Moving Charges and Magnetism Chapter Notes | Physics Class 12 - NEET PDF Download

Both electricity and magnetism have been known for more than 200 years. However, their relationship was discovered only 200 years ago. Hans Christian Oersted in 1820 discovered that a straight wire carrying current caused deflection in the magnetic compass needle. 

From here, started the discoveries in the field of 'electromagnetism', which we are going to study in detail in this document. 

Oersted Experiment: Observations and Conclusion

To learn about moving charges and magnetism, we need to understand the significant experiment performed by a Danish Physicist, Hans Christian Oersted in 1820. 

When a magnet is pointed to the compass needle, the needle moves.

• He observed accidentally that, a current in a straight wire caused a noticeable deflection in a nearby magnetic compass needle. 
• 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. 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.
• Oersted concluded that moving charges or currents produced a magnetic field in the surrounding space.

• A magnetic field is like an invisible area around something that's magnetic. It helps us understand how the magnetic force spreads around the object.
• When an electric charge or current moves close to a magnet, it creates a magnetic field. Tiny particles, like electrons with a negative charge, move around and make this magnetic field. 
• These fields can start inside the atoms of magnetic things or in wires that have electricity flowing through them.

Magnetic Field is a region of space around a magnet or current-carrying conductor or a moving charge in which its magnetic effect can be felt.

• A magnetic field depicts how a moving charge flows around a magnetic object. Magnetic force is a force that arises due to the interaction of magnetic fields. It can be either a repulsive or attractive 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 the presence of both the electric field E (r) and constant magnetic field B (r).

• The force on an electric charge q due to both of them can be written as: 

• This force was given first by H.A. Lorentz based on the extensive experiments of Ampere and others. It is called the Lorentz force.

Biot-Savart's Law

Biot-Savart’s law is an equation that gives the magnetic field produced due to a current-carrying segment. This segment is taken as a vector quantity known as the current element.

• The magnetic field dB due to this element is to be determined at a point P which is at a distance r from it. Let θ be the angle between dl and the displacement vector r. 
• According to Biot-Savart’s law, the magnitude of the magnetic field dB is proportional to the current I, the element length |dl|, and inversely proportional to the square of the distance r. Its direction* is perpendicular to the plane containing dl and r.

Magnetic filed due to a current element

Also, where  is the drift velocity of the charge.

Direction of dB: The direction of magnetic field is given by the right-hand thumb rule as shown in the image below.Right-hand thumb rule

Magnetic Field due to Circular Current-Carrying Coil

B = μ0NI a2/2(a2 + x2)3/2

or,    

The direction of  is given by Right-hand screw rule.

Now,  , 

Also, we have. 

Magnetic Field At The Centre Of A Circular Coil

B = μ0NI / 2a 

Magnetic Field At The Centre Of A Circular Arc Carrying Current

Ampere's Circuital Law

i.e. 

• The direction of the magnetic field at a point on one side of a conductor of any shape is equal in magnitude but opposite in direction of the field at an equidistant point on the other side of the conductor.
• If the magnetic field at a point due to a conductor of any shape is Bo if it is placed in vacuum then the magnetic field at the same point in a medium of relative permeability μr is given by.
• If the distance between the point and an infinitely long conductor is decreased (or increased) by K-times then the magnetic field at the point increases (or decreases) by K-times.
• The magnetic field at the center of a circular coil of a radius smaller than another similar coil with a greater radius is more than that of the latter.
• For two circular coils of radii R1 and R2 have the same current and the same number of turns,where B1 and B2 are the magnetic fields at their centers.
• The magnetic field at a point outside a thick straight wire carrying current is inversely proportional to the distance but the magnetic field at a point inside the wire is directly proportional to the distance.

Applications of Ampere's Circuital Law 

Magnetic Field in a Solenoid

Magnetic Field in Solenoid depends on various factors such as the number of turns per unit length, the current strength in the coil, and the permeability of the material used in the solenoid. The magnetic field of a solenoid is given by the formula:

B = μ_oIN/L

where,

• μ_o is the permeability constant with a value of 1.26 × 10⁻⁶ T/m,
• N is the number of turns in the solenoid,
• I is the current passing through the coil,
• L is the coil length.

Note: The magnetic field in a solenoid is maximum when the length of the solenoid is greater than the radius of its loops.

Force on a Conductor in Magnetic Field

Force on a conductor in Magnetic Field

Torque on a coil in Magnetic Field

Force Acting on a Charged Particle in a Uniform Magnetic Field

Here,  q is the magnitude of the charge, vsinθ is the component of the velocity that is acting perpendicular to the direction of the magnetic field and the B is the magnitude of the applied magnetic field.

F=kqvB sinθ

Here, k is the proportionality constant, whose value is equal to the 1 (k=1). Thus,

F=qvB sinθ

In vector form,

F=qv × B

This is the required expression for the force acting on a moving charge in a uniform magnetic field. There are two cases, for the magnetic force value:

1. Velocity and magnetic field are parallel – In this equation, if the velocity and magnetic field are parallel, then the angle made between the velocity and the magnetic field is zero. This representation of the magnetic force acting on a moving charge in a uniform magnetic field can be expressed as follows.

           F=qvB sin θ

2. Velocity or the speed and magnetic field are normal – In case the speed of a particle which is charged becomes normal to the magnetic field. Then, the angle made between them will be equal to 90°. Thus,

F=qvB sin(90)

F=qvB(1)

Fmax=qvB

Maximum force is acting on a moving charge in a uniform magnetic field in this case.

           F=qvB(0)

           F=0

There is no force in this case.

Moving Coil Galvanometer

The moving coil galvanometer was first devised by Kelvin and later modified by D'Arsonval. This is used for the detection and measurement of small electric currents. 

