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?
6 Ù and 12 Ù in parallel have an equivalent resistance between two vertices, Req, that given by
1/6 + 1/12 = 1/Req
3/12 = 1/Req = 1/4
Req = 4 Ù
My cut and pasted "Ohm" symbols came out as Ù
To calculate the equivalent resistance between any two vertices of the triangle, we can use the concept of series and parallel resistors.
Step 1: Start by labelling the resistors as A, B, and C.
Step 2: At one vertex of the triangle, we have two resistors in parallel (A and B). The total resistance for resistors in parallel is given by the formula:
1/R_total = 1/R₁ + 1/R₂
In this case:
1/R_AB = 1/6 + 1/6 = 2/6 = 1/3
So the equivalent resistance between vertices A and B is 3 Ω.
Step 3: Move to the next vertex. We have a resistor (C), and the equivalent resistance between vertices B and C will be:
R_BC = R_C = 6 Ω
Step 4: Finally, move to the last vertex. We have two resistors in parallel (A and C). The total resistance for resistors in parallel is:
1/R_AC = 1/6 + 1/6 = 2/6 = 1/3
So the equivalent resistance between vertices A and C is also 3 Ω.
Therefore, the equivalent resistance between any two vertices of the triangle is 3 Ω.
To find the equivalent resistance between any two vertices of the triangle, we can use the concept of parallel and series resistances.
In this case, since the resistors are connected in a way that forms a triangle, it means that each resistor is connected to the other two resistors. This forms three parallel branches of resistors.
When resistors are connected in parallel, the equivalent resistance (R_eq) can be calculated using the formula:
1/R_eq = 1/R₁ + 1/R₂ + 1/R₃
Where R₁, R₂, and R₃ are the resistances of the three parallel branches.
Given that each resistor has a resistance of 6 Ω, we can substitute this value into the equation:
1/R_eq = 1/6 Ω + 1/6 Ω + 1/6 Ω
Simplifying this equation, we get:
1/R_eq = 3/6 Ω
= 1/2 Ω
Now, to find R_eq, we take the reciprocal of both sides of the equation:
R_eq = 2 Ω
Therefore, the equivalent resistance between any two vertices of the triangle is 2 Ω.