6/2 RC AC Curcuits

In todays class we explored AC RC and LRC circuits. In DC circuits, ohm's law is applied. In a AC circuit the same law applies, the only difference is that capacitors, resistors and inductors create resistance.

We began by defining a few formulas such as root mean square current, root mean square voltage,  capacitance reactance, inductive reactance, and impedance.

we began our first experiment by connecting the function generator across the resistor and capacitor with a current probe across the resistor and capacitor. 

We then started Logger Pro and connected the sensors. We measured Voltage and current. Using the stats button we got our values for maximum voltage and maximum current. We were given different frequency which we used to conduct the experiment.

Here is a close up of our values for Vmax and I max.

Here we have our second frequency.

and our second set of values for Vmax and Imax.

Using Vmax and Imax we were able to calculate experimental impedance. We then compared it to the theoretical impedance. For 10hertz, our theoretical impedance was 159.3 ohms and our experimental was 153.5 ohms with a 3.64 % error. The percentage is reasonable. For 1000 hertz, the theoretical impedance was 10.13 ohms and the experimental was 32.9 with a 224.8% error. This error was high because there must have been some inconsistency with the wiring or procedure. We also used the incorrect frequency with could've been a major factor in the high percent error. 

We also computed the phase angle using the formula above (inverse tangent *inductance reactance minus capacitance reactance divided by resistance).
In conclusion,  When frequency is low, the impedance of the capacitor is high, so most current will flow through the resistor. As the frequency increases, more current is diverted through the capacitor, and less to the rest of the circuit. Thus, the response is low pass.

5/28 AC Circuits

In today's class, we examined AC circuits and how capacitors, inductors and resistors behave with this type of power supply.

We began class by calculating root mean square voltage of a oscillating voltage with Vmax.

Alternating current with a resistor

In our first experiment, we use logger pro to graph current and voltage using an AC power supply with a resistor. The graph that is produced is sinusoidal. From here we can get our Vmax value. 

After finding the slope of the current, we found that the average current is zero.

Using the graph from logger pro, we were able to get Vmax and Imax so that we could calculate root mean square voltage. 

Alternating current with a capacitor

Here we derived the formula for voltage with capacitance and we see that there is a phase shift.

In this experiment, we connect the AC power supply to a capacitor and measure the voltage and current using logger pro. We found that when there is a large voltage across the capacitor it is indicating that the capacitor is fully charged. We see that current is zero when the voltage is at its maximum.

Here we have our voltage vs current graph, which formed a circle. This indicates the motion of the current when it is at a certain voltage, and indicated that power is conserved because of the capacitor.

Using our graphs from above we calculated reactant capacitance using Vrms and Irms. We calculate the experimental and theoretical then compared both. We had a 79% error which is really high, this is because we did not account for the resistance within the power supply. We also calculated the phase shift and found it to be 0.23.

Alternating current with inductors

For our next experiment, we connected the AC power supply to an inductor in order to see how its behavior. 

Here we derived the formula for inductive reactance, which is equal to omega times inductance.

Here we have our graph of current vs. voltage, we see that the graph is consistent with the phase shift 

Using our graph from logger pro we were able to calculate the experimental value of capacitance reactance and compare it to the theoretical value. We  got a 25% error. We also calculate the phase shift which ended up to be 0.35.

5/19 Electromagnetic Induction and Induction

Today we explored electromagnetic induction and induction. Using active physics we were able to get a better understanding of how they work.




We began with the study of magnetic flux and induced EMF. Using active physics, we seen the when there is a big change in magnetic flux, the induced EMF is just as big. We also concluded that negative EMF means there is a positive change in magnetic flux.

Our next assignment is a copper rod given current. We were asked to predict in which direction the copper rod would move when given current. 

What we found is that when we gave the rod a certain current, the rod would move in the direction of that current. However, when we changed the direction of the current, the rod moved in the opposite direction. This is because the current through the rod is going through a magnetic field (pictured above) which causes a force. Knowing this we can predict in which direction the rod would move.

We continued class with the study of inductors. We we found that inductance is negative EMF over the change in current with respect to time. In addition, we see that voltage is equal to inductance multiplied by the change in current with respect to time.

We continue to do some derivations and we find that current is equal to capacitance times the change in voltage over time.

Using what we found above, we are able to find the relationship between EMF and flux. Using length, number of turns, and area. 

here we have the derivation for inductance which we will use for Inductancts of solenoid. 

Given a length, area and numbers of turns, we were able to calculate inductance of solenoid.

Here we derived units of inductance, which we found to be Henry's .

For our next activity, we were asked what the current would be in a circuit after  a significant  amount of time has passed.

We found that current would plateau and goes towards infinity (pictured above).

Using ActivePhysics, we continued our study of RL circuits, which are circuits that contain inductors and resistors. We found that a change in flux would induce EMF. We also concluded that when the switch was closed the EMF would go in the opposite direction and the induced EMF opposes EMF of the battery. We also seen that when time was increased the inductiveness increased. The results of our experiment are pictured above.

