Faraday Report

8/17/2019 Faraday Report

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Faraday’s Effect

Jishnu Rajendran

1211027

School Of Physical Sciences,NISER

March 17, 2016


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Introduction

Faraday rotation is a magneto-optical phenomenon—that is, an interaction between light and a magnetic fieldin a medium. The Faraday effect causes a rotation of the plane of polarization which is linearly proportional tothe component of the magnetic field in the direction of propagation.Any transparent substance whose moleculesare not mirror symmetric and which is not a racemic mixture of its enantiomorphic stereoisomers is ’opticallyactive’ to some degree, i.e. it exhibits the phenomenon of circular birefringence whereby the plane of polarization

of a beam of linearly polarized light passing through the substance is rotated by an amount proportional to thedistance traveled. Dextrose (otherwise known as d-glucose, one of the 16 stereoisomers of pentahydroxyaldehyde)and d-tartaric acid (used in soft drinks) are nutritious examples of optically active substances. Pure or insolution, they both rotate the plane of polarization counterclockwise relative to the direction of propagation;their mirror images do the opposite and, incidentally, are not nutritious. Some crystals, e.g. natural quartz,exhibit both optical activity and birefringence. In 1845 Faraday (1846) discovered that a magnetic field in thedirection of propagation of a light beam in a transparent medium produces the effects of circular birefringence.Thus he established for the first time a direct connection between optics and electromagnetism - a connectionhe had long suspected to exist and for which he had searched for many years. However, a half century passedbefore an explanation of the Faraday effect was formulated in terms of Maxwell’s electromagnetic theory of light(developed in the 1860’s) and the concept of the atomicity of charge which had evolved during the 19th centuryunder the influence of electrochemistry and been validated by Thomson’s discovery of the electron in 1897.

Objectives

• To determine the magnetic flux density between the pole pieces using axial hall probe of the teslameterfor different coil currents. The mean flux density is calculated by numerical integration and the ratiomaximum flux-density over mean flux density established.

• To measure the maximum flux density as a function of the coil current and to establish the relationbetween the mean flux density and coil current anticipating that the ratio found in aim no. 1 remainsfixed.

• To determine the angle of rotation as a function of the mean flux-density using different colour filters. Tocalculate the corresponding verdet’s constant in each case.

• To evaluate verdet’s constant as a function of wavelength.

Apparatus

1. Glass rod(SF6) 30mm

2. Coil, 600 turns - 2 no.s

3. Iron core, U-shaped, laminated

4. Halogen lamp5. Double condenser, f = 60mm

6. Var. transformer, 25V AC/20V DC, 12A

7. Commutator switch

8. Tesla-meter and Axial Hall probe

9. Colour filter(440nm, 505nm, 525nm, 595nm)

10. Polarizing filter with Vernier

11. Lens, f = 150mm

12. Lens holders(5) and slide mount

13. connecting cords, 750mm - 3 no.s

14. Multimeter

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Figure 1: Addition of right and left circularly polarized components on two planes defined by x = 0 and x = D,respectively. Because nl > nr, E l is more retarded in phase at x = D relative to x = 0 than E r, with the resultthat their sum at x = D is rotated clockwise by the angle ψ = π(nl − nr)D/λ.

Theory

Representation of Linearly polarized light as a superposition of circularly polarizedcomponents

The electric field vector E of a plane wave of linearly polarized light of frequency traveling in the +X directionwith its polarization in the Y direction can be represented by the expression

E(x, t) = Er + El (1)

whereEr(x, t) = {0, (A/2) cos[ω(t − nx/c)], (A/2)sin[ω(t − nx/c)]} (2)

andEl(x, t) = {0, (A/2) cos[ω(t − nx/c)],−(A/2)sin[ω(t − nx/c)]} (3)

represent the electric field vectors in plane waves of right and left circularly polarized light, respectively. Theamplitude of E is A, and the velocity of propagation is c/n where n is the index of refraction of the medium.Thus a beam of linearly polarized light can be considered as a linear superposition of right and left circularlypolarized components.

