AC on Capacitors and Inductors

Alternating Current on Capacitors and Inductors

After connecting a function generator to a 220E-6 F capacitor and setting up LoggerPro for voltage and current data collection, we made some initial calculations.

We calculated the reactance of the capacitor (X_c) at a frequency of 10 Hz to be 72.34 ohms. The calculation is shown above under the "theoretical" heading. Next we turned the frequency on the function generator to 10 Hz and collected the data on LoggerPro.

From the sinusoidal current and potential graphs, we obtained the maximum voltage (V_max) and maximum current (I_max). [Unfortunately the screenshot above does not explicitly show how we obtained the values. The next trial shows in detail how we obtained our data].
We calculated the experimental reactance ("exp") using the formula shown above. The reactance is the ration of the root mean square voltage (V_rms) divided by the root mean square current (I_rms). V_rms is equal to V_max divided by sqrt(2) and similarly, I_rms is equal to I_max divided by sqrt(2). We then clean up the equation to get an equation we can use with our measured values, X_c = V_max/I_max. We plugged in our "max's" to this equation and found the experimental reactance to be 69.69 ohms. When we compared this value to our theoretical value, we got a 4% error.

We repeated the calculations for a frequency of 20 Hz. We found the theoretical reactance to be 36.17 ohms. Above we see the collection from LoggerPro with greater detail. 

For our experimental values we used highlighted a part of the graph and used the "Statistics" option in LoggerPro. A box in each graph shows the maximum potential (V_max) and maximum current (I_max). We saw that V_max = 3.21 V and I_max = 0.08862 A. From this we calculated X_c = V_max/I_max = (3.21 V)/(0.08862 A) = 36.22 ohms. Compared to the theoretical value 36.17 ohms, there is a 0.1% error.

Now we connected an inductor of an unknown inductance. We used the formula below to calculate the inductance of the inductor. Again we used LoggerPro data to find V_max and I_max. We saw that we could take various trials at different frequencies and compare the experimental values for inductance to find the inductor's inductance We did this for 10, 20 and 40 Hz, respectively and found that the values averaged 1.32 H.

We repeated the experiment with a bolt inserted in the center of the inductor. We calculated the values for inductance the same way we did the first time and found that it lowered significantly.

Electromagnetic Induction

Measuring Inductance

We wired up a circuit according to the schematic above. We used a function generator to provide 10.00 V with an internal resistance r, of 50 ohms. A circuit with resistance 347 ohms and an inductor with internal resistance R_L and inductance L were all wired in series with the function generator. Our goal is to measure the inductance of the inductor directly, then compare it with two experimental values obtained from the physical characteristics of the inductor and from data obtained from an oscilloscope. The true inductance of the inductor was 8.22 mH.

 Here we see the inductor we used, it consisted of N = 440 turns.

We adjusted the oscilloscope to get an output that allowed us to collect our half time decay. We counted the divisions on the oscilloscope and measured the half time to be 6.67 microseconds.

From the data collected from the oscilloscope, we were able to calculate the inductance of the inductor. This value was 9.63 mH.

Now, we can calculate the inductance from the inductor's physical attributes. We measured the cross-sectional area to be A = (0.04 m)^2. The number of turns was written on the inductor, N = 440 turns. The measured length l = 0.055 m. Plugging all of this into the original formula gave us a calculated inductance of 7.08 mH.

Solenoids and Magnetic Fields

Magnetic Fields Inside Current-Carrying Loops

We made some predictions about how the earth's magnetic field changes with respect to angle along three axes. Prof. Mason connected a cylindrical magnetic field sensor to LoggerPro software and projected the results on a screen. Our predictions are in black ink and the actual results.

Magnetic Field Inside a Coil with Varying Loops

We started with a magnetic sensor that was connected to a computer so that we could collect data on LoggerPro.

We then connected a wire to a power source and monitored the current using a multimeter.

Next we coiled the wire around a hollow plastic cylinder. The picture above shows us preparing for our first run, with one loop around the hollow cylinder.

We switched the power source on and measured a current of 1.93 A (the picture shows a different value, in actuality the reading fluctuated a bit).

We carefully held the loop in place with the aid of the hollow cylinder and positioned it to where the sensor on the sensor probe was in line with the wire loop. We collected the data on LoggerPro. After a few seconds of collection, we turned the power off so that we could obtain a "zero" point of reference.

To calculate the magnetic field from our graphs, we used averages. We took the mean of the magnetic field from the time when the power was passing through the wire using the "Statistics" option in LoggerPro. Next we took the mean at a time period when the power was already turned off. We subtracted calculated the difference between the "on" and "off" averages and that became our experimental magnetic field recorded. The graph above shows that we recorded an mean of 0.021 mT when the current was on and 0.007 mT when it was off. So we took the magnetic field to be B_loop = 0.021 mT - 0.007 mT = 0.014 mT. This was our experimental value for 1.93 A and 1 loop.

We repeated the same process as above to calculate the magnetic field. In this trial we have 2 loops, 1.93 A, and 0.023 mT.

This trial shows 3 loops, 1.93 A, 0.027 mT.

Our final trial shows 4 loops, 1.93 A, 0.044 mT.

This table shows the data we collected for all trials. We kept the current constant at 1.93 A for all four trials. In doing so, we can see a direct relationship between the number of loops in the coil to the magnetic field it produces. We can see that as the number in loops increases, so does the magnetic field.

Magnetic Field in Motion

We connected a solenoid which contained many loops to a device that could measure current.

We passed a magnet through the center of the solenoid and observed the meter.

We observed that the reading was affected by the velocity of the magnet. The slower the magnet went, the smaller the reading on the meter.

On the contrary, the faster the magnet was passed through the center the higher the reading. This told us that current was created when passing a magnetic field through the center of looped wires at a velocity. Passing the magnet in the loop quickly caused the meter to go one way and then pulling the magnet back out would cause it to go the other way. When we tried it again with poles reversed, the same thing happened except that the directions were switched from the first orientation.

In this horrible photograph, we can see (or pretend to see) three things that will maximize current output. Firstly, the velocity of the magnet going though will maximize the current flow as observed from our experiments. Second, the strength of the magnet will also increase the current. This can be seen by doubling-up the magnets we were using and passing them through the coils. Lastly, the number of loops in the coil will also increase the current. From our experiment with the magnetic sensor we proved this result.

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.