MOVING CHARGES AND MAGNETISM || MOVING CHARGES AND MAGNETISM Class 12 Notes || Class 12 Notes || Notes ||

                     1. INTRODUCTION

This topic deals with the moving charges and the magnetic effect. Both Electricity and Magnetism have been known for more than 2000 years. However, it was only about 200 years ago, in 1820, that it was realised that they were intimately related. During a lecture demonstration in the summer of 1820, Danish physicist Hans Christian Oersted noticed that a current in a straight wire caused a 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 its centre 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. Iron filings sprinkled around the wire arrange themselves in concentric circles with the wire as the centre. 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 realised 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 19th century. A remarkable scientific and technological progress took place in the 20th century. This was due to our increased understanding of electromagnetism and the invention of devices for production, amplification, transmission and detection of electromagnetic waves. 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. A current or a field (electric or magnetic) emerging out of the plane of the paper is depicted by a dot (¤). A current or a field going into the plane of the paper is depicted by a cross (⊗ ). 

 

           2. MAGNETIC FORCE 

1. Sources and fields

Before we introduce the concept of a magnetic field B, we shall recapitulate what we have learnt in the electric field E. We have seen that the interaction between two charges can be considered in two stages. The charge Q, the source of the field, produces an electric field E, where

                     E = Q r / (4πε0 )r²

where r ˆ is unit vector along r, and the field E is a vector field. A charge q interacts with this field and experiences a force F given by 

           F = q E = q Q r ˆ / (4πε0 ) r²

The field E is not  just an artefact but has a physical role. It can convey energy and momentum and is not established instantaneously but takes finite time to propagate. The concept of a field was specially stressed by Faraday and was incorporated by Maxwell in his unification of electricity and magnetism. In addition to depending on each point in space, it can also vary with time, i.e., be a function of time. In our discussions in this chapter, we will assume that the fields do not change with time. The field at a particular point can be due to one or more charges. If there are more charges the fields add vector i ally. You have already learnt in Chapter 1 that this is called the principle of superposition. Once the field is known, the force on a test charge is given by the above equation . Half marathon distance = 0.5 marathon = "21.0982 kilometres (km). Just as static charges produce an electric field, the currents or moving charges produce (in addition) a magnetic field, denoted by B (r), again a vector field. It has several basic properties identical to the electric field. 

It is defined at each point in space (and can in addition depend on time). Experimentally, it is found to obey the principle of superposition: the magnetic field of several sources is the vector addition of magnetic field of each individual source. 

2. Magnetic Field, Lorentz Force

Let us suppose that there is a point charge q (moving with a velocity v and, located at r at a given time t) in presence of both the electric field E (r) and the magnetic field B (r). The force on an electric charge q due to both of them can be written as 

F = q [ E (r) + v × B (r)] ≡ F electric +F magnetic 

This force was given first by H.A. Lorentz based on the extensive experiments of Ampere and others. It is called the Lorentz force. You have already studied in detail the force due to the electric field. If we look at the interaction with the magnetic field,we find the following features:

(i) It depends on q, v and B (charge of the particle, the velocity and the magnetic field). Force on a negative charge is opposite to that on a positive charge. 

(i i) The magnetic force q [ v × B ] includes a vector product of velocity and magnetic field. The vector product makes the force due to magnetic field vanish (become zero) if velocity and magnetic field are parallel or anti-parallel. The force acts in a (sideways) direction perpendicular to both the velocity and the magnetic field. Its direction is given by the screw rule or right hand rule for vector (or cross) product. 

(i i i) The magnetic force is zero if charge is not moving (as then |v|= 0). Only a moving charge feels the magnetic force. The expression for the magnetic force helps us to define the unit of the magnetic field, if one takes q, F and v, all to be unity in the force equation F = q [ v × B] =q v B sin θ n ˆ , where θ is the angle between v and B . The magnitude of magnetic field B is 1 SI unit, when the force acting on a unit charge (1 C), moving perpendicular to B with a speed 1m/s, is one newton. Dimensionally, we have [B] = [F/q v] and the unit of B are Newton second / (coulomb metre). This unit is called tesla (T) named after Nikola Tesla (1856 – 1943). Tesla is a rather large unit. A smaller unit (non-SI) called gauss (=10–4 tesla) is also often used. The earth’s magnetic field is about 3.6 × 10–5 T. Table 4.1 lists magnetic fields over a wide range in the universe. 

