Question

In the given circuit the equivalent resistance between the points \(\mathrm{A}\) and \(\mathrm{B}\) in \(\mathrm{ohm}\) is. (A) 9 (B) \(11.6\) (C) \(14.5\) (D) \(21.2\)

Short answer

The equivalent resistance between points A and B in the given circuit is approximately \(11.667\, \Omega\). Thus, the correct answer is (B).

Step-by-step solution

Identify the layout of the circuit and the resistors connection

For the given circuit, the resistors are connected in a combination of series and parallel. We have to observe the circuit to determine the connections and simplify the resistors step by step.

Firstly, simplify the parallel connected resistors

In the circuit, we can see that there are some resistors connected in parallel. Let's simplify them using the parallel resistors formula: For the resistors \(R_1=14\) and \(R_2=7\), connected in parallel: \[\frac{1}{R_{eq1}} = \frac{1}{14} + \frac{1}{7}\]

Calculate the equivalent resistance for the parallel resistors

Now, let's calculate the equivalent resistance for \(R_{eq1}\) from the calculation above: \[\frac{1}{R_{eq1}} = \frac{1+2}{14} = \frac{3}{14}\] Therefore, \(R_{eq1} = 4.667\, \Omega\).

Next, simplify the series connected resistors

Now, we have the simplified circuit with resistors \(R_{eq1}\), \(R_3=5\), and \(R_4=2\) connected in series. Let's calculate the equivalent resistance using the series resistors formula: \[R_{eq} = R_{eq1} + R_3 + R_4\]

Calculate the overall equivalent resistance

Finally, let's find out the overall equivalent resistance: \[R_{eq} = 4.667 + 5 + 2 = 11.667\, \Omega\] The equivalent resistance between points A and B in the given circuit is approximately \(11.667\, \Omega\). Thus, the correct answer is (B).

Key concepts

Circuit Analysis

Circuit analysis is a fundamental skill in electrical engineering and physics that involves determining the behavior and characteristics of various circuits. By analyzing a circuit, you can understand the flow of electricity, calculate the voltage drops, and find the equivalent resistance, which is crucial for predicting how a circuit will respond to different inputs.
Understanding the layout and connections of components is the first step. Here, recognize whether resistors are connected in series or parallel. In complex circuits, a combination of these configurations is typical. Breaking down these circuits into simpler series or parallel parts helps streamline the analysis.
When analyzing circuits, start by identifying all components and their connections. Note that components in series carry the same current, while those in parallel have the same voltage across them. This understanding is essential in calculating the net resistance and overall functioning of the circuit.

Series and Parallel Circuits

In electrical circuits, resistors can be connected in series, parallel, or a combination of both, impacting the overall resistance. Understanding these setups aids in analyzing and simplifying circuits.

• Series Circuits: Resistors in series are connected end-to-end, providing a single path for current. The total resistance is the sum of the individual resistances, using the formula:\[R_{total} = R_1 + R_2 + R_3 + \ldots\]

Series circuits have the same current flowing through every component, but the voltage is divided among them.

• Parallel Circuits: Here, components are connected across the same two nodes, offering multiple paths for the current. The total resistance is found using the formula:\[\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots\]

In parallel circuits, the voltage across each component is the same, while the current is divided based on resistance.
Combining these rules allows for simplification of complex circuits through step-by-step reduction, making calculations more manageable.

Resistor Calculations

Performing resistor calculations is essential for determining the equivalent resistance in a circuit, which influences its performance. Here’s how to approach it:

• Identify the Configuration: Determine if resistors are connected in series, parallel, or a mix of both. This affects the calculation approach.
• Series Resistances: Simply add up to find the total resistance. For example, if two resistors with values 4Ω and 6Ω are in series, their total resistance is 10Ω:

\[R_{eq} = 4 + 6 = 10\, \Omega\]

• Parallel Resistances: Use the reciprocal formula to find the equivalent resistance. Suppose two resistors 14Ω and 7Ω are parallel, calculate as follows:

\[\frac{1}{R_{eq}} = \frac{1}{14} + \frac{1}{7}\]
This simplifies to the equivalent resistance: \[R_{eq} = \frac{1}{\left(\frac{1}{14} + \frac{1}{7}\right)} = 4.67\, \Omega\]
Advanced circuits often involve finding the equivalent resistance in parts before combining them as seen in the original exercise, making step-by-step calculations pivotal. Through practice, interpreting and simplifying complex resistor networks becomes easier.