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).

Capacitance

Capacitance

In this activity, we created capacitors using two sheets of aluminum foil, separation distance (provided by sheets of paper). We carefully cut two square pieces of aluminum foil and measured the area to be 0.0316 m^2 for each. We then measured the thickness of a single page. We did this by measuring the thickness of 280 pages (making sure not to include the cover sheets and dividing the total number of pages by 2 since a single sheet has two "pages" front and back) and dividing by amount of sheets. We calculated that a single sheet measured 6.357E-2 m.

We connected a multimeter to the two sheets of foil via alligator clips. The positive end on one of the aluminum sheets and the negative end on the other aluminum sheet. We separated the two sheets of foil by a single sheet of paper and measured the capacitance with the multimeter. We did this again for 2, 10, and 15 sheets of paper and recorded the capacitance. Then we folded the aluminum sheets such that they had half of their original surface area (0.0158 m^2) and measured the capacitance again for 1, 2, 10, and 15 sheets of paper separation distance, respectively.

 When collecting data for capacitance, we pressed down on the pages so that there would be the least separation distance possible. That is, that the only separation distance is the thickness of the paper sheets, not of air or deformation in the pages which will create a higher separation distance and thus data with greater inaccuracy.

Here is the data we collected from our trials. We took this data and used excel to plot a Capacitance vs. Separation Distance graph as shown below. The blue curve corresponds to the data taken from the original foil surface area and the orange curve corresponds to half of the original surface area.

 The following observations were found to be true from the data we collected.

Resistance in Circuits

Resistance in Parallel Circuits

We were given three 150 Ω resistors and wired them in parallel. Theoretically, the total resistance should be given by the formula in blue above. The theoretical resistance of the three 150 Ω resistors in parallel is 50 Ω. We took a multimeter and measure the resistance to be 49.4 Ω which is within 1% error.

Here we analyzed a circuit where some resistors are in parallel and some are in series. We simplified the circuit in steps by combining resistors. We created a symbolic equation of the total resistance of the circuit as shown in black on the bottom right of the picture above.

Using our new skills, we created a symbolic equation for a new circuit (above). We were given the resistance of each resistor. We found that the theoretical value for the total resistance in the circuit is 52.2 Ω.

We now took the resistors and wired them up according the schematic given in the previous picture. We measured the value to be 53.6 (although it fluctuated). We subtracted the internal resistance of the multimeter, 1.4 Ω, and found that the experimental value for total resistance was the same as the calculated value.

We found that resistors add directly when they are wired in series and add in inverse when wired in parallel. We also saw that it was easier to break up a circuit into simpler circuits when trying to obtain the total resistance for the circuit.

Testing the Loop using Kirchoff's Rule

Here we applied Kirchoff's Law to find the current at different points in the circuit, across the resistors. We ended up with three equations and three unknowns for the currents. We used a matrix to solve for the individual currents (in mA). The individual values for the currents as labeled are i_1 = 1.137 mA, i_2 = 0.999 mA, i_3 = 0.138 mA.

We then set up the circuit on a breadboard as shown above. We used a potentiometer as resistor #2 and adjusted it until the resistance was 2.15 kΩ (The potentiometer was very sensitive and it was very difficult to turn it to a value of exactly 2.00 kΩ).

Next, we measured the resistance across resisors R_1, R_2, R_3, the potential differences, and currents i_1, i_2, and i_3. The data is shown in the table below.

As we can see, the % discrepancy was incredibly large (130% for the third resistor!). There is a huge source of error in the potentiometer. Therefore, we ran the experiment again, except that this time we swapped the potentiometer with a resistor that had a measured resistance of 2.13 kΩ (shown in the picture below).

We took our new measurements as shown in the table below.

We can see that our new values were much more accurate than the previous ones. The largest sources of error were in the resistors and in the battery. The battery provided 1.45 V instead of 1.50 V, and the resistors did not all match the theoretical resistances that we had used to calculate our theoretical currents.