circular loop that carries a constant current. The image shows that the circular loop is placed in the y-z plane with its center at the origin O and its radius "R". The X axis is the loop axis.

x = distance P from center O of loop.

dl= a conducting element of the loop.

The Biot-Savart law gives the magnitude of the magnetic field (dB) due to current element (dl)

 

Any element of the loop will be perpendicular to the displacement vector from the element to the axial point.

For example, the element dl present in the figure is in the y-z plane. Whereas, the displacement vector r from dl to the axial point P is in the x-y plane.

The direction of dB is as shown in the figure and is perpendicular to the plane formed by dl and r. It has an x-component dBx and a component perpendicular to the x-axis, dB

 

A zero result is obtained when the components perpendicular to the x-axis are added and canceled.

The dB component due to dl is canceled by  the diametrically opposite dl element. This is shown in the image above.

So only the x component is preserved. The net effect in the x-direction can be obtained by integrating dBx = dBcosθ over the loop.

 

 

Since the circular wire forms a closed loop, the magnetic field lines are produced as shown in the figure below. The direction of the magnetic field created is given by the right-hand thumb rule.