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Lorentz Force from Magnetic Field
The Lorentz Force on an electric charge occurs when the charge moves through a magnetic field. This force is
perpendicular to the direction of the charge and also perpendicular to the direction of the magnetic field. It is
a vector combination of the two forces.
This Lorentz Force was first formulated by James Clark Maxwell in 1865, then by Oliver Heaviside in 1889,
and finally by Hendrick Lorentz in 1891.
Since electrons are moving in a wire, this force also applies to an electric current. The direction of the force is
demonstrated by the Right Hand Rule.
Cause of Lorentz Force
A magnetic field is created by the motion of an electrically charged particle—such as a proton or electron. If
that electrical charge is moving through an external magnetic field, there will be a magnetic attraction or
repulsion force, depending on how the two magnetic fields interact.
(See Basics of Magnetism for more information.)
The relationship between the force on the moving particle, the velocity of the particle through the magnetic field, the
strength of that magnetic field and the force on the particle, and the angle between the directions of the particle and
magnetic field is:
F = qvB*sinθ
where:






F is the force in Newtons
q is the electric charge in Coulombs
v is the velocity of a positive (+) charge in meters/second
B is the strength of the magnetic field in Teslas
sinθ is the sine of the angle between v and B
θ is Greek letter theta
Note: The direction of the magnetic field B is defined as from N to S. Also, the direction of the electrical charge is
from (+) to (−). An electron would move in the opposite direction.
The Lorentz Force equation implies that if the velocity of the particle is zero (v = 0), then F = 0. Also, if the particle
is moving in a direction parallel to B, again F = 0.
Current through wire
Since an electrical current in a wire consists of moving electrons, the Lorentz Force also applies to a current
in a magnetic field. When the current is perpendicular to the direction of the magnetic field, the force
equation is:
F = BIL
where:



F is the force in Newtons
B is the strength of the magnetic field in Teslas
I is the electrical current in Amperes

L is the length of the wire through the magnetic field in meters
Note: Remember that the convention for current direction in the wire is opposite the direction of the motion of the
electrons.
Lorentz Force on wire in magnetic field
This force on the wire can be measured in an experiment.
Right Hand Rule
The direction of the Lorentz force for a given direction of current and magnetic field can be remembered by the
Right Hand Rule. If you took your right hand and stuck your thumb up, your forefinger (first finger) forward and
your second finger perpendicular to the other two, then the direction of the force would be as indicated in the
drawing below.
Right Hand Rule for force on moving charge through magnetic field
The Right Hand Rule is supposed to help you remember which way things are pointing for the force on a moving
charge. But personally, I think it is confusing. Still, you should be aware of it, because some teachers include it in
tests.
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The Lorentz force
The flow of an electric current down a conducting wire is ultimately due to the motion of electrically charged
particles (in most cases, electrons) through the conducting medium. It seems reasonable, therefore, that the force
exerted on the wire when it is placed in a magnetic field is really the resultant of the forces exerted on these moving
charges. Let us suppose that this is the case.
Let
be the (uniform) cross-sectional area of the wire, and let
conductor. Suppose that the mobile charges each have charge
be the number density of mobile charges in the
and velocity
. We must assume that the
conductor also contains stationary charges, of charge
and number density
(say), so that the net charge
density in the wire is zero. In most conductors, the mobile charges are electrons and the stationary charges are
atomic nuclei. The magnitude of the electric current flowing through the wire is simply the number of coulombs per
second which flow past a given point. In one second, a mobile charge moves a distance , so all of the charges
contained in a cylinder of cross-sectional area
and length
flow past a given point. Thus, the magnitude of the
current is
current is
. The direction of the current is the same as the direction of motion of the charges, so the vector
. According to Eq. (229), the force per unit length acting on the wire is
(232)
However, a unit length of the wire contains
moving charges. So, assuming that each charge is subject to an
equal force from the magnetic field (we have no reason to suppose otherwise), the force acting on an individual
charge is
(233)
We can combine this with Eq. (169) to give the force acting on a charge
field
and a magnetic field
:
moving with velocity
in an electric
(234)
This is called the Lorentz force law, after the Dutch physicist Hendrik Antoon Lorentz who first formulated it. The
electric force on a charged particle is parallel to the local electric field. The magnetic force, however, is
perpendicular to both the local magnetic field and the particle's direction of motion. No magnetic force is exerted on
a stationary charged particle.
The equation of motion of a free particle of charge
and mass
moving in electric and magnetic fields is
(235)
according to the Lorentz force law. This equation of motion was first verified in a famous experiment carried out by
the Cambridge physicist J.J. Thompson in 1897. Thompson was investigating cathode rays, a then mysterious form
of radiation emitted by a heated metal element held at a large negative voltage (i.e., a cathode) with respect to
another metal element (i.e., an anode) in an evacuated tube. German physicists held that cathode rays were a form of
electromagnetic radiation, whilst British and French physicists suspected that they were, in reality, a stream of
charged particles. Thompson was able to demonstrate that the latter view was correct. In Thompson's experiment,
the cathode rays passed though a region of ``crossed'' electric and magnetic fields (still in vacuum). The fields were
perpendicular to the original trajectory of the rays, and were also mutually perpendicular.
