Three resistors, each of resistance 6 Ω, are connected in such a way that that they form the three sides of a triangle. What is the equivalent resistance between any two vertices of the triangle?
To find the equivalent resistance between any two vertices of the triangle, we can use the concept of resistors in parallel and series.
First, let's label the three vertices of the triangle as A, B, and C. We want to find the equivalent resistance between any two vertices, say A and B.
To do this, we need to determine the different combination of resistors that connect points A and B. There are two such combinations:
1. The direct connection between A and B
2. The connection through the third resistor (the one not directly connected to A and B).
Let's analyze each combination:
1. Direct Connection (AB):
When two resistors are connected directly in parallel, the equivalent resistance is given by the formula:
1/Req = 1/R1 + 1/R2
In this case, we have R1 = R2 = 6 Ω, hence:
1/Req = 1/6 Ω + 1/6 Ω = 2/6 Ω = 1/3 Ω
Taking the reciprocal of both sides, we find:
Req = 3 Ω
So, the equivalent resistance between A and B, when directly connected, is 3 Ω.
2. Connection through the third resistor (ACB or BCA):
When three resistors form a closed loop, the equivalent resistance is equal to their sum.
In this case, the sum of the resistors is:
R_total = R1 + R2 + R3 = 6 Ω + 6 Ω + 6 Ω = 18 Ω
So, the equivalent resistance between A and B, when connected through the third resistor, is 18 Ω.
Therefore, the equivalent resistance between vertices A and B of the triangle is 3 Ω when connected directly and 18 Ω when connected through the third resistor.
To find the equivalent resistance between any two vertices of the triangle, we can use the concept of series and parallel resistors.
Let's label the three vertices of the triangle as A, B, and C. We can assume that resistors R1, R2, and R3 are connected between vertices A and B, B and C, and C and A, respectively.
Now, let's break down the connections:
1. Resistor R1 is connected in series with resistor R2.
2. The combination of R1 and R2 is connected in parallel with resistor R3.
Using these connections, we can calculate the equivalent resistance step-by-step:
Step 1: Calculate the equivalent resistance of resistors R1 and R2, connected in series.
R1 + R2 = 6 Ω + 6 Ω = 12 Ω
Step 2: Calculate the equivalent resistance of the combination of R1 and R2 in parallel with R3.
The formula to find the equivalent resistance of two resistors in parallel is given by: 1/Req = 1/R1 + 1/R2
1/Req = 1/12 Ω + 1/6 Ω
1/Req = (2 + 4)/12 Ω
1/Req = 6/12 Ω
1/Req = 1/2
Req = 2 Ω
Therefore, the equivalent resistance between any two vertices of the triangle is 2 Ω.