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\begin{document}
\title{Measurement of Faraday Rotation in SF57 glass at 670 nm \textbf{Third and Final Draft}}
\author{Ning Zhu}
\affil{Affiliation not available}
\author{Emily A Kaplan}
\affil{Affiliation not available}
\date{\today}
\maketitle
\section{Abstract}
We performed an experiment to measure the Faraday rotation of polarized light passing through a magnetic field, as well as measuring the Verdet constant of an SF57 glass tube with a length of $0.1$ m. Our results are consistent with the general idea of Faraday rotation, which suggests that linearly polarized light experiences rotation when applying a magnetic field. We used three different methods to find Verdet constants, which are Direct Fit, Slope Fit and Lock-in Method. The values we found are $21\pm 5 \frac{radians}{T \cdot m}$, $21.095\pm0.003 \frac{radians}{T \cdot m}$ and $20.43\pm0.06 \frac{radians}{T \cdot m}$ respectively, and those values are consistent with each other within uncertainty.
%Linearly polarized light from a HeNe laser was sent through a magnetic field created by a solenoid. The light then passed through an adjustable polarizer and was detected by a photodiode. Using the photodiode, we measured how the intensity of the light changed when turning the polarizer through 360 degrees. We performed this experiment both with and without a glass tube inside the solenoid. We plotted the data as a function of voltage versus angle in plotly in order to find the angle where the voltage measured by the photodiode changed most rapidly. By keeping the polarizer at its most sensitive angle, we plotted voltage versus magnetic field and measured how the voltage changed with magnetic field. We used the lock-in amplifier to make the voltage measurement more accurate.
\section{Aims} % Section titles are automatically converted to all-caps.
% Section numbering is automatic.
1. To observe Faraday effect in this lab, which says that the rotation of plane of polarization of light changes when applying a magnetic field, which can be described using Equation \ref{1}
\begin{equation}
\label{1}
I=I_{0}cos^{2}(\theta_1-\theta_0-\phi(B))
\end{equation}
Equation \ref{2} descries the transmission of polarized light through a second polarizer:
\begin{equation}
\label{2}
I=I_{0}cos^{2}(\theta_1-\theta_0)
\end{equation}
where $I_{0}$ is the intensity of the light after passing through the first polarizer and $I$ is the light intensity passing through both polarizers at angles $\theta_{1}$ and $\theta_{0}$.
%\newline
%2.To experimentally determine the Verdet constant of a glass tube, which describes the strength of Faraday effect for a particular material:
%$$\varphi_{B}=C_{v}BL$$
%where $\varphi$ is the shift in polarization, B is the strength of applied magnetic field and L is the length of the glass tube.
2.To experimentally determine the Verdet constant of a tube made of SF57 glass, which describes the strength of Faraday effect within the glass tube:
\begin{equation}
\label{3}
\varphi_{B}=C_{v}BL
\end{equation}
where $\varphi$ is the shift in polarization, $B$ is the strength of applied magnetic field and $L$ is the length of the glass tube. The magnetic field causes a change in polarization, and measuring the Verdet constant of the glass tells us how much change in polarization there was within the glass due to the magnetic field.
\section{Introduction}
The Faraday effect was first observed by Michael Faraday in 1845, before light and matter interaction was understood. Light waves contain both a magnetic field and electric field. The electric and magnetic fields are transversely polarized with respect to the direction of propagation. Linearly polarized light refers to light that is polarized in one plane, the most simple examples being vertical polarization and horizontal polarization. Vertically polarized light refers to light whose $E$ field vector oscillates in the vertical direction, and horizontally polarized light refers to lights whose $E$ field vector oscillates in the horizontal direction. Circular polarization occurs when light has two different polarizations orthogonal to each other, but with a phase difference of 90 degrees. The resulting polarization vector oscillates circularly and can be right- or left-handed, depending on whether the phase difference is +90 degrees or -90 degrees.\newline
When light passes through a magnetic field in certain media, propagating in the same direction as the field, the magnetic field can cause different refractive indices for right- and left-circularly polarized light. This causes the right and left polarized light to have different phases. Linearly polarized light can also be thought of as a superposition of right- and left-circularly polarized light, so when linearly polarized light passes through a magnetic field, the polarization of the light will have rotated by some angle
$$\varphi=\frac{\pi v}{c}L(n_{R}-n_{L})$$
Where c is the speed of light, $\nu$ is the frequency of the light, $n_{R}$ is the refractive index of the material for right polarized light, $n_{L}$ is the refractive index of the material for left polarized light, and L is the length of the material. The angle of rotation is also proportional to the magnetic field B, so it can also be described by Equation~\ref{3}.
