FARADAY ROTATION Gennady Voronov In this experiments we experimentally determine the Verdet constant of a glass rod with specification SF-59 to be rad/mT*cm. We observed a rotation in the plane of polarization of a linearly polarized monochromatic beam of light, which we obtained by shining a red laser through an analyzer polaroid. The rotation occurs as the beam propagates through a dielectric in the presence of a magnetic field and the magnitude of the rotation is related to both the Verdet constant and projection of the magnetic field in the direction propagation of the laser beam. In this manner we determine the Verdet constant of SF-59. Finally to achieve a greater accuracy we model the magnetic field to take into account that the field is non constant inside the solenoid. Finally we conduct the experiment once without a hood and once with a hood to keep out stray light to check if it makes a difference. We discovered that the hood did indeed make a difference. APPARATUS The goal of the experiment is to determine the Verdet constant of transparent dielectric. The experimental apparatus that allows us to determine the above value consists of 4 basic components: a linearly polarized monochromatic light source, an analyzer polaroid, a solenoid, and an optical detector. A linearly polarized monochromatic light source is obtained by shining a red laser through an analyzer polarizer. The solenoid is 150mm long, and 10 layers. The solenoid is constructed with #18 double insulated copper wire. Finally the detector is simply a photodiode. The apparatus setup is shown in Fig. 2. The Faraday effect, first observed by Michael Faraday in 1845, provided the first evidence of a connection between light and magnetism. Faraday found that linearly polarized light propagating through a dielectric parallel to a static magnetic field had an observable rotation in its plane of polarization. This is represented pictorially below in Fig. 1. Fig.1 Plane of polarization of light rotated as it propagates through dielectric in presence of parallel magnetic field. This effect is a result of a magnetic field induced circular birefringence in the linearly polarized laser beam in a dielectric material. In a constant magnetic field the rotation may be quantitatively described by where Δθ is the rotation angle of the plane of polarization, ν is the Verdet constant, B is the magnitude of the magnetic field, and l is the length of the glass rod. The Verdet constant is a material property of dielectrics and it is the constant of proportionality between the angle of rotation and the magnitude of the magnetic field. The Faraday effect is used in the fabrication of optical isolators to prevent unwanted backreflections. The susceptibility of materials and carrier densities in semiconductors may also be inferred by measuring the magnitude of the Faraday rotation. Fig 2. Note the apparatus use specifically in our experiment uses a red laser as a light source and a glass rod for the dielectric. PROCEDURE In order to determine the Verdet constant of material SF-59, we propagate a laser beam through this material in the presence of a known magnetic field. We then measure the rotation of the plane of polarization using an analyzer polaroid. To determine the Verdet constant accurately we do not treat the solenoid as ideal. By doing so the magnetic field that induces the rotation is non constant. Equation 1 generalizes as such (1) To determine the Verdet constant, then we must model the magnetic field inside the solenoid. We do this by treating the solenoid as a series of current loops with 10 radial layers. Then we calculate the field at a point inside the solenoid by summing over every loop, the contribution to the field from each particular loop. The field as well as the integral in equation 2 are evaluated numerically. One additional piece is necessary to compute the magnetic field, the current running through the solenoid. We can continuously measure the voltage across the solenoid and via, (2) (3) where V is the voltage, R is the resistance, and I is the current, we may determine the current if we know R. We measured the voltage and current of across the solenoid for a few points and then plot voltage as a function of current. From this we can conclude that the resistance is Ω. The solenoid heats up when current is running through it, increasing the resistance. However we turn on the current for short enough time intervals so that the increase in resistance is negligible. Now all we must do is measure voltage across the solenoid and rotation of the plane of polarization and we may obtain the Verdet constant. We plot our results below. First we show the determined Verdet constant as a function of average B field over the glass dielectric. Our first set of data was taken without a hood. Fig 3. Measured voltage as a function of current used in determining the resistance of solenoid. Fig 4. Verdet constant as a function of Avg. B field without hood. Fig 5. Verdet constant as a function of Avg. B field with hood. First thing we notice is that the Verdet constant varies strongly with a weak magnetic field. We think this is a result of electronic noise in the photodetector which becomes insignificant as we raise the B field. The Verdet constant settles to a constant for values of avg. B field greater than We determine the Verdet constant to be rad/mT*cm with a variance of rad/mT*cm. We repeated the measurement with a hood to reduce stray light entering the photodetector and as we did for the measurement with a hood we determine a Verdet constant of rad/mT*cm with a variance of rad/mT*cm. The variance with a hood is greater however that is more likely due to the fact that the hood measurements were taken after the no hood measurements, where the measurements were taken will little time in between and the solenoid over a few runs did heat up enough for a non-negligible change in the resistance. We measured a Verdet constant of rad/mT*cm and rad/mT*cm with and without a hood respectively. Clearly using the hood impacts the determination of the constant. In the future this experiment could be improved by letting the solenoid cool in between data runs to insure that the resistance remains the same throughout the experiment. REFRENCES: Frank J. Loeffler, “A Faraday rotation experiment for undergraduate physics laboratory.” Am. J. Phys. 51 (7) July 1983 Aloke Jain et. al., “A simple experiment for determining Verdet constant using alternating current magnetic fields.” Am. J. Phys. 67(8) August Meyrath, Todd, “Electromagnet Design Basics for Cold Atom Experiments.” Phys. Rev. A 35 November 2004.