General Physics (calculus based) Class Notes

Dr. Rakesh Kapoor, M.Sc., Ph.D.

Former Faculty-University of Alabama at Birmingham, Birmingham, AL 35294

Magnetic Field


In this chapter we will learn

How the magnetic Field is produced?

The definition of Magnetic field.

Motion of a charged particle in magnetic field.

Hall effect.

Cyclotron particle accelerator.

Magnetic force on a current-carrying wire.

Magnetic torque on a wire loop.

Magnetic dipole, magnetic dipole moment.

Electric Motor.

Permanent magnets and magnetic force.

We are familiar with the magnet, we use to hang notes on a refrigerator.

Compass needle is a permanent magnet on a low - friction pivot.

We know when a permanent magnet is brought close to iron nails, it attracts them (exert force on the iron nails).

When a permanent magnet is brought close to the compass needle, it also experience a force.

This force is not electric force but is called magnetic force.

How we know that it is not simply an electrostatic force?

Magnet is made up of neutral iron or steel and compass needle is also made up of neutral iron or steel.

There exists no electrostatic force between two neutral objects.

Compass needle also points toward Earth's magnetic north pole, while neither a charged object nor an electric dipole do this.

These observations clearly indicates that magnetic force is not electrostatic force.

How the magnetic Force is produced?

A stationary point charge generates an electric field given by


Move the charge (click play)

A moving charge not only produce an electric field MagneticField_3.gif but also a Magnetic field MagneticField_4.gif

A Magnetic field MagneticField_5.gif  curls around the moving charge.

MagneticField_6.gif MagneticField_7.gif

According to right hand rule if thumb points to the direction of motion of a charge, curled fingers point to the direction of magnetic field MagneticField_8.gif.

Biot - Savart Law

"Biot-Savart" is pronounced as bee-oh sah-VAR.


According to this law the magnetic field MagneticField_10.gif at a position  MagneticField_11.gif due to a charge q, moving with velocity MagneticField_12.gif is given as


Where the value of constant MagneticField_14.gif is


MagneticField_16.gif is called permeability constant.

Since the magnetic field MagneticField_17.gif is proportional to the cross product (×) of vector MagneticField_18.gif and MagneticField_19.gif, therefore the direction of MagneticField_20.gif will always be perpendicular to MagneticField_21.gif and MagneticField_22.gif.

Two moving point charges not only exert electric force on each other but also exert a magnetic force on each other.
For the presence of magnetic force, both the charges should be in motion with respect to each other.

Magnetic force and definition of MagneticField_23.gif

If MagneticField_24.gif is the magnetic field at a point due to some moving charge (charges), the magnetic force MagneticField_25.gif on an other charge q at that point moving with velocity MagneticField_26.gif is given as


The magnitude of the force can be written as


where φ is the angle between velocity vector MagneticField_29.gif and magnetic field MagneticField_30.gif.

In other words magnetic field is related to the magnetic force as


Magnetic field is the "force per unit positive charge per unit velocity".

SI units for magnetic field are Tesla.




Direction of Magnetic Force

MagneticField_34.gif MagneticField_35.gif

The direction of the magnetic force MagneticField_36.gif is given by the right hand rule as shown in the above figure (a) and (b).

When q is negative the direction of MagneticField_37.gif will be opposite to that of MagneticField_38.gif (figure c).


The force  MagneticField_40.gif acting on a charged particle moving with velocity MagneticField_41.gif through a magnetic field  MagneticField_42.gif is always perpendicular to MagneticField_43.gif and MagneticField_44.gif.

Since MagneticField_45.gif is always perpendicular to MagneticField_46.gif therefore it can not change the magnitude of MagneticField_47.gif, it can only change the direction of MagneticField_48.gif.

Interactive Checkpoint - 1 (Direction of Magnetic Force)

A proton moving from south to north enters a uniform magnetic field MagneticField_49.gif directed into the page.
(a) What will be the direction of magnetic Force MagneticField_50.gif?
(b) If an electron replaces the proton, what will be the direction of magnetic Force MagneticField_51.gif?
(c) What happens to MagneticField_52.gif for the proton if the magnetic field MagneticField_53.gif points out of the page?