\section{Method}
Our setup consisted of a solid state diode laser with a wavelength of $670$ nm, a solenoid with a coil constant of $11.1 \frac{mT}{A}$, a rotatable polarizer, and a photodiode. The current through the solenoid was controlled by a programmable bipolar operational power supply to produce either a DC current, AC current, or AC current with a DC offset. This allowed us to produce wither a constant magnetic field $B_{0}$ or a time varying magnetic field $B = B_{0} + B_{1} \cos(\omega t)$ as needed.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/P1060257/faraday1}
\caption{{\label{fig:YourApparatus}
A solid state diode laser beam passes through a polarizer, then passes through the solenoid. The solenoid can either contain nothing or contain an SF57 glass tube. The beam then passes through an adjustable polarizer and is detected by a photodiode.%
}}
\end{center}
\end{figure}
Linearly polarized light was sent through the solenoid containing a glass tube but without any current flowing through it, passed through the polarizer, and detected by the photodiode. We rotated the polarizer around $360$ degrees, using $15$ degree steps, and recorded the voltage read by the photodetector. We then repeated the previous procedure, but with a current of $-3.0$ amps running through the solenoid, therefore changing the magnetic field. Magnetic field can be calculated from current using Equation \ref{magneticfield}
\begin{equation}
\label{magneticfield}
B=11.1I \frac{mT}{A}
\end{equation}
We changed the magnetic field because we want to measure the field-induced-rotation in polarization angle of the light as it passes through the glass rod. This allows us to find the change in Verdet constant since Verdet constant is a measure of how large a change in polarization angle can be produced for a given field.
%ADD A PARAGRAPH EXPLAINING (1) WHY YOU WOULD VARY THE MAGNETIC FIELD (TO MEASURE THE FIELD-INDUCED-ROTATION IN POLARIZATION ANGLE OF THE LIGHT AS IT PASSES THROUGH THE GLASS ROD) AND (2) WHAT THAT ALLOWS YOU TO DO (DEMONSTRATE THAT MAGNETIC FIELD INDUCES A CHANGE, AND DETERMINE THE VERDET CONSTANT, WHICH IS A MEASURE OF HOW LARGE A CHANGE IN POLARIZATION ANGLE CAN BE PRODUCED FOR A GIVEN MAGNETIC FIELD).
%THIS PARAGRAPH SHOULD BE IN ANALYSIS: We then fit our data to a function of the form $V=V_{0}sin(\phi)^2$. We could find $\frac{\Delta V}{\delta \phi}$ by taking the derivative and using ph=45 degrees, and we could find DV by taking the difference of the voltage read by the photodetector when the laser is on (at maximum voltage read) and off. We could then find Dph by calculating how much the angle of maximum transmission through the polarizer shifter. With all of this information, we could find dB/dphand use the equation $\frac{\Delta B}{\Delta \phi}=\frac{1}{L}\times\frac{1}{C_{v}}$ to find the Verdet constant of the glass tube.
%The previous method mentioned allowed us to find the angle of greatest sensitivity of the polarizer, which is the point of inflection on the V vs ph graph. To verify the value of the Verdet constant, we used a second method where the polarized stayed at the angle of greatest sensitivity and we varied the current of the solenoid, going in 0.5A steps from -3A to 3A, thereby varying the magnetic field within the solenoid between - XXXXX mT and + XXXXX mT. We could then graph voltage vs magnetic field, which results in a linear graph. The slope of the graph is $\frac{\Delta V}{\Delta B}$. We calculated $\frac{\Delta V}{\Delta \phi}$ previously, so we can find $\frac{\Delta B}{\Delta\phi}$ and therefore the Verdet constant.
%A third method for finding the Verdet constant of a material involves using a lock-in amplifier to better read the signal from the photodiode. The lock-in we used was set at a frequency range was 100 Hz, the fine frequency range was 10 Hz. The lock-in amplifier had a gain of 10, the pre-amplifier a gain of 500, the bandpass filter a gain of 2, and the low pass filter a gain of 1.We used a function generator to sinusoidally drive the voltage going into the solenoid, using a frequency of 100 Hz. We varied the RMS value of the voltage driving the solenoid and recorded the RMS value of the corresponding photodiode voltage.