Under certain circumstances the velocities of right and left circularly polarized light may be different, asin d-glucose or in a substance in which there is a magnetic field. In such a circumstance one can calculateseparately the propagation of the two components and then recombine them to obtain the electric vector of theresultant linearly polarized wave. The net effect, as we now show, is a rotation of the plane of polarization. Toindicate a difference in velocities for right and left circularly polarized waves we replace the index n in equations2 and 3 by nr and nl, respectively. As illustrated in Figure 1, the electric vector on the y-z plane at x = D isthen

E(D, t) = Er(D, t) + El(D, t) (4)

Using the identities

cos(α) + cos(β ) = 2 cos(α/2 + β/2) cos(α/2 − β/2) (5)

andsin(α) − sin(β ) = 2cos(α/2 + β/2) sin(α/2 − β/2) (6)

we find for the electric field at x = D the expression

E(D, t) = {0, A cos[ω(t− (nr + nl)D/2c)] cos[ω(nl − nr)D/2c], A cos[ω(t− (nr + nl)D/2c)]sin[ω(nl −nr)D/2c]}(7)

This represents an oscillating electric field of amplitude A inclined with respect to the y- axis by an angle

ψ = tan−1(E zE y

) = ω(nl − nr)D

2c =

π(nl − nr)D

λ (8)

where λ is the wavelength of the light in vacuum. Thus circular birefringence can be explained as an effect of

a difference in the propagation velocities of right and left circularly polarized light. The difference is generallyvery small, amounting to only a few parts per million in optically active materials as well as in the Faradayeffect at moderate magnetic field strengths. However, in a typical setup D/λ l so that even minute differencesbetween nl & nr yield values of ψ which are large enough to be measured accurately.

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Magnetic circular birefringence - Becquerel’s theory of the faraday effect

Experiments established early on that the Faraday rotation angle ψ is proportional to the product of D andthe magnetic field B. Thus

ψ = V ·D ·B (9)

where V is a proportionality factor called the Verdet constant. The question remained as to how one mightcalculate the value of V from the fundamental physics of light, magnetism, and matter.

Up to this point in our development we have only used the two ideas; 1)light is a linearly superposabletransverse wave with velocity c/n, and 2) nr = nl. The first step in understanding the Faraday effect in termsof electromagnetic theory and the atomic structure of matter was taken by H. Becquerel, in 1897. His theorywas based on the concept of the newly discovered electron, but it did not include the idea of quantized energystates introduced by Planck in 1900, or the concept of light quanta, developed by Einstein in 1905. Even thoughthe Becquerel theory does not make use of quantum principles, it does provide a simple conceptual frameworkfor a preliminary understanding of the phenomenon, and yields quantitative predictions for the Verdet constantthat are remarkable close to the measured values. The next substantial steps beyond this theory were not takenuntil the early 1930’s, but then only in gases (Serber, 1932; Van Vleck, 1932). A large amount of theoreticaland experimental work has been done on the Faraday effect in solids since 1960, because of its application tosemiconductors and the construction of microwave switching devices. Recent theoretical treatments are verycomplex, as a perusal of reviews (e.g. Mavroides 1972) will show. Incidentally, the theory of the Faraday effectin ionized gas, where the active agents are free electrons, is not so complicated. The effect is important in radioastronomy because it provides a measure of the average value of the product Brne along the line of sight to asource of linearly polarized radio emission, where Br is the component of the magnetic field along the line of sight and ne is the free electron density (typically Br ≈ 10−6 gauss, ne ≈ 0.1cm−3). Following Becquerel andsubsequent presentations of his theory (see e.g. Rossi 1957), we assume that a transparent material containsparticles of mass m and charge q embedded in a continuum of opposite charge and restrained by elastic forceto vibrate about fixed sites. We consider the situation in which there is a steady magnetic field B of magnitudeB in the +X direction and a plane right circularly polarized electromagnetic wave which, at a given point,produces a rapidly varying electric field E of constant magnitude E rotating clockwise with respect to B in aplane perpendicular to B. In the steady state the charged particles move in a circle of radius r governed by theequation (cgs units)

