3.9 Force Acting on a current carrying conductor

We have seen that a magnetic field is associated with a current carrying conductor. If a current carrying conductor is placed in a magnetic field, it experiences a force (F) depending on the magnitude of current (I), length of conductor (l) and magnitude of the magnetic field (B). The mathematical expression for the force in vector form is given by F=I(lxB)

The above equation can be derived as follows: Consider a metal of length l, cross sectional area 'A' and free electron density 'n' carrying a current I. In metals the current conduction is due to the drifting of electrons. When the conductor is placed in a magnetic field of strength B, each of the moving electron in the metal experiences a Lorentz force given by F=eVdxB where Vd is the drift speed of electrons.

The drift speed of electrons is Vd=1/nAe

Therefore the force experienced by each electron is f=eVdBsin theta=e(1/nAe)B sin theta

f=IBsin theta/nA

As the conductor is of length l the number of free electron present is N=Aln

Therefore the total force acting on a conductor of length l is

F=Nf=Aln(IB sin theta)/nA=IlB sin theta

In vector form it can be written as F=I(lxB), here the direction of l is taken as the direction of I.

Cyclotron

Rectangular Coil in a
Magnetic Field

3.5 Ampere's Circuital law

Ampere in his experiments on magnetic field due to electric current found that the sum of dot product of magnetic field and path length for any continuous closed path is equal to the product of permeability and current inside the path. He stated the above observation as a general law known as Ampere's circuital law.

According to ampere's law the line integral of magnetic field along any closed path is equal to muzero times the current enclosed by the path. Amperes circuital law is not different from Biot-Savart law. Both relates the magnetic field the current and both express the same physical consequence of electric current.

Magnetic Field due
to a circular coil

Applications of Ampere's
 Circuital Law

                                                                                                                   








      

3.4 Magnetic field due to a circular coil

Consider a circular coil of radius a carrying a steady current I. Let it be placed with its center at origin and plane perpendicular to the X axis.

Now imagine a small element AB of length dl. The magnetic field at a point P on the X axis at a distance x from the origin is given by

where r is the distance between the element and the point P.

If vector r makes an angle alpha with the X axis, then dB will be inclined to the +Y axis. Hence dB can be resolved into X and Y components given by

Now if we consider an identical element CD diametrically opposite to AB, the magnetic field due to CD will be  in a direction inclined to the Y axis.

The above expressions makes it clear that the Y component of magnetic field due to diametrically opposite identical elements cancels each other. Hence the net magnetic field at P is simply the vector sum of x components of all the elements of the ring.

Thus the magnetic field due to a coil at a point on the x axis depends on the current I, the radius of the coil and the distance of the point from the center.

If the point P selected at the center of the coil then x=0

The direction of magnetic field due to a circular coil is obtained by the right hand thumb rule which is stated as follows:

Curl the palm of right hand around the circular coil with the fingers pointing in the direction of current. Then the extended thumb gives the direction of magnetic field.

Thus an anti-clock wise current gives a magnetic field out of the coil and a clock wise current gives a magnetic field into the coil.

Current loop as a magnetic dipole

In the preceeding section we have seen that anti clock wise current gives outgoing flux, indicating south polarity. In short a current loop gives north polarity on one side  and south polarity on the other side resulting in a magnetic dipole.

The magnetic dipole moment of a current loop is defined as the product of electric current and area.Its SI unit is Am2.

Now magnetic field at a point on the axis of a circular coil given by the equation 10 can be rewritten as 

The above equation is similar to that of the electric field on the axial line of an electric dipole. This we can conclude that a current loop is equivalent to a magnetic dipole of moment equal to m=IA. This can be generalized to any geometric shape of the current loop.

Magnetic Field due to a long
straight current carrying conductor

Ampere's Circuital Law

3.3 Magnetic field due to a long straight current carrying conductor

Consider a long conductor carrying a steady current I. Let us now find the magnetic field on a point P at a distance 'a' away from the conductor. The long conductor can be imagined to be composed of very large number of small current elements. Now the magnetic field at the point is the vector sum of magnetic field due to all such current elements.

Let the perpendicular from P meet the conductor at O. Consider a small current element of length dl at a distance l below O. Let the current element make at an angle theta with the vector connecting dl and P.

Now the magnetic field at P due to the element is given by Biot-Savart law as

dB = (muzeroxIdl sin theta)/4xpiexrxr into the plane of the paper.
Now from the diagram we get

theta equal to 90-theta.
l equal to a tan theta
cos phi equal to a/r

using the above equations the variables in the Biot-Savarts law can be rewritten as

sin theta equal to sin(90- phi)=cos phi

The net magnetic field at P can be now obtained by integrating (4) between proper limits. The expression reveals that magnetic field due to a long straight conductor depends on current and the distance of the point from the conductor.

Now the direction of magnetic field can be obtained by a simple law known as right hand grip rule which is stated as follows:

Grasp the conductor in right hand with the extended thumb pointing in the direction of the current.Then the palm fingures around in the direction of the magnetic field.

