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.