−mω2r = −kr + Eq + Bqωr/c (10)

where k is the ”spring constant” of the restraining force. For left circularly polarized light the sign of magneticterm is reversed. Equation 10 is readily resolved to obtain for r the formula

r = (Eq/m)/(ω20 − ω

2 −Bqω/mc) (11)

where ω0 = (k/m)1/2.

Figure 2: The particle, with mass m and charge q , is fixed to the end of a spring, with spring constant k,attached rigidly to the x axis. The directions of light propagation and the B field are parallel to the x axis; theE field is perpendicular to the x axis and rotates in the y z plane.

Displacement of a charge q from its oppositely charged equilibrium site by the distance r creates an electricdipole of magnitude qr. If there are N such dipoles per unit volume, then the polarization P has the magnitudeN qr, and the permittivity is ε = (1 + 4πP/E ). Therefore, in a dielectric where the permeability µ = 1 and thepropagation velocity v = c/ε1/2 and in the presence of a magnetic field, we obtain for the index of refractionfor right circularly polarized light the expression

nr(ω) = c/v = [1 + (4πN q 2/m)/(ω20 − ω

2 − Bqω/mc)]1/2 (12)

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and for left circularly polarized light

nl(ω) = c/v = [1 + (4πN q 2/m)/(ω2

0 − ω2 + Bqω/mc)]1/2 (13)

Equations 12 and 13 show that the indices ”blow up” as the frequency of the light approaches a valuesuch that ω2

0 − ω2 ± Bωq/mc = 0. (One can avoid this unphysical consequence of our simple assumptions by

adding to the equation of motion a damping term representing a drag force proportional to the velocity of thecharged particle. Such a term can give a classical account of the phenomenon of absorption. Both the indicesof refraction and the absorption coefficient have maxima near ω0. Fortunately, omission of a damping term hasvery little effect on the accuracy of the theory at frequencies far from ω0, as in the present case.) The sign of the effect of the magnetic field on the indices of refraction for circularly polarized light depends on the sign of the product Bωq . In particular, for right circular polarization with B in the direction of propagation ( Bx > 0)and negatively charged particles, Bωq < 0, which implies a decrease in nrandacorrespondingincreaseinnl .Therefore, under these assumptions (nl−nr) > 0 and, according to equation 8, ψ > 0. The change of nr causedby turning on a magnetic field B is equal to the change of nr when the frequency of the light is changed fromω to ω + ∆ωr , where ∆ωr is defined by the quadratic equation

ω2 + Bωq/mc = (ω + ∆ωr)2 (14)

The solution of this equation is ∆ωr = Bq/2mc. Similarly,

ω2 −Bωq/mc = (ω + ∆ωl)2 (15)

and ∆ωl = −Bq/2mc. These ∆ωs are very small compared to ω. Thus the difference in indices can be written

with high accuracy as

nl − nr = (dn/dω)(∆ωl −∆ωr) = (dn/dλ)(λ2/2πc)(Bq/mc) (16)

where λ = 2πc/ω is the vacuum wavelength of light. Combining equations 8 and 16 we obtain for the angle of faraday rotation

ψ = dn

dλ ·

λ

2c2 ·

q

mD · B (17)

Thus the Verdet constant is related to the constants of the interacting particles by the formula

V = dndλ · λ

2c2 · q

m (18)

Equation 18 is the formula derived by H. Becquerel. It was characterized by Van Vleck (1932) as ”rathertoo simple”. Indeed, the quantum treatment of the problem in even the simplest of materials is complicated,and the results depend critically on the nature of the medium, i.e. whether it is dielectric, semiconducting,diamagnetic, paramagnetic, ferromagnetic, etc. Of course this means that measurements of the Faraday effectcan be of value in the study of the electronic structure of matter provided an adequate theoretical frameworkfor its interpretation is available. We will bury the complications by defining a constant C such that equation18 is written as