Furthermore, if a particle is subject to a force
done on the particle by the force is
and moves a distance
in a time interval
, then the work
(250)
The power input to the particle from the force field is
(251)
where
is the particle's velocity. It follows from the Lorentz force law, Eq. (234), that the power input to a
particle moving in electric and magnetic fields is
(252)
Note that a charged particle can gain (or lose) energy from an electric field, but not from a magnetic field. This is
because the magnetic force is always perpendicular to the particle's direction of motion, and, therefore, does no work
on the particle [see Eq. (250)]. Thus, in particle accelerators, magnetic fields are often used to guide particle motion
(e.g., in a circle) but the actual acceleration is performed by electric fields.
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Biot Savart Law
The mathematical expression for magnetic flux density was derived by Jean Baptiste Biot and Felix Savart. Talking
the deflection of a compass needle as a measure of the intensity of a current, varying in magnitude and shape, the
two scientists concluded that any current element projects into space a magnetic field, the magnetic flux density of
which dB, is directly proportional to the length of the element dl, the current I, the sine of the angle and θ between
direction of the current and the vector joining a given point of the field and the current element and is inversely
proportional to the square of the distance of the given point from the current element, r. This is Biot Savart law
statement.
Where, K is a constant, depends upon the
magnetic properties of the medium and system of the units employed. In SI system of unit,
Therefore, final Biot Savart law derivation is,
Let us consider a long wire
carrying an current I and also consider a point p. The wire is presented in the below picture by red color. Let us also
consider an infinitely small length of the wire dl at a distance r from the point P as shown. Here, r is a distance
vector which makes an angle θ with the direction of current in the infinitesimal portion of the wire.
If you try to visualize the condition, you can easily understand the magnetic field density at that point P due to that
infinitesimal length dl of wire is directly proportional to current carried by this portion of the wire. That means
current through this infinitesimal portion of the wire is increased the magnetic field density due to this infinitesimal
length of wire, at point P increases proportionally and if the current through this portion of wire is decreased the
magnetic field density at point P due to this infinitesimal length of wire decreases proportionally. As the current
through that infinitesimal length of wire is same as the current carried by the wire itself.
It is also very
natural to think that the magnetic field density at that point P due to that infinitesimal length dl of wire is inversely
proportional to the square of the straight distance from point P to center of dl. That means distance r of this
infinitesimal portion of the wire is increased the magnetic field density due to this infinitesimal length of wire, at
point P decreases and if the distance of this portion of wire from point P, is decreased, the magnetic field density at
point P due to this infinitesimal length of wire increases accordingly.
Lastly, field
density at that point P due to that infinitesimal portion of wire is also directly proportional to the actual length of the
infinitesimal length dl of wire. As θ be the angle between distance vector r and direction of current through this
infinitesimal portion of the wire. The component of dl directly facing perpendicular to the point P is dlsinθ,
Now combining these three statements, we can write,
This is the basic form of Biot Savart's Law Now putting the value of constant k (which we have already introduced
at the beginning of this article) in the above expression, we get
Here, μ0 used
in the expression of constant k is absolute permeability of air or vacuum and it's value is 4π10 -7 Wb/ A-m in SI
system of units. μr of the expression of constant k is relative permeability of the medium. Now, flux density(B) at
the point P due to total length of the current carrying conductor or wire can be represented as,
If D is the
perpendicular distance of the point P form the wire, then
Now, the expression
of flux density B at point P can be rewritten as,
As per the figure above,
Finally the expression of B comes as,
This angle θ depends upon the length of the wire and the
position of the point P. Say for certain limited length of the wire, angle θ as indicated in the figure above varies from
θ1 to θ2. Hence, flux density at point P due to total length of the conductor is,
Let's imagine the wire is infinitely long, then θ will vary from 0 to π that is θ1 = 0 to θ2 = π. Putting these two values
in the above final expression of Biot Savart law, we get,
This is nothing but the expression of Ampere's Law.
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AMPERES LAW
Introduction
The magnetic field at a distance r from a very long straight wire, carrying a steady current I, has a magnitude equal
to
(31.)
and a direction perpendicular to r and I. The path integral along a circle centered around the wire (see Figure 31.1) is
equal to
(31.2)
Here we have used the fact that the magnetic field is tangential at any point on the circular integration path.
Figure 31.1. Magnetic field generated by current.
Any arbitrary path can be thought of as a collection of radial segments (r changes and [theta] remains constant) and
circular segments ([theta] changes and r remains constant). For the radial segments the magnetic field will be
perpendicular to the displacement and the scalar product between the magnetic field and the displacement is zero.
Consider now a small circular segment of a trajectory around the wire (see Figure 31.2). The path integral along this
circular segment is equal to
(31.3)
Figure 31.2. Path integral along a small circular path.
