Magnetism

Prepared by

Md. Amirul Islam

Lecturer

Department of Applied Physics & Electronics

Bangabandhu Sheikh Mujibur Rahman Science &

Technology University, Gopalganj – 8100

Many historians of science believe that the compass, which uses

a magnetic needle, was used in China as early as the 13th

century B.C., its invention being of Arabic or Indian origin.

The early Greeks knew about magnetism as early as 800 B.C.

They discovered that the stone magnetite (Fe3O4) attracts pieces

of iron.

Legend ascribes the name magnetite to the shepherd Magnes,

the nails of whose shoes and the tip of whose staff stuck fast to

chunks of magnetite while he pastured his flocks.

Frenchman Pierre de Maricourt shows through experiments

that every magnet, regardless of its shape, has two poles, called

north pole and south pole.

Reference: Physics II by Robert Resnick and David Halliday, Topic – 29.0, Page – 905

The poles received their names because of the way a magnet

behaves in the presence of the Earth’s magnetic field. If a bar

magnet is suspended from its midpoint and can swing freely in a

horizontal plane, it will rotate until its north pole points to the

Earth’s geographic North Pole and its south pole points to the

Earth’s geographic South Pole.

Although the force between two magnetic poles is similar to the

force between two electric charges, there is an important

difference. Electric charges can be isolated (witness the electron

and proton), whereas a single magnetic pole has never been

isolated. That is, magnetic poles are always found in pairs.


The region of space surrounding any moving electric charge

contains a electric field as well as a magnetic field. Historically,

the symbol B has been used to represent a magnetic field. The

direction of B at any location is the direction in which a compass

needle points at that location.


We can define a magnetic field B at some point in space in terms

of the magnetic force FB that the field exerts on a test object, for

which we use a charged particle moving with a velocity v. From

experimental result we get that:

The magnitude FB of the magnetic force exerted on the particle

is proportional to the charge q and to the speed v of the particle.

The magnitude and direction of FB depend on the velocity of the

particle and on the magnitude and direction of the magnetic

field B.


When a charged particle moves parallel to the magnetic field

vector, the magnetic force FB acting on the particle is zero.

When the particle’s velocity vector makes any angle θ≠0, with

the magnetic field, the magnetic force acts in a direction

perpendicular to both v and B; that is, FB is perpendicular to

the plane formed by v and B as shown on the figure left.


The magnetic force exerted on a positive charge is in the

direction opposite the direction of the magnetic force exerted on

a negative charge moving in the same direction as shown on the

figure right.


The magnitude of the magnetic force FB exerted on the moving

particle is proportional to sinθ, where θ is the angle the

particle’s velocity vector makes with the direction of B.


We can summarize these observations by writing the magnetic

force in the form,


Following figure shows the right-hand rule for determining the

direction of the cross product v×B

You point the four fingers of

your right hand along the

direction of v with the palm

facing B and curl them toward

B. The extended thumb, which

is at a right angle to the fingers,

points in the direction of v×B. If

the charge is negative, the

direction of FB is opposite to the

thumb direction.

The electric force acts in the direction of the electric field,

whereas the magnetic force acts perpendicular to the magnetic

field.


The electric force acts on a charged particle regardless of

whether the particle is moving, whereas the magnetic force acts

on a charged particle only when the particle is in motion.

The electric force does work in displacing a charged particle,

whereas the magnetic force associated with a steady magnetic

field does no work when a particle is displaced.

SI unit of magnetic field is the newton per coulomb-meter per

second, which is called the tesla (T):


Because a coulomb per second is defined to be an ampere, we

see that:

A non-SI magnetic-field unit in common use, called the gauss

(G), is related to the tesla through the conversion: 1T = 104G.

Example of some magnetic field and their field magnitude:

Reference: Physics II by Robert Resnick and David Halliday, Table – 29.1, Page – 910

A charge moving with a velocity v in the presence of both an

electric field E and a magnetic field B experiences both an

electric force qE and a magnetic force qv×B. The total force

acting on the charge is then,


This force is called Lorentz Force.

Later

Reference: Later

A current-carrying wire also experiences a force when placed in

a magnetic field as because current is a collection of many

charged particles in motion. The resultant force exerted by the

field on the wire is the vector sum of the individual forces

exerted on all the charged particles making up the current.


(a) A wire suspended

vertically between the poles

of a magnet. (b) the

magnetic field (blue crosses

signs) is directed into the

page. When there is no

current in the wire, it

remains vertical. (c) When

the current is upward, the

wire deflects to the left. (d)

When the current is

downward, the wire deflects

to the right.

Let us consider a straight segment of wire of length L and cross-

sectional area A, carrying a current I in a uniform magnetic

field B, as shown in Figure. The magnetic force exerted on a

charge q moving with a drift velocity vd is qv×B. If n is the

number of charge per unit volume then total charge on that

section of conductor is nAL.