5/26 LR Circuits

We began class with the discussion of current verses time, and we found that as current changes over time it induces EMF in the inductor.

Our first activity is to calculate inductance given a number of turns in a coil, the area and resistance. We found the inductance to be 760 micro henry's. We also discussed engineering notation which uses 10^-6 for values.

Using the inductance we calculated above we were able to calculate resistance is an 18 gauge coil. Which resulted in .48 ohms.

Finally, using the inductance and resistance we were able to calculate our time constant, using the equation tao= L/R.

Using the oscilloscope, we experimentally calculated our inductance, which we used to compare to our theoretical value of 760 micro henry's.

Using the graph from the oscilloscope, we calculated our experiment inductance.

Our experimental value for inductance was 703 micro henrys. We had a 7.5 % error.

In our next activity, did a LR circuit problem, in which there are two resistors in parallel and one resistor is in series with an inductor. Using the inductor and resistor 1, we were able to calculate the the time constant using the time constant formula. Our time constant was 2.9E-4.

Using ohms law, we were able to calculate Imax, and used that to calculate the current (picture above).

Using the value for current from the pervious picture we were able to find the time.

5/14 Faradays Law and Lenzs Law

Purpose: Today will study Faradays Law and Lenzs Law. We will also cover flux 

through a loop and EMF. A little background on these laws are as follows: Faraday's la of induction is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF). Lenzs Law states: If an induced current flows, its direction is always such that it will oppose the change which produced it. Which means the the induced voltage and the charge in magnetic flux have opposite signs.

We began class with a demonstration. We have two wires with current passing through them (picture below). The wires begin to repel each other because they become magnetized by the magnetic field. However, when the current in one of the wires is reversed the wires get closer to each other.

We were then given a scenario: We were given two lines, in which each had current, magnetic field and a magnetic force. We predicted that the current would be flowing upward. Our conclusion to the question is picture below.

Professor Mason then conducted another experiment by creating alternating current using a power supply in two parallel lines. We found that there were no net forces acting on the system because of the alternating current (pictured below)

Mason then created this graph on logger pro for the experiment from above. This is the magnetic field with respect to time graph which shows us the behavior of the magnetic field due to alternating current.







We then continued class with our next experiment: The Magnetic field  At the Center of A Current Loop.
We were asked to find the proportionality of loops and magnetic fields by measuring the magnetic field by in increasing the number of loops. We measured the magnetic field of each increasing loop on logger pro and graph it. What we found was that  the number of loops and the current is proportional to the magnetic field, thus the more loops the greater the magnetic field. Our values and graph is pictured below.

Next we move to flux. Flux measures the number of magnetic field lines that pass perpendicularly through a surface. we were given a plane with surface area ab, we were told to find the magnetic flux through that surface. We found that when the magnetic field is parallel to the surface area, the magnetic flux is zero, and when it is perpendicular to the area then it is defined by the following equation:

Here we also see Flux through a loop: when both the direction and magnitude of the magnetic field does not change across the surface bounded by a closed loop, flux can be expressed by the bottom equation in the picture above.

We then continue our studies with the creation of current in a coil of wire using magnetic fields. Professor Mason then used a galvanometer and a magnet to cause magnetic fields to change which created current which could be seen on the meter. (Galvanometer pictured below).

We found four ways to maximize current on induced EMF, we said more loops on a coil, bigger loops by increasing Area, bigger magnet, and moving the magnet faster.

We concluded that current could be induced by in a coil by a changing magnetic field.

Next we move on to Lenz's Law.

Direction of the Induced emf: Lenz's Law

General statement:
The effect of the induced emf is such as to oppose the change in magnetic flux that causes the induced emf. 

Professor Mason then conducted another experiment with a transformer, which has the ability to produce different voltages. The current passed through the coils and the magnetic field was generated from the metal pole sticking out the middle of the transformer.

In the picture below, we can see that the metal ring around the black pole is levitating and the reason for this is that there is a  magnetic field generated from this pole causing the ring to move in the opposite direction. We then conducted another experiment but this time the ring had a gap in it, this made the current unable to flow, thus the magnetic field was also affected. There was no magnetic field so the ring did not move. (no picture)

In the pictures below, a piece of metal is placed on top of the magnet which caused the light bulb to light up. 

When the north pole of the magnet is going toward a loop, the flux increases and an another upward magnetic field is created causing a counterclockwise current. This causes an upward force.

For our next experiment we were given to scenarios,  a magnet was to be dropped down and aluminum tube then down a plastic tube. What we found was that the magnet traveled at a much slower rate when traveling down the aluminum tube.

We concluded that the magnet moved slower down the aluminum tube because the magnetic field created a magnetic force going up. Thus, the induced current generated a magnetic field which then created a force that opposed the induced EMF, which caused it to move at a slower rate.

We continued with a mathematical representation of Faraday's Law (pictured below)

We took our definition of EMF and magnetic flux through a single loop of coil, and derived Faraday's law relating to electro motive force to flux.

Finally, we ended class by taking a look at the EMF verses time and compared it to the magnetic field verses time, we seen that when magnetic field is at a low EMF is at a high and vice versa.

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