Biot-Savart's Law

Magnetic Field due to 
a circular coil

3.2 Biot-Savart's Law

Soon after the qualitative study of magnetic effects of electric current, two French physicists Jean Baptiste Biot and Felix Savart performed qualitative study on the magnetic effect of electric current. They experimentally found that the magnetic effect at a point due to an electric current depends on the current strength, distance of the point from the current carrying conductor and geometry of the current carrying conductor. They stated their observations in the form of a law known as Biot-Savart's Law.

According to the Biot-Savart's law the magnitude of magnetic field dB at a point due to a current element ldl is directly proportional to

1.  the magnitude of current (I)
2. the length of current element (dl)
3. sine of the angle between the current element and the vector connecting the current element and the point  and
4. Inversely proportional to the square of the distance between them.

dB directly proportional to (Idl sin theta)/ r*r

or

The direction of magnetic field is perpendicular to the plane containing dl and r and is given by the right hand screw rule.

In the above expression muzero/4pie is the constant of proportionality and muzero is called the permeability of vaccum. Its value is 4piex 10 raised to -7 TmA raised to -1.

To represent a magnetic field the following convention is followed.

A magnetic field acting perpendicularly in to the plane of the paper is represented by the symbol o and a magnetic field acting perpendiularly out of the same of the paper is represented by the symbol dot.

The magnetic field at any point on the right of the conductor is into the plane of the paper and that on the left is out of the paper.

Biot-Savart law combined with the superposition principle is used to determine the magnetic field due to any current carrying conductor. In the following sections we shall obtain expressions for magnetic field due to a straight long conductor and also due to a circular coil carrying current using Biot-Savart law.

Magnetic Field due to a long
straight current carrying conductor

Introduction

3.1 Introduction

The electrostatic and magnetic phenomena were observed even in BC. As early as 600 BC, Thales a Greek philosopher observed the attracting properties of amber. The attracting properties of certain stones in magnesia were observed in 800 BC.

Both electricity and magnetism has been known for more than 2000 years. However it was only about 200 years ago,in 1820, that it was realized that they were intimately related.During a lecture demonstration in the summer of 1820, the Danish Physicist Hans Christian Oersted (Hans Christian Oersted (1777-1851) was a Danish physicist and chemist, professor at Copenhagen. He observed that a compass needle suffers a deflection when placed near a wire carrying an electric current. This discovery gave the first empirical evidence of a connection between electric and magnetic phenomena ) noticed that a current in a straight wire caused noticeable deflection in a nearby magnetic compass needle. He investigated this phenomenon. He found that the alignment of the needle is tangential to an imaginary circle which has the straight wire as the center and has its plane perpendicular to the wire. It is noticeable when the current is large and the needle sufficiently close to the wire so that the earth's magnetic field may be ignored. Reversing the direction of the current reverses the orientation of the needle. The deflection increases on increasing the current or bringing the needle closer to the wire.

The science of magnetism blossomed with the publication of the famous book De Magnete in 1600 by William Gilbert. The science of electricity started to grow from the time of Benjamin Franklin. For a long time it was believed that electricity and magnetism are independent of each other

Iron filings sprinkled around the wire arranged themselves in concentric circles with the wire as the center. Oersted concluded that moving charges or currents produced a magnetic field in the surrounding space. Following this there was intense experimentation. In 1864, the laws obeyed by electricity and magnetism were unified and formulated by James Maxwell who then realized that light was electromagnetic waves. Radio waves were discovered by Hertz, and produced by J.C. Bose and G. Marconi by the end of the 19 th   century. A remarkable scientific and technological progress has taken place in the 20 th century. This is due to our increased understanding of electromagnetism and the invention of devices for production, amplification, transmission and detection of electromagnetic waves.

In 1820 in a lecture demonstration Hans Christian Oersted noticed that current in a straight wire caused a noticeable deflection in a nearby compass needle. He also found that on reversing the direction of current the direction of deflection is also reversed. On further experimental studies he found that deflection increases on increasing the current or in bringing the magnetic needle closer to the wire.Finally Oersted concluded that a moving charge or current produces a magnetic field in surrounding space.

In this chapter we will see how magnetic field exerts forces on moving charged particles, like electrons, protons and current carrying wires. We shall also learn how currents produce magnetic fields. We shall see how particles can be accelerated to very high energies in a cyclotron. We shall study how currents and voltages are detected by a galvanometer.

Oersted's discovery revolutionized the science of magnetism and electricity. By 1820s the scientific world realized the fact that electricity and magnetism are interrelated phenomena and the cause of both effect are basically electric charges and their motion.

In the subsequent chapters on magnetism, we adopt the following convention: A current or a field (electric or magnetic ) emerging out of the plane of the paper is depicted as a dot. A current or a field going into the plane of the paper is depicted by a cross.

Cyclotron

Biot-Savart's Law