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.
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To find the equivalent resistance between points A and B, we can use a combination of series and parallel resistance equations.
First, calculate the resistance between two adjacent points on the star. Since there are the two resistors connected in parallel between each pair of adjacent points, we can use the formula for the equivalent resistance of two resistors in parallel:
1/Req = 1/R1 + 1/R2
R1 = R2 = R = 6 Ω
Substituting the values into the formula:
1/Req = 1/6 Ω + 1/6 Ω
= 2/6 Ω
= 1/3 Ω
Now, let's call the resistance between two adjacent points as Req1.
Next, we need to calculate the resistance between the center point and one of the points on the star. This can be done by using the formula for the equivalent resistance of two resistors in series:
Req2 = R1 + R2
= R + Req1
= 6 Ω + (1/3 Ω)
= 6 Ω + 1/3 Ω
= (18 Ω + 1 Ω) / 3
= 19/3 Ω
Now, we have the equivalent resistance when going from the center point to one of the points on the star. Let's call this as Req2.
Finally, to find the equivalent resistance between points A and B, we have to consider that the two resistors connected in parallel between A and B have a total resistance of 2Req2:
ReqAB = 2 * Req2
= 2 * (19/3) Ω
= 38/3 Ω
Therefore, the equivalent resistance between points A and B is 38/3 Ω or approximately 12.67 Ω.