Biot-Savart Law

Magnetic Field of the Earth

In this experiment, we calculate the magnetic field of the earth using a compass and a magnetic field created a coil with current passing though it.

We first we obtained wire that has been looped around many times around a cardboard circle with a stand for a compass to be placed inside. The compass was placed and the whole wire-compass setup was rotated until the compass was "zeroed" at north. Angles were to be measured, therefore it is easier to start at zero though we could have also measured the angle displacement (final-initial).

Here we see the coils in their zeroed form. The ends of the coils were hooked up to a power supply via alligator clips. We noted on the cardboard that the diameter was measured as 4.7 cm or D =  0.047 m. Also noted on the cardboard were the number of turns, N = 39 turns, or loops of the wire.

Took a multimeter to measure the current going through the coil. We took this measurement as well as the angle displacement on the compass.

Here we see the data for 3 trials. We divided the magnetic field by the tangent of the angle displacement, as shown above, for each trial on the data. The calculation for the first trial is shown above. However, our data was 200% above the actual value, which is 2E-5. We looked through our data and finally realized that the diameter of the coil does not really look like 47 cm at all. The diameter recorded on the cardboard was incorrect!

We measured the diameter of the coil to be 14.7 cm. We recorded one more point at 32 mA for good measure and recalculated everything using the formula shown above.

We used an Excel spreadsheet for our calculations. We made sure to convert our measured degrees to radians before calculating. We saw that our new values were much closer than in the previous data set. We calculated an average and then a percent error. Although the percent error was a bit higher than what we desired, there was a huge uncertainty in our measurements of loop diameter, current measurement, and the largest, angle measurement.

Magnetic Field at Center of Solenoid

Magnetic Field at Center of Square Loop with Current

Magnetic Fields and Motors

Electric Motor

We were given the electric motor shown above. It consisted of two magnets at either end of a coil that is free to rotate and attached to a rod connected to the yellow frame. The rod contained a commutator that had two metal strips through which current could be passed through. The commutator is a cylindrical piece metallic piece which has slits, breaking up the current and reversing the direction. This is crucial for the motor to keep running. In an earlier demonstration, we had seen a loop with electric charge going through it would rotate 90 degrees when in a magnetic field, but it would just wobble in that (horizontal) position. This is because there is a net torque of zero because of its position with respect to the magnetic field. Alternating the current allows there to be torque applied continuously.

We made a few observations when experimenting with this motor. Firstly, if we reversed both magnets, the motor turned in opposite direction. Secondly, the speed of the motor was dependent on the voltage applied. The higher the voltage, the faster the motor would spin.

Creating a Simple Motor

In this experiment, we created a simple electric motor based on the principles of the motor we used previously. The materials were enamel coated wire, paper clips, sandpaper, magnets, and tape. We also used a grey power supply to power the motor.

We created "stands" with paper clips taped to a whiteboard. The wire was looped many times leaving two straight ends that sat on the paper clip loops. One end of the wire was sanded all the way around while the other only half way (this served as a commutator). Alligator clips attached to a voltage source were connected to each paper clip. a magnet was placed under the coil and another was placed on top creating a magnetic field. This allowed the coil to spin repeatedly. Unfortunately we had trouble finding the perfect position of the top magnet, therefore we kept moving it around the top on the coil until it began moving freely. The video above shows the demonstration.

Magnetism

Magnetic Field

A bar magnet was placed on a horizontal whiteboard and a compass was placed near its north pole. We noticed that the needle pointed directly at the north pole of the magnet. We repeated this for many positions around the magnet (along the blue line) and drew a red arrow on the whiteboard representing the orientation of the compass needle.

Here we see the resulting arrows at various positions around the magnet.

We figured the magnetic field of the bar magnet must look similar to the diagram above.

Prof. Mason sprinkled some iron filings evenly around a magnet and used a projector to show the results. We can see that the magnetic field looks very similar to the one we had hypothesized.