V = −dn

dλ ·

λ

2c2 · C

e

me(19)

where e and me are the charge and mass of the free electron. It is reassuring that for visible light in ordinarymolecular hydrogen gas, H 2, the value of C is 0.99. For other substances and other spectral bands the valuesof C can be substantially different from unity and, in some cases, even negative. In the case of the interstellarmedium where the effect is due to the presence of free electrons one has ω0 = 0 and C = 1.

There exists a relation between the wavelength and the Verdet’s constant

V (λ) = π

λ

n2(λ) − 1

n(λ)

A +

B

λ2 − λ20

(20)

where A = 15.71 × 10−7 rad · T −1, B = 6.34 × 1019 rad · m2 · T −1, λ0 = 156.4nm and n(λ) is the refractiveindex of the material at a given wavelength.

Experimental Set-up and Procedure

The equipment was set up as shown in 3 and 4. The 50 W experimental lamp was supplied by the 12 V ACconstant volt- age source. The DC output of the power supply which was variable between 0 and 20 V DC,was connected via an multimeter to the coils of the electromagnet which were in series. The electromagnet

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Figure 3: Experimental set-up for the quantitative treatment of the Faraday Effect

Figure 4: 1 Condenser,f =6mm, 2 Coloured glass, 3 Polariser, 4 Test specimen(flint glass SF6), 5 Analyser,6 Lens, f =15cm, 7 Screen

needed for the experiment was constructed from a laminaded U-shaped iron core, two 600-turn coils and thedrilled pole pieces, the electromagnet was then being arranged in a stable manner on the table on rod.

After the flux-density distribution was measured, the 30 mm long flint glass cylinder was inserted in the polepiece holes and the jack was raised such that the magnet was interpolated in the experimental set-up betweenthe two polarisation filters.

First of all, the experiment lamp, fitted with a condensor having a focal length of 6 cm, was fixed on theoptical bench. This was followed by the diaphragm holder with coloured glass, two polarisation filters and alens holder with a mounted lens of f = 15cm. A translucent screen was put in a slide mount at the end of theoptical bench. The ray paths have been traced in 4.

The planes of polarisation of the two polarisation filters were initially arranged in parallel. The experimentlamp was switched on and the incandescent lamp moved into the housing until the image of the lamp filamentwas in the ob jective lens plane. The electromagnet was then moved into the path of the image rays and waspositioned so that the pole piece holes with the inserted glass cylinder are aligned with the optical axis. Bysliding the objective lens along the optical bench, the face of the glass cylinder was sharply projected onto thescreen. Adjustment was completed by inserting the coloured glass in the diaphragm holder. The polarizing filterpermanently had a position of +900. In this case the analyser had a position of 00 ±∆φ for perfect extinctionwith ∆φ being a function of the coil current, respectively of the mean flux-density.

The maximum coils current under permanent use was 3.5 A. However, the current could be increased up to4 A for a very few minutes without risk of damage to the coils by overheating.

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Observations and Calculations

Table 1: Magnetic flux density as a function of position between the two pole pieces of magnetsPosition of hall probe(in cm) 0.5A 1A 1.5A 2A 2.5A 3A

-1 -0.6 -0.3 -0.2 -0.1 0 0.2-0.5 0.3 1.2 1.7 3.1 3.8 5

0 5.9 11.7 19.1 21.6 31.3 400.5 21.6 47.8 71.6 90.5 114.9 1401 31.3 64.9 93.5 122.4 150.5 179.9

