Speed Of Light Lab Write Up PDF

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Physics Lab experiment write up explaining our measurement method as well as solving Maxwell's equations that mathematically calculates the speed of light....

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Speed of Light – Physic 816 Advanced Lab

Physics 816 Advanced Lab Wichita State University Fall Semester 2017 Dr. Solomey

Speed of Light Measurement Bathiya Samarakoon Steven Roothaan Timothy Reed

Speed of Light – Physic 816 Advanced Lab

Abstract: In this experiment, our team used the time-of-travel method to measure the speed for light c. This configuration uses an oscilloscope to measure the time elapse delay of a laser that is driven by a function generator which allows for a phase shift to be measured before and after a set distance that the light travels. This distance was varied and the time delays were measured from the input time of the function generator to the input time of a photo sensor, which received the emitted laser after it had traveled over its varying lengths. We experimentally measured value of c’ = 2.6172x108 m/s fits within a 13% error of the current national institute of science and technology (NIST) value of c = 299,792,458 m/s (in a vacuum).

Maxwell’s Equations: James Clerk Maxwell (1831-1879) was a Scottish mathematical physicist whose research in kinetics and electricity laid the foundations for modern Quantum mechanics and special relativity. His insight of early electrodynamics allowed him to correlate the velocity of an electromagnetic disturbance to that of the speed of light, hypothesizing that light itself (including radiant heat, and other radiations) are an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.

Figure 1 -Wave model of EM energy: Maxwell unified existing laws of electricity and magnetism and showed oscillating electric field (E) produces a magnetic field (B) and vice versa. These electromagnetic fields are perpendicular to each other and propagates the EM wave.

Maxwell took a set of known experimental laws in electromagnetics (Gauss’ Law, Faraday's Law, and Ampere’s Law) and unified them into a coherent set of equations known as Maxwell's Equations. They describe how electric and magnetic fields propagate, interact, and are influenced by objects. Maxwell was one of the first to determine the speed of propagation of electromagnetic (EM) waves was the same as the speed of light, concluding that EM waves and visible light were really the same thing. Maxwell also showed that light doesn't need a medium (no ether) but only changing Electric and Magnetic fields. In a vacuum, light always travels at a constant c ≈ 300,000 kilometers per second, 186,000 miles per second, or 670 million miles per hour; although the speed of light can slow down in the case of certain transparent media such as glass, water, and air.

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Speed of Light – Physic 816 Advanced Lab

The Electromagnetic Laws and Maxwell’s’ Equations [1]: Starting with the four equations relating electric ( E) and magnetic (B) vector fields:

 E 

 0

Gauss’ law for electricity: the electric flux out of any closed surface is proportional to the total charge enclosed within the surface; meaning a charge will radiate a measurable field of influence around it.

  B 0 Gauss’ law for magnetism: the net magnetic flux out of any closed surface is zero; meaning there are north and south magnetic poles (monopoles have not been known to exist) and the flux of the system as a whole is zero.

 E 

B t

Faraday’s Law of Induction: the line integral of the electric field around a closed loop (the curl of E) is equal to the negative of rate of change of the magnetic flux through the area enclosed by the loop.

B   0J   0 0

E t

Ampère’s Law: the line integral of the magnetic field around a closed loop (the curl of B) is proportional to the electric current flowing through the loop.  is the divergence operator.

x is the curl operator.

ρ = net charge inside J = current density -12 0 is electric permittivity (8.854188×10 A2 s4 kg-1 m-3): The ability to transmit an electric field through free space. 0 is magnetic permeability (1.2566371×10-6 kg m A-2 s-2): Relationship of electric current and strength of the associated magnetic field.

Working with Maxwell’s’ Equations [1]: 3

Speed of Light – Physic 816 Advanced Lab

In a region of space where there is no electric currents or charges (free space), the current density (J) and charge (ρ) are zero, therefore the previous equations become:

B t

 E 0

E 

 B 0

B  00

E t

These equations constitute a set of coupled, first-order, partial differential equations for E and B. They can be decoupled by applying the “curl” (measure of strength):

   B  E  t      B [ E] [ ] t

Change the order of differentiation:

Substituting for

:

Apply the vector identity

:

The same result is obtained for the magnetic field B:

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Speed of Light – Physic 816 Advanced Lab

Now we have second order decoupled differential equations for E and B. In a vacuum, each Cartesian component of E and B sat nal wave equation describing the propagation of a sinusoidal wave:

⇒ Solving for v, Maxwell’s equations imply that empty space supports the propagation of electromagnetic waves, traveling at a velocity of:

The velocity of propagation depends upon vacuum electric permittivity 0 and magnetic permeability  0, both of which can be determined by electrical measurements and will calculate v to be the speed of light, c.

