INTRODUCTION

Concepts of both electricity and magnetism was known for almost 2000 years. These two phenomena were considered as independent phenomena. But in 1820  a Danish physicist noticed that a current-carrying wire caused a noticeable deflection in the magnetic compass needle placed near that current-carrying wire. So he concluded that moving charges or currents produces magnetic fields.

Later another scientist  Oersted did some experiments with a magnet and a coil and found that the motion of the magnet near the coil is producing current in the coil. Later Faraday made laws of electromagnetic Induction based on the Oersted experiment’s observation. He concluded that change in a magnetic field produces current.       

In 1864, the laws obeyed by electricity and magnetism were unified and formulated. James Maxwell then realized that light is an electromagnetic wave.  After the unification of electricity, magnetism and electromagnetic waves as a single unit called electrodynamics, remarkable scientific and technological progress took place in the 20th century. 

What will we learn in this chapter?

In this chapter, we will study magnetostatics. It is a phenomenon associated with steady currents. When a steady current flows through a wire, it produces a constant magnetic field around it. We shall see how particles can be accelerated to very high energies in a cyclotron.

We will see how currents and voltages are detected by a galvanometer.

Magnetic force

Let’s first recapitulate whatever we have learned so far. First, we studied about the static property of the charges in “Electrostatics”, we understood how charges exert force on each other, we discussed electric fields, electric flux, and electric potentials. Then in the later chapter “ current electricity’’ we studied the phenomena related to charges in motion. We have discussed currents, mobility, drift velocity, resistivity and many more.

  1.  Magnetic forces: Sources and fields

In this unit, we will study another property of moving charges.

Moving charges produce magnetic fields around, this property is called the magnetic effect of current. 

  • Also when a current carrying wire is placed in a magnetic field, it will experience some magnetic force on it.
  • Just like static charges produce an electric field, moving charges produce magnetic fields. Magnetic field is again a vector field.
  • The basic property of a magnetic field is very similar to an electrical field.
  • Magnetic fields are also found to obey the principle of superposition.
  • A charge particle moving in magnetic field also experience magnetic force
  1. Magnetic field and Lorentz force

Suppose we have a charge ‘q’  in an electric field ‘E’ so it will experience an electric force on it   

Electric force  Fe= q E   ….(1)

Now suppose we have switched on the magnetic field also and the charge is moving with a speed ‘v’ in this magnetic field ‘B’

So it will experience magnetic force on it. 

Magnetic force Fm= q ( v ×B)    ….(2)

So total force on the charged particle when it moves in the electric and magnetic force is given by the following expression

Lorentz force F= F electric + F magnetic

F= q E + q ( v×E) = q ( E+ v×B)

The above force F is the total force on the charged particle and is called Lorentz force.

Features of magnetic force   Fm= q ( v ×B)

  • It depends on q, v and B.  Magnetic force on a negative charge is opposite to magnetic force on a positive charge.
  • It involves the cross-product of velocity and magnetic field. so if the velocity and magnetic field are parallel to each other, the magnetic force will be zero.  magnetic force will be maximum when velocity is perpendicular to field B.
  • If charge is at rest so V=0 then magnetic force will be zero. Only a moving charge feels magnetic force
  • Magnetic force is a conservative force, It do not change the energy of the system
  • Magnetic force only changes direction of motion of charges and does not change the speed of the charges. S.I unit of Magnetic force is Tesla.
  1. Magnetic force on current carrying conductor

We can extend our analysis for force due to the magnetic field on a single moving charge to a straight rod carrying current. Inside a current-carrying conductor, charges are moving inside the conductor and when this current-carrying conductor is placed in magnetic force, moving charge inside the conductor experience magnetic force as discussed above  Fm= q ( v ×B), and so the whole conductor also feels that magnetic force. Derivation of the magnetic force on the current carrying charge.

Fm= q ( v ×B) This formula is for a single charge ‘q’ but when in conductor we have a large no of such charges.

Suppose we have ‘n’ no of charges per unit volume of the conductor of length ‘l’ and cross-section area ‘A’.

total charges inside the conductor = nlA

Force on 1 charge is  Fm= q ( v ×B)

Force on  total charges inside the conductor = nlA * e ( vd×B) ..(1)

Here ‘vd’ = drift velocity of charges inside the conductor, q= e, a charge of carrier inside the conductor.

Now we know that current  I= neAvd  …(2)

So using 2 in 1 we get  F= I (l×B)

The above formula is the formula for force on a current carrying conductor of length ‘L” and having current ‘I’  placed in magnetic field B.

Here  l= a vector of magnitude l and direction is that of current.

Motion in a magnetic field

Magnetic forces do not do any work on the charged particle or you can say that work done by magnetic forces on the charged particle moving in the magnetic field is zero.

You have learned in mechanics that  W= F ds = F ds cosθ

So if force and the displacement is perpendicular to each other then work done= 0

In the case of the magnetic motion of a charged particle in a magnetic field, The magnetic field and the velocity vector are always perpendicular to each other as shown. So no work is done.

Also  from work-energy theorem, we have  work done = change in Kinetic Energy

So If the work done=0, then the change in kinetic energy of the charged particle would also be equal to zero. This means that the Kinetic energy of the charged particle moving in the magnetic field is constant. Also, the magnitude of the velocity of the charged particle moving in the magnetic field is constant. But the direction of velocity can be changed.