1.5 31.3 64.1 93.6 123.5 151.4 180.52 30.1 61.5 89.9 119.4 143.4 171.9

2.5 18.1 35.2 48.8 71.3 80.5 95.33 3.1 6 9.5 12 10.2 12.1

3.5 0.2 0.2 0.6 1.1 1.5 1.54 -0.6 -0.5 -0.3 -0.3 -0.2 -0.2

Average 20.2 41.6 60.85714 80.1 97.45714 117.1

Figure 5: Variation of magnetic field with distance and current

Not all data points are mentioned

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Figure 6: Calibration curve for magnetic field as a fucntion of current


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Filter = 595 nm

Table 2: Data set for λ = 595 nmCurrent(A) Magnetic field(mT) δφ1 δφ2 ∆φ

0 0 0 0 0

0.5 20.2 0.4 0.6 0.51 41.6 1 0.8 0.91.5 60.857 1.8 1.6 1.72 80.1 2.2 2.2 2.2

2.5 97.457 3 3.2 3.13 117.1 4.2 3.8 4

Figure 7: linear variation of ∆φ with magnetic field for λ = 595nm.


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Filter=580nm

Table 3: Data set for λ=580nmCurrent(A) Mag. Field(mT) δφ1 δφ2 ∆φ

0 0 0 0 0

0.5 20.2 1.2 0.8 11 41.6 1.4 1.8 1.61.5 60.857 2.6 3.2 2.92 80.1 3.6 4 3.8

2.5 97.457 4.6 4.6 4.63 117.1 4.6 5.2 4.9



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Filter = 525nm

Table 4: Data set for λ=525nmCurrent(A) Mag. field(mT) δφ1 δφ2 ∆φ

0 0 0 0 0

0.5 20.2 1.2 0.8 11 41.6 2.2. 1.6 1.91.5 60.857 3.6 2.4 32 80.1 5.2 3.8 4.5

2.5 97.457 6 5 5.53 117.1 7 5.2 6.1



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Table 5: Data set for λ = 505nmCurrent(A) Mag. field(mT) δφ1 δφ2 ∆φ

0 0 0 0 00.5 20.2 0.6 0.6 0.61 41.6 1.2 3 2.1

1.5 60.857 2.7 4 3.352 80.1 4 4.2 4.1

2.5 97.457 4.8 6 5.43 117.1 5.4 6.8 6.1


Determination of Verdet’s constant as a function of wavelength

The graphs of the above section give us the values of the slope which is the product of the Verdet’s constantand the length of the glass rod. To obtain the Verdet’s constant, we divide the slopes with the distance of theglass rod which is 30 mm.

Results

The change in angle of polarization was seen to be increasing linearly with increasing magnetic field.The verdet’s constant was seen to be decreasing with increasing magnetic field.The change in angle of polarisation became opposite when the magnetic field polarity was flipped.


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Table 6: Calculated verdet’s constant as a function of wavelengthWavelength(nm) Slope in (deg. T −1) Verdet’s const. (deg · T −1 · m−1)

505 55.43 1847.67525 54.61 1820.33580 42.48 1416595 35.36 1178.7

Figure 11: variation of Verdet’s constant with wavelength

Sources of error

This experiment relies on detecting changes in the intensity of the spot of light formed after it emerges out of the polariser through the lens. Hence the maximum error can occur in the measurement of slight changes instate of polarization by our bare eyes. This error can be minimized if we use a highly sensitive detector whichcan project the images on a computer screen having enough resolving power to detect even slight changes in

intensity. More errors can occur due to loose fittings. The polariser and analyser rotators may become loosein the process of making measurements. It should be minimized by using tight screws and also taking care notto turn it in opposite directions alternately. Even a slight loosening can change the angle that we measure onthe analyser. At no point the intensity drops to zero (we can see a bluish dim light). This is a manufacturingdefect of the polariser and analyser.

Precautions

1. Care should be taken while handling the glass rod. Its ends should be clean.

2. The polariser and analyser should be fixed firmly on the optical bench.

3. At high currents, the coil will heat up fast, the current has to be manually adjusted continuously.

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