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.

Electric Potential

Electric Potential

In this lab, we took conductive paper that had a line and dot painted on it with metallic paint. A voltage supplier created electric potential difference between the line and dot via alligator clips. This voltage was measured as 15.02 V (please disregard the reading on multimeter as the picture was from a previous, and failed, attempt at the experiment).

We then measured two points on the higher and lower voltage, respectively. When we measured two points on the higher voltage, the reading came out to be -0.62 V. When we measured two points on the lower voltage, we got a reading of 1.37.

Starting from the point on the right, we measured the potential difference at 1 cm intervals going towards the line. We used excel to record the data and to create a  Potential vs Position graph as shown below.

Immersion Heater

Immersion Heater

In this activity we took a 3.41 W immersion heater and submerged it in a given amount of water for 10 minutes. We were to calculate how much the change in temperature of the water would be after 10 minutes had elapsed. While we performed the calculations, the water was heated and the temperature change with respect to time was tracked using LoggerPro software. At the end we compared our experimental results with the results of the LoggerPro data. The change in temperature was indeed within the uncertainty range that we had calculated.

Ohm's Law, Relating Electric Potential, Current, and Resistance

Measuring Electric Current

Here we connected a battery to power a light bulb via alligator clips. We see that all the components are functional and the bulb lights up.

We connected an ammeter to a position before and after the bulb as shown in the diagram below to measure the electric current at those positions. We found that the current was 110 +/- 5 mA for both positions. This means that the current remains constant.

Measuring Electric Potential, Current and Resistance

(Picture of the resistor used)

Our next set up was a bit different and consisted of a voltage supplier and a resistor shown in the picture below. An ammeter was again used to measure the current and a voltmeter was used to measure electric potential.

We plotted a graph showing the current vs voltage for the data that we obtained from our resistor and the data that our neighboring group obtained from their resistor. The two lines were plotted and fitted with linear trendlines which suggests that there is a linear relationship between current (I) and voltage (V), (they are proportional). We can say that I=kV, where k is a constant. We learned that solving for k (k=I/V), we get the resistance (R). So now we can say that I=RV. From our graph, the slopes of the trendlines are actually the resistance of the resistors. The two resistors were different and had a different resistance. This is shown by the slopes of the two lines, they are both different.

Other Variables

In this part of the experiment, we measured the resistance of 7 different wires with unique characteristics. They varied in material, length, and diameter. Plotting a graph of resistance vs length we get the curve shown above. The majority of the points follow a positive linear slope, however two of the points make the trendline inaccurate for this graph. We found the following relationships to hold true following our analysis.

Area (A)  is inversely proportional to the resistance (R), the wire length (L) is directly proportional to the resistance, and the resistivity of the material (rho) is directly proportional to the resistance. The data point that does not follow the trendline in the resistance vs length graph has to do with it having a larger cross sectional area (larger diameter).