Question

You make a parallel combination of resistors consisting of resistor A having a very large resistance and resistor B having a very small resistance. The equivalent resistance for this combination will be: a) slightly greater than the resistance of the resistor A. b) slightly less than the resistance of the resistor \(\mathrm{A}\). c) slightly greater than the resistance of the resistor B. d) slightly less than the resistance of the resistor B.

Short answer

Question: When a resistor A has a very large resistance (almost infinity) and a resistor B has a very small resistance (almost zero) are connected in parallel, the equivalent resistance of the parallel resistors will be _____: a) equal to the resistance of the resistor A b) equal to the resistance of the resistor B c) slightly greater than the resistance of the resistor B d) slightly smaller than the resistance of the resistor A Answer: c) slightly greater than the resistance of the resistor B

Step-by-step solution

Formula for parallel resistors

Recall the formula for calculating the equivalent resistance when two resistors are in parallel: \(R_{eq} = \frac{R_1 \times R_2}{R_1 + R_2}\)

Understand the problem

We are given that resistor A has a very large resistance (almost infinity) and resistor B has a very small resistance (almost zero). Let's consider their respective resistances as \(R_A\) and \(R_B\).

Apply the formula

Apply the formula for equivalent resistance with the given values: \(R_{eq} = \frac{R_A \times R_B}{R_A + R_B}\)

Analyze the result

Considering that \(R_A\) is very large and \(R_B\) is very small, the denominator (\(R_A + R_B\)) will be very close to \(R_A\) and the numerator (\(R_A \times R_B\)) will be a very small value compared to \(R_A\). Therefore, when divided, the equivalent resistance will be slightly greater than the very small resistance \(R_B\). Thus, the correct answer is: c) slightly greater than the resistance of the resistor B.

Key concepts

Equivalent Resistance

In electrical circuits, equivalent resistance simplifies complex connections by representing combined resistors as a single resistance value. For parallel resistors, the equivalent resistance is always less than the resistance of the smallest resistor in the group.
This occurs because, in a parallel circuit, each additional resistor provides another path for current to flow, effectively increasing the total current and thus reducing the overall resistance.
For two resistors placed in parallel, the equivalent resistance is calculated using the formula:

• Formula for parallel resistors: \( R_{eq} = \frac{R_1 \times R_2}{R_1 + R_2} \)

The equivalent resistance value allows us to replace the entire resistor network with a single resistor, simplifying analysis and understanding of the circuit's total behavior.

Resistor Combinations

Resistors can be combined in several ways to achieve desired resistance levels. The two common methods are series and parallel combinations. Each method impacts the total resistance in a circuit differently.

• Series Combination: Here, resistors are connected end-to-end, and the total resistance is simply the sum of individual resistances. The formula is \( R_{eq} = R_1 + R_2 + \ldots + R_n \).
• Parallel Combination: In this setup, resistors are connected across the same two points, sharing both nodes. The total resistance in a parallel circuit is less than the smallest individual resistor. The formula used is: \( \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_n} \).

Understanding these combinations helps in designing circuits to achieve particular current and voltage characteristics, as seen in most electronic devices.

Resistance Calculation

Calculating resistance, especially equivalent resistance in complex circuits, requires careful application of mathematical formulas. Here, attention to detail is critical.
For parallel combinations, the formula \( R_{eq} = \frac{R_1 \times R_2}{R_1 + R_2} \) simplifies the process, enabling straightforward calculations even if the resistors vary widely in value.
When assessing a circuit with both very high and very low resistance, like in the exercise, consider how each resistor contributes to the total. With one very large and one very small resistor:

• The larger resistor contributes minimally to the numerator \( R_A \times R_B \), making it close to the smaller resistance \( R_B \).
• The denominator \( R_A + R_B \) primarily reflects the larger resistance \( R_A \), keeping the overall equivalent resistance only marginally greater than the smallest \( R_B \).

This analytical approach assists in predicting how changes in resistor values can impact total circuit resistance.