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 _____ Ohms.

Feb 27, 2025 by ADMIN 188 views

Understanding the Basics of Parallel Resistors

When two resistors are connected in parallel, the total resistance of the circuit is less than the individual resistances of the two resistors. This is because the current can flow through both resistors simultaneously, reducing the overall resistance. In this article, we will explore the equation for calculating the total resistance in a circuit with two parallel resistors and use it to find the value of $R_2$ .

The Equation for Total Resistance in a Parallel Circuit

The equation for total resistance in a parallel circuit is given by:

\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2}

where $R_T$ is the total resistance, $R_1$ is the resistance of the first resistor, and $R_2$ is the resistance of the second resistor.

Given Values and the Equation for $R_2$

We are given that the total resistance in the circuit is 2 ohms, and $R_1$ is 6 ohms. We need to find the value of $R_2$ using the equation for $R_2$ in terms of $R_T$ and $R_1$ .

Deriving the Equation for $R_2$

To derive the equation for $R_2$ , we can start with the equation for total resistance in a parallel circuit:

\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2}

We can rearrange this equation to isolate $R_2$ :

\frac{1}{R_2} = \frac{1}{R_T} - \frac{1}{R_1}

Taking the reciprocal of both sides, we get:

R_2 = \frac{1}{\frac{1}{R_T} - \frac{1}{R_1}}

Simplifying the expression, we get:

R_2 = \frac{R_1 R_T}{R_T - R_1}

Substituting the Given Values

We are given that $R_T$ is 2 ohms and $R_1$ is 6 ohms. Substituting these values into the equation for $R_2$ , we get:

R_2 = \frac{6 \times 2}{2 - 6}

Simplifying the expression, we get:

R_2 = \frac{12}{-4}

R_2 = -3

However, since resistance cannot be negative, we need to take the absolute value of the result:

R_2 = |-3|

R_2 = 3

Therefore, the value of $R_2$ is 3 ohms.

Conclusion

In this article, we derived the equation for $R_2$ in terms of $R_T$ and $R_1$ and used it to find the value of $R_2$ in a circuit with two parallel resistors. We found that the value of $R_2$ is 3 ohms.

Key Takeaways

The equation for total resistance in a parallel circuit is given by $\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2}$ .
The equation for $R_2$ in terms of $R_T$ and $R_1$ is given by $R_2 = \frac{R_1 R_T}{R_T - R_1}$ .
The value of $R_2$ is 3 ohms when $R_T$ is 2 ohms and $R_1$ is 6 ohms.
Frequently Asked Questions (FAQs) about Parallel Resistors =============================================================

Q: What is the total resistance in a circuit with two parallel resistors?

A: The total resistance in a circuit with two parallel resistors is given by the equation $\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2}$ , where $R_T$ is the total resistance, $R_1$ is the resistance of the first resistor, and $R_2$ is the resistance of the second resistor.

Q: How do I calculate the total resistance in a circuit with two parallel resistors?

A: To calculate the total resistance in a circuit with two parallel resistors, you can use the equation $\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2}$ . Rearranging this equation, you can solve for $R_T$ :

R_T = \frac{R_1 R_2}{R_1 + R_2}

Q: What is the relationship between the total resistance and the individual resistances in a parallel circuit?

A: In a parallel circuit, the total resistance is less than the individual resistances. This is because the current can flow through both resistors simultaneously, reducing the overall resistance.

Q: Can I use the equation for total resistance in a series circuit to calculate the total resistance in a parallel circuit?

A: No, you cannot use the equation for total resistance in a series circuit to calculate the total resistance in a parallel circuit. The equation for total resistance in a series circuit is $R_T = R_1 + R_2$ , while the equation for total resistance in a parallel circuit is $\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2}$ .

Q: How do I find the value of $R_2$ in a circuit with two parallel resistors?

A: To find the value of $R_2$ in a circuit with two parallel resistors, you can use the equation $R_2 = \frac{R_1 R_T}{R_T - R_1}$ . Rearranging this equation, you can solve for $R_2$ :

R_2 = \frac{R_1 R_T}{R_T - R_1}

Q: What is the value of $R_2$ when $R_T$ is 2 ohms and $R_1$ is 6 ohms?

A: To find the value of $R_2$ when $R_T$ is 2 ohms and $R_1$ is 6 ohms, you can substitute these values into the equation $R_2 = \frac{R_1 R_T}{R_T - R_1}$ :

R_2 = \frac{6 \times 2}{2 - 6}

Simplifying the expression, you get:

R_2 = \frac{12}{-4}

R_2 = -3

However, since resistance cannot be negative, you need to take the absolute value of the result:

R_2 = |-3|

R_2 = 3

Therefore, the value of $R_2$ is 3 ohms.

Q: What is the significance of the equation for $R_2$ in a parallel circuit?

A: The equation for $R_2$ in a parallel circuit is significant because it allows you to calculate the value of $R_2$ when the total resistance and the resistance of the first resistor are known. This equation is useful in a variety of applications, including electronics and electrical engineering.

Q: Can I use the equation for $R_2$ in a parallel circuit to calculate the total resistance in a series circuit?

A: No, you cannot use the equation for $R_2$ in a parallel circuit to calculate the total resistance in a series circuit. The equation for total resistance in a series circuit is $R_T = R_1 + R_2$ , while the equation for $R_2$ in a parallel circuit is $R_2 = \frac{R_1 R_T}{R_T - R_1}$ . These equations are not equivalent and cannot be used interchangeably.