Our Experimental Method: In this experiment, we used the “time-of-travel” (also called “time-of-flight”) method to measure the speed of light. The setup requires a light source and a steady path of know length in order to measure the time that elapses while the light signal travels down its course (see Figure 2) Most modern methods for speed of light experiments generally involve the use of a well collimated beam of light. Collimated light consists of parallel rays which do not decrease intensity with distance, allowing minimally spread out. Rays from conventional sources (lightbulbs) are difficult to collimate, but a laser (Light Amplification by Stimulated Emission of Radiation) emits a beam that is already well collimated and can be collimated even further using simple optics. Making the laser the best method for the light source in a speed of light measurement.

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Figure 2 - Implementation Flow Chart: See PASCO AP-8586 Laser Speed of Light Apparatus User Manual for detailed setup.

Speed of Light – Physic 816 Advanced Lab

Another important criteria for this measurement is to distinguish between the origination signal and the signal carrier after the light has traveled its distance. With a single, continuous beam, you would not know when to start and stop timing, but by using a modulated laser signal, we can add information to the carrier. For example, change the signal to the form of a square wave for the initial light signal and a sinusoidal wave for our traveled light signal. A function generator with 2 separate outputs is used to modulate the traveled lights signal. One output for the initial source (d=0) displayed as a square wave and the second output (d= ∇ x ) is used to modulate the light from the laser at ~3 MHz (see Figure 3). The laser light is reflected onto a concaved mirror (not shown) allowing it to be redirected and focused onto a light receiver. An oscilloscope is used to observe Figure 3 – Equipment configuration: the modulated light (d= ∇ x ) and the phase Picture showing the setup used to measure c’. difference as compared to the initial (d=0) signal Equipment used: (see Figure 4). BK Precision 4011A Function Generator Tektronix 2213 Oscilloscope Speed of Light Diode Laser OS-8475 Component Carrier OS-9107

By moving the mirror over various lengths ( ∇ x ), the experiment can be repeated for several distances. It is important to note that the distance the light travels is 2 times the mirrors length from the laser. After the light travels to the mirror, it is then redirected, and once again must travel the mirrors length back to the photodetector. The difference in length from the front of the laser to the front of the light receiver was taken into account as to minimize error in the calculations for c’. Figure 4 – Oscilloscope view of phase shift: Display on the dual trace oscilloscope showing the distinct shift between peaks of the initial signal generator (square wave) and the signal of the reflected beam (sine wave) for 2 different distances light traveled.

For each carefully measured distance ( ∇ x ), we used the oscilloscope to measure the change in the time delay of light initiated by the function generator in comparison with the light

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Speed of Light – Physic 816 Advanced Lab

emitted by the laser after it traveled its specific distance.

Our Measurements & Conclusion: Using the experimental methods as described above, the necessary data (Distance & Time) was collected for each mirror position and graphed in Microsoft Excel (see Figure 5). By plotting Distance versus Time and then generating a linear trend line for our overall data points, our experimental measured speed of light is c’ = 2.6172x108 m/s.

Experimentally Measured Speed of Light 20

Distance Measured (Meters)

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f(x) = 2.62 x + 0.05 R² = 0.99

16 14 12 10 8 6 4 2 0 0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

Time of Flight (Nano Seconds)

Figure 5 – Plot of Distance over Time of experimental measurements:

The theoretical value of the speed of light constant according to the latest national institute of science and technology (NIST) is c = 299,792,458 m/s (in a vacuum). Given our experimental results above, the calculated % error for our measurement is 12.8 % ( ). This discrepancy can be attributed to several possible systematic sources of error; the error associated with accurately measuring the overall distance the light travels, how precise was the oscilloscope screen read for the time delay. Even the time delay of the circuitry within the oscilloscope and photodetector as well as the cables connecting the equipment, all have a certain delay associated to them. How about a time delay associated with the redirection of light on the mirror? All these experimental imperfections, given enough time and effort, can be improved or minimized and included in the overall speed of light experiment to improve the % error.

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Speed of Light – Physic 816 Advanced Lab

References [1] David J. Griffiths, Introduction to Electrodynamics (3rd edition, Addison Welsley, 1999), Chap. 9, p. 375-376.

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