Checkpoint 1

The figure shows three situations in which a charged particle with velocity MagneticField_55.gif travels through a uniform magnetic field MagneticField_56.gif. In each situation, what is the direction of the magnetic force MagneticField_57.gif on the particle?


Magnetic Field Lines

Magnetic field lines are similar to electric field lines.

The direction of the tangent of a magnetic field line at any point gives the direction of MagneticField_59.gif at that point.


The spacing of the lines represent the magnitude of MagneticField_61.gif. Magnitude of MagneticField_62.gif at point P is more that at point Q.


Magnets are called magnetic dipoles, one pole is called north pole and other is called south pole.

The field lines of a magnet always enters the south pole, pass through the body of the magnet and leave through the north pole.

MagneticField_64.gif MagneticField_65.gif
Magnetic dipole Electric dipole

Contrary to field lines of an electric dipole, inside the magnet, field lines move from south to north pole.

Opposite magnetic poles attract each other, and like magnetic poles repel each other.

Magnetic monopole

There is a dramatic difference between the electric and magnetic dipoles.

The electric dipole is made of individual negative and positive charges ("monopoles") and can be separated from each other.

One might expect a magnetic dipole to be made of positive (north) and negative (south) magnetic monopoles, but no one has ever found an individual magnetic monopole.

When you cut a magnet in two, you do not get two magnetic monopoles, you just get two magnets.

Motion of a charged particle in magnetic field.

The Magnetic force MagneticField_66.gif on the charged (q) particle moving with  velocity MagneticField_67.gif in magnetic field MagneticField_68.gif is given as



This force is always perpendicular to the velocity of the particle like a centripetal force, therefore charged particle moves in a circular path and φ=90°.

As per Newton's second law, the magnitude of centripetal acceleration a is given as


Where m is the mass of the charged particle.

In circular motion speed v of the charged particle is related to its centripetal acceleration.


where r is the radius of the circular path.

Equating above two values of acceleration a gives us  radius of the circular path of the charged particle.


Radius decreases with increase in charge or magnetic field.

Radius increases with velocity or mass of the particle.

Now the time period T (the time to complete one revolution) is equal to the circumference divided by the speed:


The frequency (number of revolutions per second) will be given as


Angular frequency ω of the motion is given as


Interactive Checkpoint - 2 (Motion of charged particle in Magnetic field)

A positron (positive charged particle with same mass as electron) moving horizontally from south to north enters a uniform magnetic field MagneticField_77.gif directed out of the page.
(a) Will the positron moves in clockwise or anti-clockwise circular path?
(b) What happens to the radius if we increase the speed of the particle?
(c) What happens to the radius if we increase the magnitude of the magnetic field MagneticField_78.gif?
(d) Will the radius change if we change it to an electron?

Interactive Checkpoint - 3 (Motion of charged particle in Magnetic field)

Two charged particles (one negative and other positive) of same speed v, moving  horizontal from south to north enters a uniform magnetic field MagneticField_80.gif directed out of the page.
(a) If the mass of the negative charge particle is half of the positive particle which particle will have larger radius and how much?
(b) Which particle will have more number of revolutions per second and by what factor?

Helical Path

We have seen when velocity of a charge particle is perpendicular to the magnetic field it moves in a circular path.

What happens if angle is 0°?

What happens if angle 0°<φ<90°?

Perpendicular velocity component will cause a circular motion while parallel component will cause a linear motion, therefor the resulting path will be a helical path.

Watch how the pitch of the helix changes with angle θ between  MagneticField_83.gif and MagneticField_84.gif ?

Aurora borealis in northern hemisphere can be explained by this phenomenon.

Crossed Fields

When electric field MagneticField_85.gif and magnetic field MagneticField_86.gif are perpendicular to each other, they are called crossed fields.

Discovery of Electron

Let us see what happens when a charged particle moves in a crossed fields. We will do simulation of J. J. Thompson experiment.

When a negative charge (electron) moves through the plates with only electric field on, it experience a upward force.

When a negative charge (electron) moves through the plates with only magnetic field on, it experience a downward force.