%In order to better read the signals coming out from the photodiode, we applied a lock-in amplifier. A lock-in amplifier, also known as phase sensitive amplifier, is a powerful AC voltmeter. It only picks up frequencies that are around the reference frequency. In this way, we can filter out any white noise that would come from other sources, such as the 60 Hz signal from overhead lights, because we can specify the frequency range which we want the lock-in to amplify.
%In this way, when the wavelength of our laser drifts away a little bit, the lock-in amplifier can fix this unexpected change.
In order to reduce the noise of our signal and therefore better read the signal coming from the photodiode, we used a lock-in amplifier. Lock-in amplifiers, also known as phase sensitive amplifiers, are powerful AC voltmeters. Lock-ins only pick up frequencies that are around our reference frequency, can be set to the frequency of our signal. Lock-ins can filter our white noise coming from other sources, such as the 60 Hz signal coming form the overhead lights. White noise is entirely random in both frequency and phase, so when averaging using the lock-in, the signal of the white noise averages to zero. Since lock-ins are also phase sensitive, and signal that is drifting in phase will be filtered out, which we would want because our signal should not be drifting in phase.
\section{Results}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/Voltage-vs-Angle-for-Faraday-Rotation/Faraday-Rotation-B-field-comparison-(1)}
\caption{{\label{fig:directmethoddata} Shown are the values of the photodiode voltage versus angle of the polarizer. The polarizer was rotated through $360^{\circ}$ with a step size of $15^{\circ}$.The red curve is voltage when a magnetic field of $-33.3$ mT was produced by the solenoid. The blue curve is the voltage when a magnetic field of $0$ mT was produced by the solenoid. There is a phase shift between the red line and the blue line, and this phase shift is used in our calculation for Direct Fit Method.
%Values of voltages obtained by changing the rotation angles over a 360 degrees range with a step size of 15 degrees. (0) means there is no magnetic field applied, and (-3) means there is a magnetic field applied, and the current goes through the solenoid is -3A.Replace this text with your caption%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/V-vs-B1/V-vs-B1}
\caption{{\label{fig:VB} Voltage coming out from the photodiode versus magnetic field with the polarizer at $\theta_1-\theta_2$ = 45 degrees. Theory predicts that there should be a linear relationship between the voltage and the magnetic field. The blue line represents a linear fit to our data points. The slope of this line ($\frac{dV}{dB}$) is used in our calculation for "Slope Fit Method", which equals to $0.000443\frac{V}{mT}$. From $\frac{dV}{dB}$, we can find $\frac{d\theta}{dB}$, which allows us to find the Verdet constant.%
}}
\end{center}
\end{figure}
\section{Analysis}
\subsection{Direct Fit Method}
We plot the photodiode voltage versus angle of the polarizer with and without a magnetic field applied as shown in figure \ref{fig:directmethoddata}. It can be seen from the graph that the curve shifts a little bit, which is a direct result of the magnetic field. By applying curve fit to our data points, we find the phase shift between two curves is $4$ degrees($d\theta$).
%We got 45 degrees after applying trigonometry to our previous fits since we used $cos$ function instead of $cos^2$ to fit our data in \ref{fig:directmethoddata}. This value is consistent with the value we got when we set the second derivative of $\frac{V}{\theta}$ to zero (the inflection point).
The current provided to induce a magnetic field is $-3$A, which can be translated to $dB$ using equation \ref{magneticfield}. With these numbers, we are able to calculate Verdet constant using equation :
\begin{equation}
c_v=\frac{1}{L}\frac{d\theta}{dB}
\end{equation}
where $L$ is the length of the solenoid, which is $0.1$m in our case.
\newline
We can't guarantee that we turn the polarizer to the angel we want exactly, and we estimate there will be an uncertainty of $0.5$ degrees in $d\theta$. By plugging in the maximum($4.5$ degrees) and minimum($3.5$ degrees) into our calculation, we are able to get an uncertainty of the Verdet constant.\newline
%23.58, 18.34
In this way, we get:
$$V_{c}=21\pm 5\frac{radians}{T \cdot m}$$
%We fit our data to a function of the form $V=V_{0}sin(\phi)^2$. We could find $\frac{\Delta V}{\delta \phi}$ by taking the derivative and using ph=45 degrees, and we could find DV by taking the difference of the voltage read by the photodetector when the laser is on (at maximum voltage read) and off. We could then find Dph by calculating how much the angle of maximum transmission through the polarizer shifter. With all of this information, we could find dB/dphand use the equation $\frac{\Delta B}{\Delta \phi}=\frac{1}{L}\times\frac{1}{C_{v}}$ to find the Verdet constant of the glass tube.