3. Magnetic force on a current-carrying conductor

We can extend the analysis for force due to magnetic field on a single moving charge to a straight rod carrying current. Consider a rod of a uniform cross-sectional area A and length l. We shall assume one kind of mobile carriers as in a conductor (here electrons). Let the number density of these mobile charge carriers in it be n. Then the total number of mobile charge carriers in it is nl A. For a steady current I in this conducting rod, we may assume that each mobile carrier has an average drift velocity v d . In the presence of an external magnetic field B, the force on these carriers is: 

                     F = (nl A)q v d × B 

where q is the value of the charge on a carrier. Now n q v d is the current density j and (n q v d )A is the current I (see Chapter 3 for the discussion of current and current density). Thus, 

F = [(n q v d )lA] × B = [ j Al ] × B

   = I l × B 

where l is a vector of magnitude l, the length of the rod, and with a direction identical to the current I. Note that the current I is not a vector. In the last step we have transferred the vector sign from j to l. In this equation, B is the external magnetic field. It is not the field produced by the current-carrying rod. If the wire has an arbitrary shape we can calculate the Lorentz force on it by considering it as a collection of linear strips d l. 

   3. MOTION IN A MAGNETIC FIELD

We will now consider, in greater detail, the motion of a charge moving in a magnetic field. We have learnt in Mechanics that a force on a particle does work if the force has a component along (or opposed to) the direction of motion of the particle. In the case of motion of a charge in a magnetic field, the magnetic force is perpendicular to the velocity of the particle. So no work is done and no change in the magnitude of the velocity is produced (though the direction of momentum may be changed). [Notice that this is unlike the force due to an electric field, q E, which can have a component parallel (or antiparallel) to motion and thus can transfer energy in addition to momentum.] We shall consider motion of a charged particle in a uniform magnetic field. First consider the case of v perpendicular to B. The perpendicular force, q v × B, acts as a centripetal force and produces a circular motion perpendicular to the magnetic field. The particle will describe a circle if v and B are perpendicular to each other. If velocity has a component along B, this component remains unchanged as the motion along the magnetic field will not be affected by the magnetic field. The motion in a plane perpendicular to B is as before a circular one, thereby producing a helical motion. If r is the radius of the circular path of a particle, then a force of m v² / r, acts perpendicular to the path towards the centre of the circle, and is called the centripetal force. If the velocity v is perpendicular to the magnetic field B, the magnetic force is perpendicular to both v and B and acts like a centripetal force. It has a magnitude q v B. Equating the two expressions for centripetal force, m v 2/r = q v B, which gives 

                    r = m v / qB 

for the radius of the circle described by the charged particle. The larger the momentum, the larger is the radius and bigger the circle described. If ω is the angular frequency, then v = ω r. So, 

                    ω = 2π ν = q B/ m

which is independent of the velocity or energy . Here ν is the frequency of rotation. The independence of ν from energy has important application in the design of a cyclotron. The time taken for one revolution is T= 2π/ω ≡ 1/ν. If there is a component of the velocity parallel to the magnetic field (denoted by v|| ), it will make the particle move along the field and the path of the particle would be a helical one. The distance moved along the magnetic field in one rotation is called pitch p. Using, we have 

             p = v||T = 2πm v|| / q B 

The radius of the circular component of motion is called the radius of the helix.

      4. AMPERE’S CIRCUITAL LAW

There is an alternative and appealing way in which the Bi ot-S a v art law may be expressed. Ampere’s circuital law considers an open surface with a boundary. The surface has current passing through it. We consider the boundary to be made up of a number of small line elements. Consider one such element of length d l. We take the value of the tangential component of the magnetic field, B t , At this element and multiply it by the length of that element d l [Note: B t d l=B.dl]. All such products are added together. We consider the limit as the lengths of elements get smaller and their number gets larger. The sum then tends to an integral. Ampere’s law states that this integral is equal to µ 0 times the total current passing through the surface, i.e., “B..d I d l l = µ 0 where I is the total current through the surface. The integral is taken over the closed loop coinciding with the boundary C of the surface. The relation above involves a sign-convention, given by the right-hand rule. Let the fingers of the right-hand be curled in the sense the boundary is traversed in the loop integral “B.dl. Then the direction of the thumb gives the sense in which the current I is regarded as positive. For several applications. proves sufficient. We shall assume that, in such cases, it is possible to choose the loop (called an amper ian loop) such that at each point of the loop, either

(i) B is tangential to the loop and is a non-zero constant B, or 

(i i) B is normal to the loop, or

(i i i) B vanishes.

Now, let L be the length (part) of the loop for which B is tangential. Let I e be the current enclosed by the loop.