Understanding Parallel Resistances and Their Equivalent Values

In this article, we will explore a problem related to parallel resistances. The problem focuses on finding the individual resistances when they are in a specific ratio and their combined equivalent resistance is given. This is a common scenario in electrical engineering and circuit analysis, and understanding it can help in solving more complex electrical circuit problems.

Problem Statement and Solution

Two resistances are in the ratio 1:2. When these resistances are connected in parallel, their equivalent resistance becomes 8 ohms.

Let the two resistances be ( R_1 ) and ( R_2 ). According to the problem, the ratio of these resistances is 1:2.

Mathematically, this can be expressed as:

( frac{R_1}{R_2} frac{1}{2} )

This implies:

( R_2 2R_1 )

When these resistors are connected in parallel, the formula for equivalent resistance ( R_{eq} ) is:

( frac{1}{R_{eq}} frac{1}{R_1} frac{1}{R_2} )

Substituting ( R_2 2R_1 ) into the formula, we get:

( frac{1}{R_{eq}} frac{1}{R_1} frac{1}{2R_1} )

Simplifying the above expression:

( frac{1}{R_{eq}} frac{1}{R_1} frac{1}{2R_1} frac{2}{2R_1} frac{1}{2R_1} frac{3}{2R_1} )

Thus, the equivalent resistance ( R_{eq} ) can be written as:

( R_{eq} frac{2R_1}{3} )

According to the problem, the equivalent resistance is 8 ohms:

( frac{2R_1}{3} 8 )

Solving for ( R_1 ), we multiply both sides by 3:

( 2R_1 24 )

Dividing by 2:

( R_1 12Omega )

Substituting ( R_1 12Omega ) back into ( R_2 2R_1 ):

( R_2 24Omega )

Verification

To verify the solution, we can check the equivalent resistance:

( frac{12}{12} frac{1}{24} frac{24 12}{24} frac{36}{24} frac{3}{2} )

Multiplying by 3, we get:

( frac{3}{2} times 3 8 Omega )

The solution checks out.

Conclusion

Understanding how resistances combine in parallel and how to solve for their individual values based on given ratios and equivalent resistance is crucial in electrical circuit analysis. This problem provides a fundamental example of such an analysis.

Keywords: parallel resistances, equivalent resistance, circuit analysis