%$$V_{c}=\frac{1}{L}\times \frac{\Delta\theta}{\Delta B}$$
%$$\frac{\Delta\theta}{\Delta B}=\frac{\Delta V}{\Delta B} \times \frac{\Delta \theta}{\Delta V}$$
%$$\theta_{B}=104 degrees; \theta_{0}=108 degrees$$
%$$\Delta\theta=\theta_{B}-\theta_{0}=-0.069 radians$$
%$$\Delta B=-3A\times 11.1\frac{mT}{A}=-33.3mT$$
%$$\frac{\Delta\theta}{\Delta B}=\frac{-0.069radians}{-33.3mT}=0.00207\frac{radians}{mT}=2.07\frac{radians}{T}$$
%$$V_{c}=\frac{1}{L}\times \frac{\Delta\theta}{\Delta B}=\frac{1}{0.1m}\times 2.07\frac{radians}{T}=20.7\frac{radians}{T \cdot m}$$
\subsection{Slope Method}
The direct fit method allowed us to find the angle of greatest sensitivity of the polarizer by looking at the shift in phase as shown in Figure \ref{fig:directmethoddata}.
%We got 45 degrees after applying trigonometry to our previous fits since we used $cos$ function instead of $cos^2$ to fit our data in \ref{fig:directmethoddata}.
%which is the point of inflection on the $V$ vs $\theta$ graph.
%This value is consistent with the value we got when we set the second derivative of $\frac{V}{\theta}$ to zero (the inflection point).
To verify the value of the Verdet constant, we used a second method where the polarizer stayed at the angle of greatest sensitivity, which is $45$ degrees after we applied trigonometry to our previous fits. We did this because we used $cos$ function instead of $cos^2$ to fit our data in Figure \ref{fig:directmethoddata}, but we will be expressing $V(\theta)$ in the form of $cos^2$ in the rest of our report.
%We got the angle by setting the second derivative of $\frac{dV}{d\theta}$ to zero, which corresponds to the inflection point.
We then varied the current of the solenoid, going in 0.5A steps from $-3$A to $3$A, thereby varying the magnetic field within the solenoid between $-33.3$ mT and $+33.3$ mT. We could then graph voltage vs magnetic field, which results in a linear graph as shown in Figure \ref{fig:VB}. The slope of the graph is $\frac{dV}{dB}$. We calculated $\frac{dV}{d\theta}$ by taking the first derivative and setting angle to $45$ degrees as mentioned above, so we can find $\frac{dB}{d\theta}$ and therefore the Verdet constant. In the direct fit method, we estimated that we have an uncertainty of $0.05$ degrees for d$\theta$, and this uncertainty is associated with $45$ degrees here since we applied trigonometry as aforementioned. By plugging in the maximum ($45.5$ degrees) and minimum ($44.5$ degrees) into our calculation, we are able to get the uncertainty for Verdet constant.
%Initially, we fit our data to the form $V=V_{0}(sin(\theta_{1}+\theta_{2}))^2$, which using trigonometry, is $V=V_{0}\frac{(1-cos(\theta_{1}+\theta_{2})^2)}{2}$ (the actual equation that we fit to). However, we should have been fitting to the form $V=V_{0}cos(\theta_{1}-\theta_{2})^2$, which is $V=V_{0}\frac{(1+cos(\theta_{1}-\theta_{2})^2)}{2}$. Now,
%$$sin(\theta_{1}+\theta_{2})^2=sin2(th1-(-th2))$$
%2p-th2=th2' to account for the incorrect sign in our original equation.
%th2'-p=th0 to change from sine to cosine, so we do end up with the form
%V=V0(1/2)(1+cos(2(th1-th0)))
%which is V=V0cos2(th1-th2), giving us the form we should have initially fit to.
