Question

Learning Goal:

To learn to calculate the equivalent resistance of the circuits combining series and parallel connections.

Resistors are often connected to each other in electric circuits. Finding the equivalent resistanceof combinations of resistors is a common and important task. Equivalent resistance is defined as the single resistance that can replace the given combination of resistors in such a manner that the currents in the rest of the circuit do not change.

Finding the equivalent resistance is relatively straighforward if the circuit contains only series and parallel connections of resistors.

An example of a series connection is shown in the diagram:

(Figure 1)

For such a connection, the current is the same for all individual resistors and the total voltage is the sum of the voltages across the individual resistors.

Using Ohm's law (R=VI), one can show that, for a series connection, the equivalent resistance is the sum of the individual resistances.

Mathematically, these relationships can be written as:

I=I1=I2=I3=...

V=V1+V2+V3+...

Req?series=R1+R2+R3+...

An example of a parallel connection is shown in the diagram:

(Figure 2)

For resistors connected in parallel the voltage is the same for all individual resistors because they are all connected to the same two points (A and B on the diagram). The total current is the sum of the currents through the individual resistors. This should makes sense as the total current "splits" at points A and B.

Using Ohm's law, one can show that, for a parallel connection, the reciprocal of the equivalent resistance is the sum of the reciprocals of the individual resistances.

Mathematically, these relationships can be written as:

V=V1=V2=V3=...

I=I1+I2+I3+...

1Req?parallel=1R1+1R2+1R3+...

NOTE: If you have already studied capacitors and the rules for finding the equivalent capacitance, you should notice that the rules for the capacitors are similar - but not quite the same as the ones discussed here.

In this problem, you will use the the equivalent resistance formulas to determine Req for various combinations of resistors.

1

R1 R2

2

3

4

Part A

For the combination of resistors shown, find the equivalent resistance between points A and B.

(Figure 3)

Express your answer in Ohms.

Part B

For the set-up shown, find the equivalent resistance between points A and B.

(Figure 4)

Express your answer in Ohms.

Answer 1

Equivalent resistance in series = sigma Ri figure 1 Req = sigma^3_i = 1 Ri = R_1 + R_2 + R_3 Figure 2 since the resistors are in parallel 1/R_eq = sigma^n_i = 1 1/Ri 1/R_eq = 1/R_1 + 1/R_2 + 1/R_3 Figure 3 series combination R_eq = sigma^3_i = 1 Ri = 2 + 3 + 4 = 9 Ohm Figure 4 parallel combination 1/R_eq = 1/R_1 + 1/R_2 1/R_eq = 1/3 + 1/6 R_eq = 3 times 6/(3 + 6) = 2 Ohm