6. What is Hall effect?

Derive the mathematical expression for hall effect by

considering the small piece of conductor.

Half Effect
“When a current carrying conductor is placed inside
the magnetic field potential difference is generated in a direction
perpendicular to magnetic field and current”

When a sample of conductor carrying, current is placed in a uniform magnetic field perpendicular to
the direction of the current, a transverse field will be set up across the conductor. This was first
experimentally observed by Edwik H. Hall in 1879. The field developed across the conductor is
called Hall field and corresponding potential difference is called Hall voltage and its value is found
to depend on the magnetic field strength, nature of the materials and applied current. This type of
effect is called Hall effect. The principle of Hall effect is based on the simple dynamics of charges
moving in electromagnetic fields.

Explanation of Hall Effect

To explain Hall effect, consider a sample of a block of conductor of length l, width d and thickness t,
through which electric current I is supplied along x-axis as shown in figure 1. The flow of electron is
in the opposite direction to the conventional current. If the conductor is placed in a magnetic
field B along z-axis perpendicular to the direction of current, a force Bev then acts on each electrons
in the direction from top surface to the bottom of the sample. Thus electrons accumulate on the
bottom surface of the conductor which will make the surface negatively charged and top surface will
be charged positively. Hence a potential difference opposes the flow of electrons.
The flow ceases when the potential difference across the conductor along y-axis reaches a particular
value i.e. Hall voltage (VH), which may be measured by using a high impedance voltmeter. If d be the
width of the slab of the sample, then the electric field or the Hall Field (EH) will be setup across the
sample. Hence at equilibrium condition, the force downwards due to magnetic field will be equal to
the upward electric force, i.e.

            (i)

Here v is drift velocity, which can be expressed by the relation

I = -n e v A                         (ii)

Where n is number of electrons per unit volume and A is the area of cross-section of the conductor.
The area of the cross-section in the sample is A = td. So from equation (i) and (ii) we get

          (iii)

We can take some typical values for copper and silicone to see the order of magnitude of VH. For
copper n=1029m-3 and for Si, n = 1=25 m-3. Hence the Hall voltage at B = 1T and i=10A and t = 1
mm for copper and Silicone are, 0.6µV and 6 mV respectively. The Hall voltage is much more
measurable in semiconductor than in metal i.e. Hall effect is more effective in semiconductor.

Recalling equation (iii) and expressing in terms of current density and Hall field we get,

Where   is called Hall Coefficient (RH). It is negative for free electron and positive for holes in
semiconductors. In some cases, it has been found that RH is positive for metal. It also implies that the
charge carriers are positive rather than negative. Hence we have

       (iv)

f the steady electric field E is maintained in a conductor by applying a external voltage across it, the
carriers of current attains a drift velocity v. The drift velocity acquired in unit applied electric field is
known as the mobility of the carrier and is denoted by µH and is also called Hall mobility. So we have

 (v)

Thus by measuring the resistivity of the materials and knowing the Hall coefficient, density along y-
axis and current density along x-axis. This ratio is called Hall angle. Hence we have,

        (vi)

The Hall angle measures the average number of radians traversed by a particle between collisions.
Again, from equation (iii), we get

     (vii)

The quantity R has dimension of resistance, through it is not resistance in conventional sense. It is

commonly called Hall resistance. From this relation it is expected to increase Hall resistance linearly
with the increase of magnetic field, however, German Physicist Klaus Von Klitizing in 1980 in his
experiment showed that the Hall resistance did not increase linearly with the field, instead, the plot
showed a series of stair steps as shown in figure 2. Such effect has become known as the quantized
Hall effect and Klaus was awarded the 1985 Nobel Prize in Physics for his discovery.
7. What is Faraday’s Law of electromagnetic induction, explain its
mathematical relation in detail

Faraday's law (electromagnetism)

In physics, in particular in the theory of electromagnetism, Faraday's law of induction states that a
change in magnetic flux generates an electromotive force (EMF, voltage difference). The law is named
after the English scientist Michael Faraday, who discovered in 1831 on basis of observations that a
change in a magnetic field induces an electric current. This is the phenomenon of electromagnetic
induction.

Some thirty years after Faraday's discovery, between 1861 and 1864, the Scottish mathematical
physicist James Clerk Maxwell formulated the mathematical expression relating the change in magnetic
flux to the induced EMF. This relationship, known as Faraday's law of induction (to distinguish it
from Faraday's laws of electrolysis), states that the EMF induced in a circuit is proportional to the rate of
decrease of the magnetic flux that cuts across the circuit. By Ohm's law an EMF induces an electric
current in a conductor.