Principle

The principle of a moving coil galvanometer is based on the fact that when a current-carrying coil is placed in a magnetic field, it experiences a torque.

Construction

A moving coil ballistic galvanometer is shown in the figure: 

Moving Coil Galvanometer

• It essentially consists of a rectangular coil PQRS or a cylindrical coil of a large number of turns of fine insulated wire wound over a non-conducting frame of ivory or bamboo. 
• This coil is suspended by means of phosphor bronze wire between the pole pieces of a powerful horseshoe magnet NS. The poles of the magnet are curved to make the field radial. The lower end of the coil is attached to a spring of phosphor-bronze wire. 
• The spring and the free ends of the phosphor bronze wire are joined to two terminals T2 and T1 respectively on the top of the case of the instrument. L is a soft iron core. A small mirror M is attached to the suspension wire. 
• Using lamp and scale arrangement, the deflection of the coil can be recorded. The whole arrangement is enclosed in a non-metallic case.

Theory

Let the coil be suspended freely in the magnetic field.

Suppose, n = number of turns in the coil

A = area of the  coil

B = magnetic field induction of radial magnetic field in which the coil is suspended.

Here, the magnetic field is radial, i.e.,  the plane of the coil always remains parallel to the direction of the magnetic field, and hence the torque acting on the coil

τ = niAB … (1)

• Due to this torque, the coil rotates. As a result, the suspension wire gets twisted. Now a restoring torque is developed in the suspension wire. 
• The coil will rotate till the deflecting torque acting on the coil due to the flow of current through it is balanced by the restoring torque developed in the suspension wire due to twisting. 
• Let C be the restoring couple for a unit twist in the suspension wire and θ be the angle through which the coil has turned. The couple for this twist θ is Cθ.

  In equilibrium, deflecting couple = restoring couple

  ∴   ni AB = Cθ  or  i = Cθ/ (nAB)

  or i = Kθ (where C/nAB = K) … (2)

  K is a constant for a galvanometer and is known as a galvanometer constant.

  Hence 

  Therefore, the deflection produced in the galvanometer is directly proportional to the current flowing through it.

  Current sensitivity of the galvanometer: The current sensitivity of a galvanometer is defined as the deflection produced in the galvanometer when a unit current is passed through it.

  We know that, niAB = Cθ

  ∴ Current sensitivity is = 

  where C = restoring couple per unit twist

  The SI unit of current sensitivity is radian per ampere or deflection per ampere.

  Voltage sensitivity of the galvanometer : The voltage sensitivity of the galvanometer is defined as the deflection produced in the galvanometer when a unit voltage is applied across the terminals of the galvanometer.

  ∴ Voltage sensitivity,  

  If R is the resistance of the galvanometer and a current is passed through it, then V = iR

  ∴ Voltage sensitivity, Vs = 

  The SI unit of voltage sensitivity is radian per ampere or deflection per ampere.

  Conditions for sensitivity: A galvanometer is said to be more sensitive if it shows a large deflection even for a small value of current.

  We know that, 

  For a given value of i, θ will be large if (i) n is large, (ii) A is large, (iii) B is large, and (iv) C is small.

  Regarding the above factors, n, and A cannot be increased beyond a certain limit. By increasing n, the resistance of the galvanometer will increase and by increasing A, the size of the galvanometer will increase. So, the sensitivity will decrease. Therefore, B is increased. The value of B can be increased by using a strong horseshoe magnet. Further, the value of C can be decreased. The value of C for quartz and phosphor-bronze is very small. So, the suspension wire of quartz or phosphor-bronze is used. The value of C is further decreased if the wire is hammered into a flat strip.

  Summary

• If in a coil the current is clockwise, it acts as a south pole. If the current is anticlockwise, it acts as a north pole.

  
• No magnetic field occurs at points P, Q, and R due to a thin current element .

  
• Magnetic field intensity in a thick current-carrying conductor at any point x is:

  
  

  

  

  

  
  
• Graph of magnetic field B versus x:

  

  • No force acts on a charged particle if it enters a magnetic field in a direction parallel or antiparallel to the field.
  • A finite force acts on a charged particle if it enters a uniform magnetic field in a direction with a finite angle with the field.
  • If two charged particles of masses m1and m2 and charges q1 and q2 are projected in a uniform magnetic field with the same constant velocity in a direction perpendicular to the field then the ratio of their radii (R1: R2) is given by

  

  • The force on a conductor carrying current in a magnetic field is directly proportional to the current, the length of the conductor, and the magnetic field.
  • If the distance between the two parallel conductors is decreased (or increased) by k-times then the force between them increases (or decreases) k-times.
  • The momentum of the charged particle moving along the direction of the magnetic field does not change, since the force acting on it due to the magnetic field is zero.
  • Lorentz force between two charges q1 and q2 moving with velocity v1, v2 separated by distance r is given by

  

  • If the charges move, the electric as well as magnetic fields are produced. In case the charges move with speeds comparable to the speed of light, magnetic and electric force between them would become comparable.
  • A current-carrying coil is in stable equilibrium if the magnetic dipole moment is parallel to   and is in unstable equilibrium when  is antiparallel to .
  • The magnetic moment is independent of the shape of the loop. It depends on the area of the loop.
  • A straight conductor and a conductor of any shape in the same plane and between the same two endpoints carrying equal current in the same direction, when placed in the same magnetic field experience the same force.
  • There is net repulsion between two similar charges moving parallel to each other in spite of attractive magnetic force between them. This is because of the electric force of repulsion which is much stronger than the magnetic force.
  • The speed of the charged particle can only be changed by an electric force.