$$V_{c}=\frac{1}{L}\left(\frac{d\theta}{dV_{pd}}\right)_{\theta=45}\frac{dV_{pd}}{dB}$$
In this way, we get:
$$V_{c}=21.095\pm0.003\frac{radians}{T \cdot m}$$
%As shown in Figure 2:
%$$V=0.000443B+0.127$$
%Therefore:
%$$\frac{\Delta V}{\Delta B}=0.000443\frac{V}{mT}=0.443\frac{V}{T}$$
%As shown in Figure 1:
%$$V=\frac{0.021\times (1+cos(\frac{2\pi\times(\theta-162)}{180}))}{2}$$
%Taking the derivative at $\theta=117$ degrees:
%$$\frac{\Delta V}{\Delta \theta}=\frac{0.21}{2}\times(-sin(2\frac{\pi}{180}(-45)))=0.21V$$
%$$\frac{\Delta\theta}{\Delta B}=\frac{\Delta V}{\Delta B} \times \frac{\Delta \theta}{\Delta V}=0.443\frac{V}{T}\times\frac{1 radian}{0.21V}=2.1095\frac{radians}{T}$$
%$$V_{c}=\frac{1}{L}\times \frac{\Delta\theta}{\Delta B}=\frac{1}{0.1}\times2.1095\frac{radians}{T}=21.095\frac{radians}{T \cdot m}$$
\subsection{ Lock-in Method}
A third method for finding the Verdet constant of a material involves using a lock-in amplifier to better read the signal from the photodiode.
%The lock-in we used was set at a frequency range was 100 Hz and the fine frequency range adjustment was 10 Hz.
The lock-in amplifier had a gain of $10$, the pre-amplifier a gain of $500$, the bandpass filter a gain of $2$, and the low pass filter a gain of $1$. We used a function generator to sinusoidally drive the voltage going into the solenoid, using a frequency of $100$ Hz. We varied the RMS value of the AC voltage which was driving the solenoid and recorded the RMS value of the corresponding varying photodiode voltage.
The big idea of the calculation associated with this method is illustrated in Equation \ref{bigidea} :
\begin{equation}
\label{bigidea}
V_{c}=\frac{1}{L}\left(\frac{d\theta}{dV_{pd}}\right)_{\theta=45} \frac{V_{pd, RMS}}{B_{RMS}}
\end{equation}
In order to calculate the magnetic field ($B_{rms}$)that corresponds to the applied voltage, we use equation \ref{conversion} to first convert from voltage to current. We then apply equation \ref{magneticfield} to get the field strength.
\begin{equation}
\label{conversion}
I_{rms}=0.6137\times V_{rms}
\end{equation}
%$$B=11.1\times\frac{mT}{A}\times I$$
$V_{pd, RMS}$ represents the aforementioned RMS value of photodiode voltage. $\frac{d\theta}{dV_{pd}}$ is the same as in method 2.
\newline
We applied four different magnetic fields so we ended up with four Verdet constants. We took the average and used the maximum and minimum to calculate the uncertainty.
In this way, we get:
$$V_{c}=20.43\pm0.06\frac{radians}{T \cdot m}$$
%$$Gain=G_{preamplifier}\times G_{filter}\times G_{lock-in-amplifier}\times G_{lowpass filter}=500\times2\times10\times1=10000$$
%$$V_{Photodiode}=\frac{V_{output}\times1.11}{10000}$$
%We varied the RMS value of the voltage driving the solenoid and recorded the RMS value of the corresponding photodiode voltage.
\section{Discussion}
In addition to the uncertainty coming from the measurements, the fits of the data points might also result in some uncertainties. In order to see whether our fits in Figure \ref{fig:directmethoddata} and Figure \ref{fig:VB} are reliable or not, we calculated the $\tilde{\chi}^2$ for those fits. Based on our calculation, we can conclude that our obtained Verdet constants are obtained from accurate fits from our data.
For the fits shown in Figure \ref{fig:directmethoddata}, we obtained a $\tilde{\chi}^2=1.150$ for the fit of our data with no field applied,and $\tilde{\chi}^2=1.206$ for the fit of our data with a magnetic field applied.
For the fit shown in Figure \ref{fig:VB}, we obtained a $\tilde{\chi}^2=1.018$.
As a $\tilde{\chi}^2\approx1$ indicates a good fit, we can conclude that those fits are reliable. Therefore, we can assume that our curves fit our data well, giving us reasonable results for Verdet Constant.
The detailed calculation for $\tilde{\chi}^2$ can be found on \href{http://osf.io/yegjq/files//}{Open Science Framework}.
%The issues that had the most impact on our data occurred with the lock-in method. We initially took data for currents that were much too high for our solenoid, causing possible temperature effects in our data. We also saturated the lock-in amplifier without realizing it. For future experiments, even more accuracy could be achieved by measuring in smaller angle increments around the angle that was found to be most sensitive. It would also be interesting to test the methods we have used here using a material with a much smaller Verdet constant.
%Furthermore, had we had more time and not had to retake our data due to the temperature effects on the solenoid, we could have measured the Verdet constant for different materials.
