Question

the total resistance in a circuit with two parallel resistors is 2 ohms and ( r_1 ) is 6 ohms. using the equation for ( r_2 ), in terms of ( r_t ) and ( r_1 ), what is ( r_2 )? ( r_2 ) is (square) ohms.

Explanation:

Step1: Recall parallel resistance formula

The formula for total resistance $R_T$ of two parallel resistors is:
$$\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2}$$

Step2: Rearrange to solve for $R_2$

Substitute $\frac{1}{R_1}$ to the left side:
$$\frac{1}{R_2} = \frac{1}{R_T} - \frac{1}{R_1}$$
Combine the fractions on the right:
$$\frac{1}{R_2} = \frac{R_1 - R_T}{R_T R_1}$$
Take reciprocal of both sides:
$$R_2 = \frac{R_T R_1}{R_1 - R_T}$$

Step3: Plug in given values

Given $R_T=2$ ohms, $R_1=6$ ohms:
$$R_2 = \frac{2 \times 6}{6 - 2}$$

Step4: Calculate the result

Simplify numerator and denominator:
$$R_2 = \frac{12}{4} = 3$$

Answer:

$R_2$ is 3 ohms.