204847858-Lorentz-Force

Trong vật lý học và điện từ học, lực Lorentz là lực tổng hợp của lực điện và lực từ tác dụng lên một điện tích điểm chuyển động trong trường điện từ. Oliver Heaviside là người đầu tiên suy luận ra công thức cho lực Lorentz vào năm 1889, mặc dù một số nhà lịch sử cho rằng James Clerk Maxwell đã đưa ra nó trong một bài báo năm 1865. Định luật được đặt theo tên của Hendrik Lorentz, người tìm ra công thức sau Heaviside một vài năm và ông đã nghiên cứu và giải thích chi tiết ý nghĩa của lực này.

Force

The Lorentz force is the force on a point charge due to electromagnetic fields.

If a particle of charge q moves with

velocity v in the presence of an electric field E and a magnetic field B, then it will

experience a force that is given by

* Vectors are denoted by boldface

Let us consider a change q moving in a combined electrical and

magnetic field with a velocity v.

If the electric field is E and the magnetic field is B, then, the force

experienced in the electric field alone will be qE and will be in the

direction of the electrical field E for a positive charge q

i.e FE = qE

also, a charge q moving in the magnetic field B with a velocity v

experiences a force that is equal to the cross product of v and B

multiplied by the charge q

i.e FB = q( v × B )

which is perpendicular to the plane containing v and B

Thus, acc, to the principle of superimposition the total force acting on the charged particle is

• For continuous charge distribution the force is given by

F =

where ρ = the volume charge density

and J = the current density

•

Motion of particle in electric field

Electric field lines are directed from +ve charge to -ve charge, and

hence positive charge moves in the direction of the electric field while negative charge in the direction opposite

to it (i.e of –E)

Motion of charge in magnetic field

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 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 qvB Thus, for centripetal force,

=> is the radius of the circle described by the

particle

The larger the momentum, the larger is the radius and bigger the circle described If ω is the angular

frequency, then

So,

which is independent of the velocity or energy

•

• 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

The time taken for one revolution is If there

is a component of the velocity parallel to the

magnetic field 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 which is given by

•

The Lorentz force occurs in many devices, including:

– Cyclotrons and other circular path particle

accelerators

– Mass spectrometers

– Velocity Filters

– Magnetrons

• Natural phenomenon of aurora borealis is also due

to the motion of charged particles in earth’s

magnetic field.

Ampere ‘ S Circuital

Law

In classical electromagnetism, Ampère's circuital law,

discovered by André-Marie Ampère in 1826, relates

the integrated magnetic field around a closed loop to

the electric current passing through the loop It relates

magnetic fields to electric currents that produce them

Using Ampere's law, one can determine the magnetic field associated with a given current or current associated with a given magnetic field, providing there is no time changing electric field present

According to this law “The line integral of

resultant magnetic field along a closed plane curve is equal to μ0 time the total current

crossing the area bounded by the closed

curve provided the electric field inside the

loop remains constant" Thus:

It relates magnetic fields to electric currents that produce them Using Ampere's law, one can

determine the magnetic field associated with a given current or current associated with a given magnetic field, providing there is no time changing electric field present In its historically original form, Ampère's

circuital law relates the magnetic field to its electric current source The law can be written in two forms, the "integral form" and the "differential form" It can

also be written in terms of either the B or H magnetic

fields

Integral form

In SI units ,the "integral form" of the original Ampère's circuital law is a

line integral of the magnetic field around some closed curve C The curve C in turn bounds both a surface S which the electric

current passes through, and encloses the current The mathematical statement of the law is a relation between the total amount of magnetic field around some path (line integral) due to the current which passes through that enclosed path (surface integral) It can be written in a

number of forms

In terms of total current, which includes both free and bound current,

the line integral of the magnetic B-field (in tesla, T) around closed

curve C is proportional to the total current Ienc passing through a

surface S (enclosed by C):

where J is the total current density (in ampere per square metre,

Am−2)

Alternatively in terms of free current, the line integral of the magnetic

H-field (in ampere per metre, Am−1) around closed curve C equals

the free current

If, enc through a surface S:

• where Jf is the free current density only Furthermore is the closed line integral around the closed curve C,

• dℓ is an infinitesimal element (a differential) of the curve C (i.e a vector with magnitude equal to the length of the infinitesimal

line element, and direction given by the tangent to the curve C)

• dS is the vector area of an infinitesimal element of surface S (that is, a vector with magnitude equal to the area of the

infinitesimal surface element, and direction normal to surface S The direction of the normal must correspond with the

orientation of C by the right hand rule), see below for further explanation of the curve C and surface S.

The equation only applies in the case where the electric field is constant in time, meaning the

currents are steady (time-independent, else the

magnetic field would change with time); see below for the more general form In SI units, the equation states for total current:

Differential form

and for free current

where ∇× is the curl operator.

1 Magnetic field of a Solenoid:

A solenoid is a coil of wire designed to create a strong magnetic field inside the coil By wrapping the same wire many times around a cylinder, the magnetic field due to the wires can become quite strong The number of

turns N refers to the number of loops the solenoid has

More loops will bring about a stronger magnetic field The formula for the field inside the solenoid is

B = µ I N / L

2 Magnetic field of toroid

Toroid is a hollow circular ring (like a medu vadai) on which a large number of turns of a wire are wound The field of the toroidal solenoid

is therefore confined wholly to the space enclosed by the windings.

If we consider path 2, a circle of radius r, again by symmetry the field

is tangent to the path and

Each turn of the winding passes once through the area bounded by path 2 and total current through the area is NI, where N is the total number of turns in the windings.

Using Ampere's law