To solve the problem of finding the highest resistance value among three unequal resistors connected in parallel, we first need to understand how resistors behave in parallel and how to calculate equivalent resistance. Given that two of the resistors are in the ratio of 1:2 and the total equivalent resistance is 1 ohm, we can derive the values step by step.
Understanding Parallel Resistance
When resistors are connected in parallel, the formula for calculating the equivalent resistance (R_eq) is given by:
- 1/R_eq = 1/R1 + 1/R2 + 1/R3
In this case, we know that R_eq = 1 ohm. Let's denote the resistors as R1, R2, and R3. Since R1 and R2 are in the ratio of 1:2, we can express them as:
Setting Up the Equation
Now, we need to incorporate R3 into our equation. The equivalent resistance formula becomes:
To simplify this, we can find a common denominator for the first two terms:
- 1/x + 1/(2x) = 2/(2x) + 1/(2x) = 3/(2x)
Substituting this back into the equation gives us:
Rearranging the Equation
Next, we can rearrange this equation to isolate R3:
To combine these fractions, we need a common denominator:
Taking the reciprocal gives us:
Finding the Values of Resistors
Now we have expressions for R1, R2, and R3 in terms of x. To find the maximum resistance, we need to ensure that all resistances are positive. This means:
Solving this inequality gives:
Now, let's choose a value for x that satisfies this condition. If we set x = 2, we can calculate the resistances:
- R1 = 2 ohms
- R2 = 4 ohms
- R3 = (2*2)/(2*2 - 3) = 4/1 = 4 ohms
Identifying the Highest Resistance
With these values, we find:
- R1 = 2 ohms
- R2 = 4 ohms
- R3 = 4 ohms
Among these, the highest resistance value is 4 ohms. Therefore, the answer to your question is that the highest resistance value among the three resistors is:
4 ohms
