Question

In the circuit shown in fig the potential difference across \(3 \Omega\) is. (A) \(2 \mathrm{~V}\) (B) \(4 \mathrm{~V}\) (C) \(8 \mathrm{~V}\) (D) \(16 \mathrm{~V}\)

Short answer

The potential difference across the \(3\Omega\) resistor is \(2.4V\), which is closest to option (A) \(2V\). Therefore, the answer is (A) \(2V\).

Step-by-step solution

Identify the series and parallel components of the circuit

Observe the circuit and identify which resistors are in series and which are in parallel. In this case, we can see that the \(3\Omega\) and \(6\Omega\) resistors are in parallel, and the combination is in series with the \(12\Omega\) resistor.

Calculate the equivalent resistance of the parallel components

For resistors connected in parallel, the equivalent resistance is given by: \[\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}.\] In our case, \(R_1 = 3\Omega\) and \(R_2 = 6\Omega\). Substitute the values and calculate the equivalent resistance of the parallel combination: \[\frac{1}{R_{eq}} = \frac{1}{3} + \frac{1}{6} = \frac{1}{2},\] so the equivalent resistance is: \[R_{eq} = 2\Omega.\]

Calculate the total resistance of the circuit

Now that we have found the equivalent resistance of the parallel combination, the total resistance in the circuit is the sum of the resistance of this combination and the resistance of the \(12\Omega\) resistor: \[R_{total} = 2\Omega + 12\Omega = 14\Omega.\]

Calculate the current flowing through the circuit

Using Ohm's Law (\(V = IR\)), we can find the current flowing through the circuit. We are given that the total voltage in the circuit is \(28V\): \[I = \frac{V}{R} = \frac{28V}{14\Omega} = 2A.\] The same current flows through the \(12\Omega\) resistor and the parallel combination of the \(3\Omega\) and \(6\Omega\) resistors.

Calculate the voltage across the parallel combination

The voltage across the parallel combination is the same as the voltage across the \(12\Omega\) resistor, as they are in series. Using Ohm's Law, we can calculate the voltage across the \(12\Omega\) resistor: \[V_{12\Omega} = I \times R_{12\Omega} = 2A \times 12\Omega = 24V.\]

Calculate the current through the \(3\Omega\) resistor

Using the current divider rule for parallel circuits, the current through the \(3\Omega\) resistor is given by: \[I_{3\Omega} = I_{total} \times \frac{R_{eq}}{R_{3\Omega} + R_{eq}} = 2A \times \frac{2\Omega}{3\Omega + 2\Omega} = 2A \times \frac{2}{5} = \frac{4}{5}A.\]

Calculate the potential difference across the \(3\Omega\) resistor

Finally, we can use Ohm's Law to calculate the potential difference across the \(3\Omega\) resistor: \[V_{3\Omega} = I_{3\Omega} \times R_{3\Omega} = \frac{4}{5}A \times 3\Omega = \frac{12}{5}A = 2.4V.\] The potential difference across the \(3\Omega\) resistor is \(2.4V\), which is closest to option (A) \(2V\). Therefore, the answer is (A) \(2V\).

Key concepts

Resistors in Parallel

Resistors in parallel share the same two nodes. This configuration allows them both to establish an independent voltage drop while sharing equally divided portions of a common current. To identify resistors in parallel in a circuit, look for nodes that connect directly to both ends of the resistors. The voltage across each pair of parallel resistors is identical.

The formula to calculate the equivalent resistance for resistors in parallel is given by:\[\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}\] This formula indicates how multiple pathways for the current make the equivalent resistance always lesser than the smallest resistor in the group. It highlights how parallel resistors decrease the total resistance in a circuit.

Current Divider Rule

The current divider rule is a straightforward application useful in determining the current flowing through individual resistors in parallel. Once you've calculated the equivalent resistance of the parallel resistors, you can easily distribute the current across each resistor based on their resistances.

The current through any resistor, say the one labeled as \(R_1\), among the parallel resistors is given by:\[I_{R1} = I_{total} \times \frac{R_{eq}}{R_1 + R_{eq}} \] Using this rule helps in identifying how currents select paths based on resistance values, with lower resistance paths carrying higher proportions of the total current.

Equivalent Resistance

Equivalent resistance simplifies complex circuits by reducing multiple resistors into a single value, making calculations easier and more intuitive. To find the equivalent resistance of resistors in parallel, use the reciprocal of the sum of reciprocals formula. The process allows conversion of the diverse resistive properties into one concise resistance figure.

In our given case with resistors of \(3\Omega\) and \(6\Omega\), applying the formula: \[ \frac{1}{R_{eq}} = \frac{1}{3} + \frac{1}{6} = \frac{1}{2} \] results in an equivalent resistance of \(2\Omega\). This calculation serves as the bedrock for further determining total resistance and current flows in the circuit.

Potential Difference

Potential difference, often recognized as voltage, represents the work needed to move a charge between two points in a circuit. It's essential to understand that the potential difference measured across parallel resistors is consistent for all components within that part of the circuit.

You determine it by calculating the current through the circuit using Ohm's Law and then multiplying that current by individual resistances. For our \(3\Omega\) resistor:\[ V_{3\Omega} = I_{3\Omega} \times R_{3\Omega} = \frac{4}{5} A \times 3\Omega = 2.4V \]This information highlights how potential difference arises directly from current and resistance attributes along the circuit's path.