\section{Conclusion}
It's experimentally confirmed that the plane of polarization changes when applying a magnetic field, and the Verdet constants obtained from three different methods are consistent with each other with their uncertainties. The weighted average of our three results is $21\frac{radians}{Tm}$, and the value quoted by TeachSpin for SF57 glass is $23\frac{radians}{Tm}$, which is close to our value but not quite. This does not mean our value is wrong since TeachSpin manual uses a different wavelength ($650$nm). The Verdet constant is wavelength dependent, as mentioned previously, so we think our values are correct because we did see a change of polarization and our values agree with each other within uncertainty. \cite{Melissinos_2003}
\section{Appendix}
%Figure one can be found here:https://plot.ly/36/~ekaplan39/
%Figure two can be found here: https://plot.ly/87/~ekaplan39/
%Excel spreadsheet can be found on the Google doc we shared with you
Important equations that we have used in our analysis for three methods:
\begin{eqnarray}
c_{v} & = & \frac{1}{L}\frac{d\theta}{dB} \\
method 1& = & \frac{1}{L}\frac{\Delta \theta}{\Delta B} \\
method 2& = & \frac{1}{L}\left(\frac{d\theta}{dV_{pd}}\right)_{at ?}\frac{dV_{pd}}{dB}\\
method 3& = & \frac{1}{L}\left(\frac{d\theta}{dV_{pd}}\right)_{at ?} \frac{V_{pd, RMS}}{B_{RMS}}
\end{eqnarray}
\subsection{Direct Method Calculation}
\begin{eqnarray}
V_{c}&=&\frac{1}{L}\cdot \frac{\Delta\theta}{\Delta B}\\
%$$\frac{\Delta\theta}{\Delta B}=\frac{\Delta V}{\Delta B} \times \frac{\Delta \theta}{\Delta V}$$
\theta_{B}&=&104 degrees; \theta_{0}=108 degrees\\
\Delta\theta&=&\theta_{B}-\theta_{0}=-0.0698 radians\\
\Delta B&=&-3A\cdot 11.1\frac{mT}{A}=-33.3mT\\
\frac{\Delta\theta}{\Delta B}&=&\frac{-0.0698radians}{-33.3mT}=0.002096\frac{radians}{mT}=2.096\frac{radians}{T}\\
V_{c}&=&\frac{1}{L}\cdot \frac{\Delta\theta}{\Delta B}=\frac{1}{0.1m}\cdot 2.096\frac{radians}{T}=21\frac{radians}{T \cdot m}
\end{eqnarray}
\subsection{Slope Fit Calculation}
\begin{eqnarray}
V &=& 0.000443B+0.127\\
\frac{\Delta V}{\Delta B} &=& 0.000443\frac{V}{mT}=0.443 \frac{V}{T}\\
V &=& \frac{0.021 \cdot (1+cos(\frac{2\pi\cdot(\theta-162)}{180}))}{2}\\
\frac{\Delta V}{\Delta \theta} &=& 0.21\cdot(-sin(2 \frac{\pi}{180} (-45)))=0.21V\\
\frac{\Delta \theta}{\Delta B} &=& \frac{\Delta V}{\Delta B} \cdot \frac{\Delta \theta}{\Delta V}=0.443\frac{V}{T}\cdot \frac{1 radian}{0.21V}=2.1095\frac{radians}{T}\\
V_{c} &=& \frac{1}{L} \cdot \frac{\Delta \theta}{\Delta B}=\frac{1}{0.1} \cdot 2.1095 \frac{radians}{T}=21.095 \frac{radians}{T \cdot m}
\end{eqnarray}
\textbf{This section appears perfectly if you click "preview". We couldn't figure out why it looks like this after we save it.} The problem --- now solved --- was that you can't have blank lines in an equation array.
\subsection{Lock in Method}\selectlanguage{english}
\begin{table}
\begin{tabular}{ c c c }
Vrms/V(Applied Voltage) & Vrms-Photodiode/V & Verdet Constant/(rad/(Tm)) \\
0.03 & 0.79 & 20.43 \\
0.04 & 1.05 & 20.37 \\
0.05 & 1.32 & 20.48 \\
0.06 & 1.58 & 20.43 \\
\end{tabular}
\end{table}
\section{Acknowledgement}
We would like to thank Professor Nathanael Fortune for his guidance and help over the past month. We also would like to express our gratitude to Dana Parsons, who has helped setting up the lab equipments. At the same time, this lab could not have been completed without the help and support from our classmates.
\selectlanguage{english}
\FloatBarrier
\bibliographystyle{plain}
\bibliography{bibliography/converted_to_latex.bib%
}
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