Question

in electricity theory, the formula $\frac{1}{r}=\frac{1}{r_1}+\frac{1}{r_2}$ is used to find the total resistance $r$ of a circuit when two resistors with resistance $r_1$ and $r_2$ are connected in parallel. in such a parallel circuit, if the total resistance $r$ is 3 ohms and $r_2$ is 4 times $r_1$, find the resistances $r_1$ and $r_2$. r1 is ohms r2 is ohms question help: ebook written example

Explanation:

Step1: Substitute given values into formula

Given $R = 3$ ohms and $R_2=4R_1$. The formula for parallel - resistors is $\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}$. Substitute $R_2 = 4R_1$ and $R = 3$ into the formula: $\frac{1}{3}=\frac{1}{R_1}+\frac{1}{4R_1}$.

Step2: Combine the right - hand side terms

On the right - hand side, $\frac{1}{R_1}+\frac{1}{4R_1}=\frac{4 + 1}{4R_1}=\frac{5}{4R_1}$. So, $\frac{1}{3}=\frac{5}{4R_1}$.

Step3: Cross - multiply

Cross - multiplying gives $4R_1=15$.

Step4: Solve for $R_1$

Dividing both sides by 4, we get $R_1=\frac{15}{4}=3.75$ ohms.

Step5: Solve for $R_2$

Since $R_2 = 4R_1$, then $R_2=4\times3.75 = 15$ ohms.

Answer:

$R_1$ is $3.75$ ohms, $R_2$ is $15$ ohms