Consider a five-pointed star made of resistors as shown below. Each resistor has resistance R=6 Ω. Find the equivalent resistance in Ohms between the points A and B.
To find the equivalent resistance between points A and B in the given star structure, we need to first analyze the circuit and understand its configuration.
Let's label the resistors in the star structure as R1, R2, R3, R4, and R5, from top to bottom, respectively.
To simplify the analysis, we can start by considering the resistors in pairs. Let's label the points where the resistors are connected in pairs as C, D, E, F, and G, as shown below:
```
A
|
R1---C---R2
| |
R5 B R3
| |
R4---G---R5
|
D
```
1. Resistance between points A and C (R_AC): In this case, R1 and R2 are connected in parallel. The formula for the equivalent resistance of two resistors in parallel is:
1/R_AC = 1/R1 + 1/R2
Plugging in the values:
1/R_AC = 1/6Ω + 1/6Ω (since R1 = R2 = 6Ω)
1/R_AC = 2/6Ω
R_AC = 6/2Ω
R_AC = 3Ω
2. Resistance between points A and G (R_AG): In this case, R3 and R_AC (the equivalent resistance found in step 1) are connected in parallel. Again, we use the formula for the equivalent resistance of two resistors in parallel:
1/R_AG = 1/R3 + 1/R_AC
Plugging in the values:
1/R_AG = 1/6Ω + 1/3Ω (since R3 = 6Ω and R_AC = 3Ω from step 1)
1/R_AG = 1/6Ω + 2/6Ω
1/R_AG = 3/6Ω
R_AG = 6/3Ω
R_AG = 2Ω
3. Resistance between points A and D (R_AD): In this case, R4 and R_AG (the equivalent resistance found in step 2) are connected in series. The formula for the equivalent resistance of two resistors in series is simply their sum:
R_AD = R4 + R_AG
Plugging in the values:
R_AD = 6Ω + 2Ω
R_AD = 8Ω
Finally, the equivalent resistance between points A and B is equal to R_AD since they are directly connected:
Equivalent Resistance (R_AB) = R_AD = 8Ω.
Therefore, the equivalent resistance between points A and B in the given star structure is 8Ω.