When one rotates a circuit in a static homogeneous magnetic field, the magnetic flux cutting across the
circuit is changed. Hence this rotation generates an electric current in the circuit, which means that the
work done by rotating the circuit inside a magnetic field is converted into an electric current. Thus,
Faraday's law is the theoretical basis of the dynamo and the electric generator.

Mathematical formulation

Faraday's law of electromagnetic induction relates the electromotive force (EMF)   to the time derivative
of the magnetic flux Φ. The law reads

where k = 1 for SI units and one over c (the speed of light) for Gaussian units. The EMF is defined
as
where the electric field E is integrated around a closed path C. The magnetic flux Φ through a
surface S that has C as boundary is defined as the surface integral,

where dS is a vector normal (perpendicular) to the infinitesimal surface element dS and dS is of length dS.
The dot stands for the inner product between the magnetic induction B and dS.

In vacuum the magnetic induction B is proportional to the magnetic field H. (In SI units: B = μ0 H with
μ0 the magnetic constant of the vacuum; in Gaussian units: B = H.)

If, in the definition of the EMF, C is a conducting loop, then under influence of the EMF a current iind will
run through it. The minus sign in Faraday's law has the consequence that the magnetic field generated
by iind opposes the change in Φ; this phenomenon is known as Lenz' law.

If the surface S is constant in size and direction, a change in Φ is solely due to a change in B.

The magnetic flux can be varied by changing the angle between the field and surface. For the simple case
of a planar loop bordering a constant area A, and a homogeneous field B of strength B, the flux is given
by

where   is a unit vector normal (perpendicular) to the plane S bordered by the loop. The initial condition
α(0) = 0° is applied. Clearly, the flux is maximum, Φ(0), if α = 0°, i.e., if B is perpendicular to S. In the
perpendicular case all of B passes through the surface. When B is parallel to surface, α = 90°, then there is
no flux through the surface, Φ = 0, because B only grazes the surface.

When α(t) becomes a function of time by rotation of the loop, Faraday's law shows that an EMF is
induced of the form,
When the rotation is uniform, i.e., α(t) linear in time t, it follows that the EMF has a sine dependence on
time and an alternating current will be induced.

The flux through two surfaces that together form a closed surface is equal because of Gauss' law. Indeed,
in the figure on the right the surfaces S1 and S2, which have the boundary C in common, form together a
closed surface. Hence Gauss' law states that

where the minus sign of the first term is due to the fact that the flux is into the volume enveloped by the
two surfaces. It follows that

and that the magnetic flux Φ can be computed with respect to any surface that has C as boundary.

Connection to Maxwell's equation

Application of Stokes' theorem gives

where S is a surface that has C as boundary. From rewriting Faraday's law as follows,

and the fact that S is arbitrary, we may conclude

which is one of Maxwell's equations. Recall that k = 1 for SI units and k = 1/c for Gaussian units.
8. What is Lenz’s Law; explain how rate of change of change of magnetic flux
opposes induced emf (electromotive force)?

Lenz's law
“ states that the direction of an induced e.m.f. will be such that if it were to cause a
current to flow in a conductor in an external circuit, then that current would
generate a field that would oppose the change that created it.”

Consider a solenoid and a bar magnet, as in Figure 1. Moving the bar magnet into the solenoid
induces an e.m.f. in the solenoid (according to Faraday's law), and because the circuit is closed, a
current flows and a magnetic field is induced.

First think about what would happen if the opposite of Lenz's Law were true. Then the direction
of the induced e.m.f. would be such that its magnetic field at the end of the solenoid nearest the
N pole of the magnet, would resemble that of a south pole, and so the bar magnet would
experience an attractive force directed into the solenoid. This would cause the bar magnet to
accelerate, increasing the rate of change of magnetic flux linkage in the coil and consequently
increasing the induced e.m.f., the current and the attractive force.

In this scenario energy is being produced from nothing. Due to conservation of energy this is not
possible and therefore the magnetic field due to the induced e.m.f. in the solenoid must oppose
the magnetic field due to the bar magnet, as predicted by Lenz's law, as in Figure 1. This
illustrates that Lenz's law is a result of energy conservation.

Lenz's law can be incorporated into Faraday's law to give

induced e.m.f.= −𝑁ΔΦ/Δ𝑡

since the sense of the e.m.f is opposite to the direction of the rate of change of magnetic flux
linkage.

Figure 1

Diagram showing induced e.m.f. producing a magnetic field that opposes the magnetic field of the
bar magnet as predicted by Lenz's law.