With crossed fields on (both magnetic field and electric field on), we can adjust the magnitude of MagneticField_88.gif and MagneticField_89.gif in such a way that the net force on the charged particle is zero.

If v is the velocity of the charged particle, when it enters the crossed fields with net force is zero, we can say


Or the magnitudes of both the forces should be same




Checkpoint - 3 (Crossed Fields)

The figure shows four directions for the velocity vector MagneticField_93.gif of a positively charged particle moving through a uniform electric field  MagneticField_94.gif (directed out of the page and represented with an encircled dot) and a uniform magnetic field MagneticField_95.gif. (a) Rank directions 1, 2, 3 and 4 according to the magnitude of the net force on the particle, greatest first. (b) Of all four directions, which might result in a net force of zero?


Hall Effect

In 1879, Edwin H.Hall, then a 24 - year - old graduate student at the Johns Hopkins University, showed that the charges moving inside a conductor can also be deflected by a magnetic field.

Consider copper strip of width d, carrying a current i. whose conventional direction is from the top of the figure to the bottom.

Let us apply an external magnetic field MagneticField_97.gif, pointing into the plane of the figure.

MagneticField_98.gif MagneticField_99.gif

Each drifting electron in the strip will experience a magnetic force MagneticField_100.gif towards right hand side (figure a ).

If MagneticField_101.gif is the drift velocity of the electrons, the magnetic force will be given as


Presence of magnetic deflection causes accumulation of electrons on the right hand edge of the strip.

This accumulation will develop a potential difference V between the right and left edge (figure b).

This is called Hall potential difference V, and will generate an electric field E such that


This electric field will try to deflects the electrons toward left hand side.

Electric force will appose the magnetic force.

When both these forces are equal, no further deflection will take place.


Drift velocity MagneticField_105.gif is related to the current density J and number density n of free electrons in a conductor.


where A is the cross sectional area of the strip. By substituting the value of MagneticField_107.gif in balancing equation we get



From this equation we can get the free electron density in a conductor.


If is the thickness of the strip, A = ℓd. In terms of thickness n can be re written as


Hall effect can be used to measure free charge density n in a conductor and drift velocity MagneticField_112.gif

Cyclotron particle accelerator.

High - energy particles, such as high - electrons and protons, have been extremely useful in probing atoms and nuclei to reveal the fundamental structure of matter.

Cyclotron is one kind of particle accelerator.

A cyclotron accelerator consists of two hollow D-shaped conductors (copper) in which the particles (protons, say) circulate.

A uniform magnetic field is applied out of the page. And an alternating electric potential difference is applied to these D,s.

Electric field will alternate between the gap of these D,s. The alternating frequency is synchronous to the circulating frequency decided by the applied magnetic field. This is called resonance condition.

This field will increase the speed of the particle when ever it enters from one D to other.

The particle will go in a spiral path with increasing speed or increasing kinetic energy.

Finally it exits out from the cyclotron using an other deflecting electric field at the exit.


Magnetic force on a current-carrying wire.

A charged particle moving in a uniform magnetic field experience magnetic force.

It does not matter if it is moving in a free space or a conductor.

Consider a wire segment of length L  carrying current i, placed in a uniform magnetic field MagneticField_115.gif coming out of the page.  

Let the charges move (click play).

Suppose all the conduction electrons in segment L (blue) take time t to cross the line A.

If MagneticField_116.gif is the drift velocity then



Total charges crossing plane A in time t is given as


Magnetic force MagneticField_121.gif experienced by charges in segment L will be given as




In vector form magnetic force can be written as


Here MagneticField_125.gif is length vector that has magnitude L and is directed along the direction of current.

Magnetic force on a current-carrying wire of arbitrary shape placed in Uniform Magnetic field

If the wire is not straight or the magnetic field is not uniform, we can imagine the wire broken up in small straight segments of length MagneticField_126.gif.

Magnetic force MagneticField_127.gif on each segment will be given as


Net magnetic force MagneticField_129.gif on the whole wire is then the vector some of all these forces.