Thus, total magnetic force,

As, I = nqvdA thus,

where L is a vector that points in the direction of the current I and has

a magnitude equal to the length L of the segment. This expression is

for a straight wire in a uniform magnetic field.


If the wire is not straight as shown in the figure, we divide the wire

into infinitesimal segments with length ds. Thus, force exerted on a

small segment of vector length ds is,

Total force on the wire will be,

where a and b represent the end points of the wire.

Later


Later

Reference: Physics II by Robert Resnick and David Halliday, Example – 29.5, Page – 918


In 1819, Oersted discovered that a current-carrying conductor

produces a magnetic field deflects compass needle.

(a) When no current is present in the wire, all compass needles point in the same

direction (toward the Earth’s north pole). (b) When the wire carries a strong

current, the compass needles deflect in a direction tangent to the circle, which is the

direction of the magnetic field created by the current. (c) Circular magnetic field

lines surrounding a current-carrying conductor, displayed with iron filings.


In 1819 Oerested discovered that a compass needle is deflected by a

current-carrying conductor due to the magnetic field associated with

the current. After that Jean-Baptiste Biot and Félix Savart did an

experiment and arrived at a mathematical expression that gives the

magnetic field B at some point in space in terms of the current that

produces the field.


(a) The magnetic field dB at point P due to the current I through a length

element ds is given by the Biot–Savart law. The direction of the field is

out of the page at P and into the page at P’ (b) The cross product ds× 𝒓points out of the page when 𝒓 points toward P (c) The cross product ds× 𝒓points into the page when 𝒓 points toward P’


That expression is based on the following experimental observations

for the magnetic field dB at a point P associated with a length element

ds of a wire carrying a steady current I as shown in figure:

The vector dB is perpendicular both to ds (which points in the

direction of the current) and to the unit vector 𝒓 directed from ds to P.


The magnitude of dB is inversely proportional to r2, where r is the

distance from ds to P.

The magnitude of dB is proportional to the current I and to the

magnitude ds of the length element ds.


The magnitude of dB is proportional to sinθ, where θ is the angle

between the vectors ds and 𝒓


Thus, we can summarize the observations as,

where µ0 is a constant called the permeability of free space:

dB in above equation is the field created by the current in only a small

length element ds of the conductor. To find the total magnetic field B

created at some point by a current of finite size conductor, we must

sum up contributions from all current elements Ids that make up the

current.


Consider a thin, straight finite length wire carrying a constant

current I and placed along the x axis as shown in Figure.

Determine the magnitude and direction of the magnetic field at

point P due to this current.

Let, point P is at a perpendicular

distance a from the wire.

A small section ds is at a distance

x from the origin on the direction

of negative x-axis.

r is the distance from point P to

ds. We consider a unit vector 𝐫from ds to P.

Magnitude of ds vector is dx.


Direction of ds is on the direction

of positive charge flow that is

current direction.

The direction of B at point P will

be out of the page, as because

ds × 𝐫 directs to the outward

direction of the page. This

direction is not depending on the

position of ds. If we consider a

unit vector k on this outward

direction, then,


According to Biot-Savart law, we

get,

As we now know that the

direction of B is toward k vector,

we now calculate the magnitude.

For determining total B we have to integrate the expression.

Before that, we have to express the variables r and x in terms of θ.


Now, sinθ =ar

or, r = a cosecθ

and, tanθ =a−x

or, x = - a cotθ

thus, dx = a cosec2θ dθ

Putting these values on the

previous expression, we get,

** If we considered the section ds on positive x-axis, the result would be same

because the angle θ for positive x-axis is greater than 90°.


Now, we can integrate the

expression within the limit θ1 and

θ2 as shown in figure.

This is the expression of B for a

finite length current carrying wire.

Special Case:

If the wire is infinitely long then θ1 = 0° and θ2 = 180°. Thus,

cos 0° – cos 180° = 2 and then,


Later


The line integral of B.ds around any closed path equals µ0I.

Where I is the total continuous current passing through any

surface bounded by the closed path.

For a straight wire as shown in

figure, the field lines are circular

around the wire. If we consider

such a closed circular path with

radius r, on every point on that

path magnitude of B will be

same. If we slice that path into

infinitesimal ds sections, for

every section the angle between

ds and B will be 0°.


Thus we get,

Here, magnitude of B is constant on that path and 𝒅𝒔 is the

circumference 2πr. This expression is calledAmpere’s Law.

** We can compare this equation with Gauss’s law, ε0 𝑬. 𝒅𝒔 = qin

Reference: Physics II by Robert Resnick and David Halliday

Later


Later


Later

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Magnetism