Equation (31.3) shows that the contribution of this circular segment to the total path integral is independent of the
distance r and only depends on the change in the angle [Delta][theta]. For a closed path, the total change in angle
will be 2[pi], and eq.(31.3) can be rewritten as
(31.4)
This expression is Ampere's Law:
" The integral of B around any closed mathematical path equals u0 times the current intercepted by the area spanning
the path "
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ELECTROMAGNETIC INDUCTION
Motional emf
Figure 32.1 shows a rod, made of conducting material, being moved with a velocity v in a uniform magnetic field B.
The magnetic force acting on a free electron in the rod will be directed upwards and has a magnitude equal to
(32.1)
Figure 32.1. Moving conductor in magnetic field.
As a result of the magnetic force electrons will start to accumulate at the top of the rod. The charge distribution of
the rod will therefore change, and the top of the rod will have an excess of electrons (negative charge) while the
bottom of the rod will have a deficit of electrons (positive charge). This charge distribution will produce an electric
field in the rod. The strength of this electric field will increase until the electrostatic force produced by this field is
equal in magnitude to the magnetic force. As this point the upward flow of electrons will stop and
(32.2)
or
(32.3)
The induced electric field will generate a potential difference [Delta]V between the ends of the rod, equal to
(32.4)
where L is the length of the rod. If the ends of the rod are connected with a circuit providing a return path for the
accumulated charge, the rod will be a source of emf. Since the emf is associated with the motion of the rod through
the magnetic field it is called motional emf. Equation (32.4) shows that the magnitude of the emf is proportional to
the velocity v. Looking at Figure 32.1 we observe that vL is the area swept across by the rod per second. The
quantity BvL is the magnetic flux swept across by the rod per second. Thus
(32.5)
Although this formula was derived for the special case shown in Figure 32.1, it is valid in general. It holds for rods
and wires of arbitrary shape moving through arbitrary magnetic fields.
Equation (32.5) relates the induced emf to the rate at which the enclosed magnetic flux changes. In the system
shown in Figure 32.1 the enclosed flux changes due to the motion of the rod. The enclosed magnetic flux can also be
changed if the strength of the enclosed magnetic field changes. In both cases the result will be an induced emf. The
relation between the induced emf and the change in magnetic flux is known as Faraday's law of induction:
" The induced emf along a moving or changing mathematical path in a constant or changing magnetic field equals
the rate at which magnetic flux sweeps across the path. "
If we consider a closed path, Faraday's law can be stated as follows:
" The induced emf around a closed mathematical path in magnetic field is equal to the rate of change of the
magnetic flux intercepted by the area within the path "
or
(32.6)
The minus sign in eq.(32.6) indicates how polarity of the induced emf is related to the sign of the flux and to the rate
of change of flux. The sign of the flux is fixed by the right-hand rule:
" Curl the fingers of your right hand in the direction in which we are reckoning the emf around the path; the
magnetic flux is then positive if the magnetic field lines point in the direction of the thumb, and negative otherwise.
"
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THE DISPLACEMENT CURRENT AND MAXWELLS EQUATIONS
The displacement current
The calculation of the magnetic field of a current distribution can, in principle, be carried out using Ampere's law
which relates the path integral of the magnetic field around a closed path to the current intercepted by an arbitrary
surface that spans this path:
(35.1)
Ampere's law is independent of the shape of the surface chosen as long as the current flows along a continuous,
unbroken circuit. However, consider the case in which the current wire is broken and connected to a parallel-plate
capacitor (see Figure 35.1). A current will flow through the wire during the charging process of the capacitor. This
current will generate a magnetic field and if we are far away from the capacitor, this field should be very similar to
the magnetic field produced by an infinitely long, continuous, wire. However, the current intercepted by an arbitrary
surface now depends on the surface chosen. For example, the surface shown in Figure 35.1 does not intercept any
current. Clearly, Ampere's law can not be applied in this case to find the magnetic field generated by the current.
Figure 35.1. Ampere's law in a capacitor circuit.
Although the surface shown in Figure 35.1 does not intercept any current, it intercepts electric flux. Suppose the
capacitor is an ideal capacitor, with a homogeneous electric field E between the plates and no electric field outside
the plates. At a certain time t the charge on the capacitor plates is Q. If the plates have a surface area A then the
electric field between the plates is equal to
(35.2)
The electric field outside the capacitor is equal to zero. The electric flux, [Phi]E, intercepted by the surface shown in
Figure 35.1 is equal to
(35.3)
If a current I is flowing through the wire, then the charge on the capacitor plates will be time dependent. The electric
flux will therefore also be time dependent, and the rate of change of electric flux is equal to
(35.4)
The magnetic field around the wire can now be found by modifying Ampere's law
(35.5)
where [Phi]E is the electric flux through the surface indicated in Figure 35.1 In the most general case, the surface
spanned by the integration path of the magnetic field can intercept current and electric flux. In such a case, the
effects of the electric flux and the electric current must be combined, and Ampere's law becomes
(35.6)
The current I is the current intercepted by whatever surface is used in the calculation, and is not necessarily the same
as the current in the wires. Equation (35.6) is frequently written as
(35.7)
where Id is called the displacement current and is defined as
(35.8)
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