Magnetic Flux

Magnetic flux follows some of the same principles as electric flux. We can determine if there is a net flux by counting the magnetic field lines going in and out. Here we show a magnetic field in red and three areas of interest circled in black and labeled 1,2,3. In area 1 there are two lines coming in and two coming out, therefore the magnetic flux is zero at that point. Area 2 encases the south pole and contains 0 lines coming in and 7 going out, therefore there is a net magnetic flux. In area 3, which encases both north and south poles, there are 7 lines going in and 7 coming out, therefore the net flux is zero.

Magnetic Field, Force, and Velocity

In this given problem, we were given that the magnetic field was 2.6E-3 T and the velocity of an electron was at 30 degrees from it with a magnitude of 3E6 m/s. We found the magnitude of the force using the formula shown above where q is the charge of an electron. We also used the fact that the cross product with the magnetic field can also be taken by taking the magnetic field and multiplying it by the sin of the angle between it and the charge velocity.

The diagram above shows that force is magnetic field is always perpendicular to the force and charge velocity.
Below we begin deriving equations that have to do with centripetal force.

We start by stating that the centripetal force is equal to the mass multiplied by the velocity squared all divided by the radius of the circle. We also know that the force is equal to qv X B so setting them equal to each other cancels out a factor of v from both sides. We got rid of the cross product since it is equal to sin(90)=1 in this case (not cos(90) as in picture). We solved for R and changed translation velocity into angular velocity (v = Rw). The R's cancel out and we can solve for w.

We have another formula for angular velocity which states that it is directly proportional to the frequency and a factor of 2pi. Setting these two formulas gives us a direct relationship between frequency (given), mass of an electron, charge of an electron, and magnetic field (which we need to find). Solving for magnetic field and plugging in the known values we calculate the magnetic field to be 0.0876 T.

Diodes and Transistors

Diodes

Above we have a picture of a circuit containing a diode, which has the symbol of a triangle with a bar at the point. Diodes contain two layers of semiconducting materials N-type containing electrons, and P-type containing holes. In between the layers there is an area of neutral charge called the depletion zone. In order for electrons to flow the negative side of a DC source must be connected to the negative side of the diode. This makes the depletion zone vanish and allows current to flow. In AC, the current goes from positive to negative in a sine wave (shown above). When AC goes through a diode, the negative parts of the wave make the depletion zone in the diode larger and do not allow current to flow. Therefore, only the positive current flows.

Building an Amplifier

The schematic in the picture above was given and we wired the circuit accordingly. However, we did not use a switch and we used a voltage supply instead of a 9V battery. The circuit consisted of various resistors, capacitors, and a transistor. The was a signal from a function generator was monitored on the oscilloscope and after a second, amplified, signal was also shown on the oscilloscope. This provided a direct comparison between the original and amplified voltages.

The pictures above show the completed circuit (minus the power source) from the schematic.

We changed the voltage (and frequency) on the function generator and took pictures of the various results. The smaller wave in each case represents the voltage from the function generator and the larger wave represents the output, after the wave has been amplified. We can see that the voltage increased dramatically in each case.

Building an Amplifier Using an Integrated Circuit

Once again we created an amplifier however this time we used an integrated circuit (IC) for ease. An integrated circuit is a tiny circuit in which diodes, resistors, and transistors are connected in a tiny space (the black rectangular component above).
This time, the amplifier was more rewarding because we got to play our favorite music instead of hearing a variety of "waves" (sounds created from the function generator). We used a given schematic with a series of resistors, capacitors, and an IC. A 3.5 mm male jack connected allowed a phone with music to be hooked up and outputted through a small computer speaker. A grey voltage supply provided power. When everything was connected correctly, we played music through a phone and it was amplified significantly through the computer speaker.

Oscilloscope

The Tap Key

We hooked up a tap key shown above to a battery and oscilloscope. We measured the battery to have a potential of 1V. We calibrated the oscilloscope so that when the tap key was not depressed the signal on the oscilloscope was at dead center. When the tap key was pressed the line moved up one square (picture below), showing that 1V DC was going through.