Checkpoint 2

The figure shows a current i through a wire in a uniform magnetic field MagneticField_131.gif, as well as the magnetic force MagneticField_132.gif acting on the wire. The field is oriented so that the force is maximum. In what direction is the field?


Hint: MagneticField_134.gif.

Interactive Checkpoint - 4 (Magnetic force on a current carrying wire)

A current carrying wire is placed in a uniform magnetic field MagneticField_135.gif.
(a) What will be the direction of force if current is pointing to the left and MagneticField_136.gif is pointing out of the page?
(b) What will be the magnetic force direction if current is switched to right?
(c) Will the force increase or decrease with increase in current?

A current carrying wire loop in a uniform magnetic field

Consider a rectangular wire loop of side lengths a and b carrying current i in a uniform magnetic field.


Will the loop be in equilibrium?

Net magnetic force on the wire loop.

Magnetic forces MagneticField_139.gif and MagneticField_140.gif experienced by the four sides of the loop are given as




If we add all the four forces, net force MagneticField_143.gif on the loop will be zero.


Net torque on the wire loop.

Let us now look at the net torque MagneticField_145.gif on the loop due to these forces.

Let us tilt the loop such that loop area vector MagneticField_146.gif makes an angle θ with the magnetic field MagneticField_147.gif.


For force MagneticField_150.gif and MagneticField_151.gif the normal distance from the axis is zero as both the forces are acting along the axis. Therefore there is no torque due to these two forces.

Length vector MagneticField_152.gif and MagneticField_153.gif of side 1 and side 3 is always perpendicular to the magnetic field, therefore the magnitude of both the forces are given as


For force MagneticField_155.gif and MagneticField_156.gif the normal distance from the axis of rotation is MagneticField_157.gif. Magnitude of torque MagneticField_158.gif due to this force will


Similarly the magnitude of torque MagneticField_160.gif will be given as


Both these torques will try to rotate the loop in clockwise direction, therefore the magnitude of net torque MagneticField_162.gif will be given as


Product a b = A, area of the loop so above equation can be rewritten as


If there are N numbers of loops, total torque τ will be given as.


Although the formula is derived for a rectangular loop but it is true for any shape of loop.

Magnetic dipole & magnetic dipole moment.

We know that, when an electric dipole of dipole moment MagneticField_166.gif is placed in an electric field MagneticField_167.gif it experienced a torque MagneticField_168.gif given as.


In vector form it is written as


θ is the angle between the dipole moment MagneticField_171.gif and electric field MagneticField_172.gif.

Similarly when a current carrying loop is placed in a uniform magnetic field, it also experience a torque.

By same analogy we can call a current carrying coil a magnetic dipole of magnetic dipole moment MagneticField_175.gif.

The direction of the dipole moment is taken to be the direction of area vector MagneticField_176.gif. Torque MagneticField_177.gif on a magnetic dipole can be written as


where θ is the angle between MagneticField_179.gif and MagneticField_180.gif. In vector form it can be written as


Magnitude of μ is



According to right hand rule if the curled fingers point in the direction of current, thumb points in the direction of magnetic dipole moment MagneticField_184.gif.

Potential energy of a magnetic dipole

When a magnetic dipole is placed in an external magnetic field, it has a magnetic potential energy.


θ is the angle between MagneticField_186.gif and the magnetic field MagneticField_187.gif.

A magnetic dipole has its lowest energy when it is parallel to the magnetic field MagneticField_188.gif.

A magnetic dipole has its highest energy when it is anti parallel to the magnetic field MagneticField_189.gif.

When an external torque rotates a dipole placed in a magnetic field from one position to other and dipole is stationary before and after the move, the work MagneticField_190.gif done by the applied torque will be given as


Checkpoint - 3 (Magnetic Dipole moment)

The figure shows four orientations, at angle θ, of a magnetic dipole moment MagneticField_192.gif in a magnetic field. Rank the orientations according to (a) the magnitude of the torque on the dipole and (b) the potential energy of the dipole, greatest first.


Electric Motor.

Electric motor is one of the most important instrument in this world. Following figure show a simple electric motor.

It consists of a rectangular wire loop carrying current i, placed in a uniform magnetic field MagneticField_194.gif.


A DC Electric Motor