Two Resistors, $R_1$ And $R_2$, Are Connected In Parallel. Given That $R_2 = 221.0$ Ohms And The Equivalent Resistance Of The Combination Is $120.7$ Ohms, What Is The Value Of $R_1$?(Unit = Ohm)

When two resistors, $R_1$ and $R_2$, are connected in parallel, they form a circuit where the current divides between the two resistors. The equivalent resistance of the combination is given by the formula:

$$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$$

where $R_{eq}$ is the equivalent resistance of the combination.

Given Information

We are given that $R_2 = 221.0$ ohms and the equivalent resistance of the combination is $R_{eq} = 120.7$ ohms. We need to find the value of $R_1$.

Using the Formula to Find $R_1$

We can start by substituting the given values into the formula:

$$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$$

$$\frac{1}{120.7} = \frac{1}{R_1} + \frac{1}{221.0}$$

To solve for $R_1$, we can first find a common denominator for the two fractions on the right-hand side:

$$\frac{1}{R_1} = \frac{1}{120.7} - \frac{1}{221.0}$$

$$\frac{1}{R_1} = \frac{221.0 - 120.7}{120.7 \times 221.0}$$

$$\frac{1}{R_1} = \frac{100.3}{26651.9}$$

Now, we can take the reciprocal of both sides to solve for $R_1$:

$$R_1 = \frac{26651.9}{100.3}$$

Calculating the Value of $R_1$

Using a calculator, we can find the value of $R_1$:

$$R_1 = \frac{26651.9}{100.3} = 265.5$$

Therefore, the value of $R_1$ is approximately 265.5 ohms.

Conclusion

In this problem, we used the formula for the equivalent resistance of two resistors connected in parallel to find the value of $R_1$. We were given the values of $R_2$ and $R_{eq}$, and we were able to solve for $R_1$ using algebraic manipulation. This problem demonstrates the importance of understanding the basics of circuit analysis and the use of formulas to solve problems in physics.

Additional Information

When two resistors are connected in parallel, the equivalent resistance of the combination is always less than the resistance of either individual resistor. This is because the current divides between the two resistors, reducing the overall resistance of the circuit.

Real-World Applications

The concept of parallel resistors is used in a wide range of real-world applications, including:

• Electronic circuits
• Electrical power distribution systems
• Audio equipment
• Medical devices

In each of these applications, the equivalent resistance of the combination of resistors is critical in determining the overall performance of the circuit.

Common Mistakes to Avoid

When working with parallel resistors, it's essential to avoid common mistakes such as:

• Assuming that the equivalent resistance of the combination is equal to the resistance of either individual resistor
• Failing to account for the current division between the two resistors
• Not using the correct formula to calculate the equivalent resistance of the combination

By understanding the basics of parallel resistors and avoiding common mistakes, you can ensure accurate calculations and reliable results in a wide range of applications.

Final Thoughts

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In the previous article, we discussed the basics of parallel resistors and how to find the value of $R_1$ given the equivalent resistance of the combination and the value of $R_2$. In this article, we'll answer some frequently asked questions about parallel resistors to help you better understand this important concept in physics.

Q: What is the difference between series and parallel resistors?

A: Series resistors are connected one after the other, so the current flows through each resistor in sequence. Parallel resistors, on the other hand, are connected between the same two points, so the current divides between the two resistors.

Q: How do you calculate the equivalent resistance of two resistors connected in parallel?

A: The equivalent resistance of two resistors connected in parallel is given by the formula:

$$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$$

Q: What is the formula for the equivalent resistance of multiple resistors connected in parallel?

A: The formula for the equivalent resistance of multiple resistors connected in parallel is:

$$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n}$$

Q: How do you find the value of $R_1$ given the equivalent resistance of the combination and the value of $R_2$?

A: To find the value of $R_1$ given the equivalent resistance of the combination and the value of $R_2$, you can use the formula:

$$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$$

$$R_1 = \frac{R_2 \times R_{eq}}{R_2 - R_{eq}}$$

Q: What is the significance of the equivalent resistance of a circuit?

A: The equivalent resistance of a circuit is a measure of the total resistance of the circuit. It is an important concept in physics and engineering, as it determines the overall performance of the circuit.

Q: How do you calculate the current through a resistor in a parallel circuit?

A: To calculate the current through a resistor in a parallel circuit, you can use Ohm's law:

$$I = \frac{V}{R}$$

where $I$ is the current, $V$ is the voltage, and $R$ is the resistance.

Q: What is the difference between a parallel circuit and a series circuit?

A: A parallel circuit is a circuit where the current divides between two or more resistors. A series circuit, on the other hand, is a circuit where the current flows through each resistor in sequence.

Q: How do you determine the equivalent resistance of a circuit with multiple parallel branches?

A: To determine the equivalent resistance of a circuit with multiple parallel branches, you can use the formula:

$$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n}$$

Q: What is the significance of the current division in a parallel circuit?

A: The current division in a parallel circuit is a measure of the amount of current that flows through each resistor. It is an important concept in physics and engineering, as it determines the overall performance of the circuit.

Conclusion

In this article, we answered some frequently asked questions about parallel resistors to help you better understand this important concept in physics. Whether you're working on a complex electronic circuit or a simple audio equipment, the concept of parallel resistors is essential in determining the overall performance of the circuit.