Introduction:

In electrical circuits, resistors are often connected in series or parallel configurations. The equivalent resistance of resistors in series and parallel can be calculated using specific formulas. Let's consider two resistors, R1 and R2, and denote their equivalent resistances in series and parallel as r1 and r2, respectively. We need to determine the ratio R1/R2.

Series Resistance:

When resistors are connected in series, their equivalent resistance is the sum of individual resistances. Mathematically, we can express this as:

R_series = R1 + R2

The equivalent resistance for resistors in series is always greater than the individual resistances. So, in this case, r1 > R1 and r1 > R2.

Parallel Resistance:

When resistors are connected in parallel, their equivalent resistance can be calculated using the following formula:

1/R_parallel = 1/R1 + 1/R2

To find the equivalent resistance (R_parallel), we take the reciprocal of the sum of the reciprocals of individual resistances. The equivalent resistance for resistors in parallel is always less than the smallest individual resistance. Therefore, r2 < r1="" and="" r2="" />< />

Determining R1/R2:

To find the ratio R1/R2, we divide the equations for r1 (series) and r2 (parallel):

R1/R2 = r1/r2

Substituting the equations for r1 and r2:

R1/R2 = (R1 + R2) / (1/R1 + 1/R2)

To simplify this expression, we can multiply the numerator and denominator by (R1 * R2):

R1/R2 = (R1 + R2) * (R1 * R2) / (R2 + R1)

By canceling out similar terms in the numerator and denominator, we get:

R1/R2 = R1 * R2 / (R1 + R2)

Conclusion:

The ratio R1/R2 for the equivalent resistances of R1 and R2 in series and parallel configurations is given by the expression R1 * R2 / (R1 + R2). This ratio depends on the values of R1 and R2 and can vary. It is important to note that the equivalent resistance in series is always greater than the individual resistances, while the equivalent resistance in parallel is always less than the smallest individual resistance.

R1/r2