Resistor Parallel Circuit Analysis

When multiple resistors are connected in a parallel circuit, two fundamental principles apply:

• The voltage across each individual resistor is the same. This voltage is also equal to the voltage across the entire parallel combination.
• The total current flowing into the parallel combination is the sum of the currents flowing through each individual resistor.

We are given the equivalent resistance (\(R_{eq}\)) of four resistors in parallel as \(20 \text{ ohms}\) and the current flowing through each resistor. Our goal is to find the value of each individual resistor.

Current Total Calculation

The first step is to determine the total current (\(I_{total}\)) flowing through the parallel circuit. We can find this by summing the individual currents given:

• Current through the first resistor (\(I_1\)) = \(0.6 \text{ A}\)
• Current through the second resistor (\(I_2\)) = \(0.3 \text{ A}\)
• Current through the third resistor (\(I_3\)) = \(0.2 \text{ A}\)
• Current through the fourth resistor (\(I_4\)) = \(0.1 \text{ A}\)

Using the principle of current division in parallel circuits:

\[ I_{total} = I_1 + I_2 + I_3 + I_4 \] \[ I_{total} = 0.6 \text{ A} + 0.3 \text{ A} + 0.2 \text{ A} + 0.1 \text{ A} \] \[ I_{total} = 1.2 \text{ A} \]

Voltage Across Resistors

Next, we can determine the voltage (\(V\)) across the parallel combination. Since the voltage is the same across all resistors in parallel, finding this voltage will allow us to calculate the individual resistances. We use Ohm's Law, which states \(V = I \times R\). Here, we'll use the total current and the equivalent resistance:

\[ V = I_{total} \times R_{eq} \] \[ V = 1.2 \text{ A} \times 20 \text{ ohms} \] \[ V = 24 \text{ V} \]

So, the voltage across each of the four resistors is \(24 \text{ V}\).

Resistor Individual Values

With the voltage across each resistor known, and the current through each resistor given, we can now apply Ohm's Law (\(R = V/I\)) to calculate the resistance of each individual resistor:

Therefore, the values of the four resistors are \(40 \text{ ohms}\), \(80 \text{ ohms}\), \(120 \text{ ohms}\), and \(240 \text{ ohms}\).

Verification of Equivalent Resistance

As a final check, we can verify if these calculated individual resistances yield the given equivalent resistance of \(20 \text{ ohms}\) using the formula for equivalent resistance in parallel:

\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \frac{1}{R_4} \] \[ \frac{1}{R_{eq}} = \frac{1}{40} + \frac{1}{80} + \frac{1}{120} + \frac{1}{240} \]

To sum these fractions, we find a common denominator, which is \(240\):

\[ \frac{1}{R_{eq}} = \frac{6}{240} + \frac{3}{240} + \frac{2}{240} + \frac{1}{240} \] \[ \frac{1}{R_{eq}} = \frac{6 + 3 + 2 + 1}{240} \] \[ \frac{1}{R_{eq}} = \frac{12}{240} \] \[ \frac{1}{R_{eq}} = \frac{1}{20} \] \[ R_{eq} = 20 \text{ ohms} \]

This matches the equivalent resistance given in the question, confirming our calculations for the individual resistor values are correct.

The values of the resistors are \(40 \text{ ohms}\), \(80 \text{ ohms}\), \(120 \text{ ohms}\), and \(240 \text{ ohms}\).