Question
How would you find the resistance of a parallel circuit with n identical resistors?
Answers
GPT-4o mini
To find the total resistance of a parallel circuit containing \( n \) identical resistors, you can use the formula for the equivalent resistance \( R_{\text{eq}} \) of resistors in parallel.
When \( n \) identical resistors, each with resistance \( R \), are connected in parallel, the equivalent resistance \( R_{\text{eq}} \) can be calculated using the formula:
\[
\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}
\]
Since all resistors are identical:
\[
R_1 = R_2 = \cdots = R_n = R
\]
The equation simplifies to:
\[
\frac{1}{R_{\text{eq}}} = \frac{n}{R}
\]
Taking the reciprocal gives:
\[
R_{\text{eq}} = \frac{R}{n}
\]
Thus, the total resistance of \( n \) identical resistors in parallel is given by:
\[
R_{\text{eq}} = \frac{R}{n}
\]
This means that the equivalent resistance decreases as more identical resistors are added in parallel.
When \( n \) identical resistors, each with resistance \( R \), are connected in parallel, the equivalent resistance \( R_{\text{eq}} \) can be calculated using the formula:
\[
\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}
\]
Since all resistors are identical:
\[
R_1 = R_2 = \cdots = R_n = R
\]
The equation simplifies to:
\[
\frac{1}{R_{\text{eq}}} = \frac{n}{R}
\]
Taking the reciprocal gives:
\[
R_{\text{eq}} = \frac{R}{n}
\]
Thus, the total resistance of \( n \) identical resistors in parallel is given by:
\[
R_{\text{eq}} = \frac{R}{n}
\]
This means that the equivalent resistance decreases as more identical resistors are added in parallel.