The video above shows the signal jumping when the tap key was pressing. The oscilloscope was set at a slower frequency so that the "jumps" in voltage can be easily seen as the tap key was being repeatedly pressed. However, in the video, we were not still using 1V but about 1.5V. The video helps to see how the tap key interacts with the oscilloscope when there is a voltage being applied on and off.

Types of Waves

Above we can see two types of waves, a square wave and a sawtooth wave. For the square wave, the vertical lines are not visible oscilloscope as they are on the display of the function generator. There is a discontinuity in the graph. In the sawtooth wave, the wave goes from a positive slope, hits a peak and goes to a negative slope until it hits a minimum and repeats. The sine wave was completely smooth (shown in the next section below) and resembles the smooth, round, up-and-down wave that most of us are familiar with.
At 96.000 Hz, we connected a speaker to hear the various sounds of these waves. The sine wave had a low bass sound. The triangle wave had a sound that had less intensity than the previous. The square wave however was the loudest and sounded distorted. We experimented with the various controls on the function generator and found that adjusting the frequency changed the pitch of the sound produced while changing the amplitude affected the loudness of the sound produced.

Determining the Period of a Sinusoidal Wave

We connected a function generator the an oscilloscope. We set the function generation to 96.000 Hz and sine wave output. We observed the oscilloscope to have a sine wave on the screen. The oscilloscope "Time/Div" was set at 2 ms. We can compare the output of the wave from the function generator to what we see on the screen of the oscilloscope.
The function generator output a sine wave at a frequency of 96.000 Hz. We know that the period is equal to the inverse of the frequency. Therefore, the theoretical period from the function generator is T= 1/ (96.000 Hz) = 0.010.
From the oscilloscope, we measured (from where the curve first hit the x-axis to when it completed a cycle) the length of 6.5 squares. From our settings on the oscilloscope, each square is 2 ms = 0.002 s. We can compute the experimental period as follows: T = (6.5 squares)(0.002 s) = 0.013.
The percent error was 30%, however for the purpose of the experiment, we can consider it acceptable.

Observing AC and DC Quality

We connected this 6V DC transformer to the oscilloscope and observed the results. We obtained a smooth, steady straight line above 6V. Perhaps we were not calibrated at "0" before connecting the transformer. The results show a clean power source.

Here we connected the "grey" DC power source to the oscilloscope and found the source to be clean. The line was again a smooth straight line.

Here we connected an AC source and found the output on the oscilloscope to be different. Although the shape was expected, there was "noise" around the signal. This is characteristic of a "dirty" power source. The signal looks very distorted.

Lissajous Figures

We connected an AC transformer to CH1 on the oscilloscope and the function generator to CH2. Here, both inputs are being shown simultaneously. The oscilloscope was set to xy mode. This means that the input from the AC transformer will affect the x-axis while the function generator will affect the y-axis to create the Lissajous figures above.

Mystery Box

In this activity, we were given a "mystery box" that had 5 uniquely-colored terminals. The box was sealed and we cannot see the inside configuration of it. Using an oscilloscope, we were to determine the internal configuration of the box. The total number of possible configurations are 5 nCr 2 = 10 possibilities.

We went in order connecting one terminal to all other possible connections, observing the results on the oscilloscope, and then moving to another terminal until all possible 10 configurations were tested. We either obtained a voltage gain in the oscilloscope, or noise, which signified no connection between the two terminals.

The picture above shows a completed diagram showing the internal connections of the mystery box. The terminals above represent, from left to right, red, green, yellow, blue, and black. The results are that red is connected only to black. Green is connected to both blue and black. Blue is also connected to black and yellow is not connected to any terminal.

Capacitors

Charging and Discharging Capacitors

Capacitors have the ability, like batteries in a way, to hold charge. In this activity, a capacitor was charged and discharged. LoggerPro software was used to record the potential change in real time and record the data.

The graph above shows the phase when the grey power supply was used to charge the capacitor. We can see that it reaches a max of about 4.6 V and does not go higher than that. This means that the capacitor is fully charged. The behavior of potential as a function of time is given by the equation 

This graph shows the capacitor becoming discharged. We can see that it has an exponential decay. The  behavior of this process is given by the equation V=(V_0)e^(-t/RC).