Motion of a charged particle in a Uniform Magnetic field

Case 1:  Motion of charged particle is perpendicular to Magnetic field.

Suppose a charged particle of charge ‘q’ is moving with a speed  ‘v ‘ perpendicular to the magnetic field. The magnetic force experienced by the charged particle would be.

 Fm= q( v×B)= qvBsinθ= qvBsin90= qvB

This force will provide a centripetal force to the charged particle to move in a uniform circular motion.

centripetal force Fc= mv^(2 )/r

Fc= Fm , This will give    mv^2/r= qvB

So, r= mv/qB   This is the formula of the radius of the circular path Time period = Circumference/ velocity

T= 2Πr/v = 2Πmv/qBv = 2Πm/qB

So we conclude that when a charged particle moves perpendicular to the magnetic field, the magnetic force on the charged particle acts like centripetal force and produces a circular motion perpendicular to the magnetic field.

The particle would describe a circle if ‘v’ and ‘B’ are perpendicular to each other.

Radius of circular path = r=mv/qB

Time period of motion = T=2Πm/qB

Frequency of the motion = f=1/T= qB/2Πm

Case 2: Velocity is parallel to the magnetic field

Suppose we have a charged particle of charge ‘q’ moving with a velocity ‘v’ parallel to magnetic field B. The magnetic force on the charged particle will be

Fm= q(v×B)= qvBsinθ=qvBsin0= 0

So we can conclude that when a charged particle moves parallel to the magnetic field then it will not experience any force on it and will continue its motion without any change in its motion.

Case 3: when the velocity vector makes an arbitrary angle with the magnetic field Suppose we have a charged particle with charge ‘q’ . It moves in magnetic field B  with velocity ‘v’ making an angle ‘Θ'  with the Magnetic field

Since the velocity is in an arbitrary direction we can resolve the velocity in a direction parallel to B and perpendicular to B.

v||= v cosΘ  and v⊥=v sin�

So we have two components of velocity: the perpendicular component will follow case 1 and the parallel component will follow case 2 described above.

Due to v, the charged particle will try to move in a circular path in a direction perpendicular to B and due to v||, the charged particle will continue with the same speed and in the same direction.

So the resultant of the two motions will be helical motion.

The radius of the helix is determined by v  and pitch ( horizontal distance traveled in one complete circle ) is determined by v||.

r= mv⊥/qB  =mv cosΘ/qB

Time period  T=2Πm/qB, frequency f=qB/2Πm

l=v||*T= 2Πmv||/qB= 2Πm v sinΘ/qB

Motion in combined Electric and Magnetic field

When a charged particle of charge ‘q’ moves with velocity ‘v’ in the electric and magnetic fields, Lorentz force acts on it which is due to both electric and magnetic force.

F= Fe + Fm

F= q( E+ v×B)

For simplicity let's assume that the velocity, electric field and magnetic field are mutually perpendicular to each other.

Here E= Ej , B= Bk , v=vi

Fe= q*Ej=qE j, so electric force is along y-direction for positive charge

Fm= q( vi × Bk)= qvB(-j)     as   i ×k= -j

so magnetic force is in -y-direction for positive charge.

We can see that electric and magnetic forces are in opposite directions so if we adjust E and B such that magnitude of the two forces are equal. Then total force on the charge ‘q’ will be zero and the particle will go undeflected. This happens when

Fe= Fm    ; qE= qvB ;    v= E/B

This condition can be used to select charged particles of particular velocity out of the beam containing charges with different speeds, therefore serving as a velocity filter.

Magnetic field due to the current element: Biot- Savart law

We know from the above discussion that the current carrying wire produces a magnetic field around it. This phenomenon is called the magnetic effect of current.

The relationship between current and magnetic field produced due to the current element is given by Biot-Savart Law.

According to Biot-Savart law, the magnitude of magnetic field dB due to current element dl

  • Is proportional to the length of the current element ‘dl’.
  • Is proportional to the current in the wire.
  • Is proportional to sinΘ, where Θ is the angle between dl vector and r vector ( position vector of point of observation P)
  • Is inversely proportional to the square of the distance between ‘r’ of the point P from the current element.

To remove the proportionality sign we need to put a constant.

Vector form of Biot-Savart law.

Direction of dB

Right-hand thumb rule: If we put our thumb of the right hand in the direction of the current, then the direction of the curl of our fingers will give the direction of the magnetic field.

Magnetic field on the axis of the current carrying loop.

 Suppose we have a current carrying loop, carrying current ‘I’ and radius R.

We need to find the magnetic at the axis of the coil at a distance ‘x’ from the center.

When we resolve dB along in the direction parallel to it and perpendicular to it, we will see that perpendicular components of dB due to half loop will be canceled by the other half of the current loop.

And parallel component of dB is integrated for the whole circular loop

Magnetic field due to  current carrying loop using Biot-Savart law

Suppose we have a currency-carrying loop of radius ‘r’ and carrying current ‘I’ and we have to find the magnetic field at the center of the circular loop at O.

From Biot-Savart law we have

Here sinθ=1 as θ=90 

So to get B we have to integrate dB along the whole loop