Understanding Resistors in Parallel and Current Distribution

When resistors are connected in parallel, the voltage across each resistor is the same. This is a fundamental property of parallel circuits. We are given that three resistors have resistances in the ratio 1 : 2 : 3. Let's denote their resistances as \(R_1\), \(R_2\), and \(R_3\). So, \(R_1 : R_2 : R_3 = 1 : 2 : 3\). We can write the resistances as \(R_1 = k\), \(R_2 = 2k\), and \(R_3 = 3k\) for some constant \(k\).

Applying Ohm's Law for Resistors in Parallel

To find the current through each resistor, we use Ohm's Law, which states that the voltage \(V\) across a resistor is equal to the current \(I\) flowing through it multiplied by its resistance \(R\). Mathematically, this is expressed as \(V = IR\). Rearranging Ohm's law to find the current, we get \(I = V/R\).

Since the voltage \(V\) is the same across all three resistors in parallel, the current through each resistor will depend only on its resistance:

• Current through \(R_1\): \(I_1 = \frac{V}{R_1}\)
• Current through \(R_2\): \(I_2 = \frac{V}{R_2}\)
• Current through \(R_3\): \(I_3 = \frac{V}{R_3}\)

Calculating the Current Ratio for Resistors in Parallel

Now, let's substitute the values of the resistances in terms of \(k\) into the current expressions:

• \(I_1 = \frac{V}{k}\)
• \(I_2 = \frac{V}{2k}\)
• \(I_3 = \frac{V}{3k}\)

We want to find the ratio of these currents, \(I_1 : I_2 : I_3\):

\(I_1 : I_2 : I_3 = \frac{V}{k} : \frac{V}{2k} : \frac{V}{3k}\)

Since \(V\) and \(k\) are common factors (and not zero), we can cancel \(\frac{V}{k}\) from each term in the ratio:

\(I_1 : I_2 : I_3 = 1 : \frac{1}{2} : \frac{1}{3}\)

To express this ratio as whole numbers, we need to find a common multiple of the denominators (1, 2, and 3). The least common multiple (LCM) of 1, 2, and 3 is 6. We multiply each term in the ratio by 6:

\(I_1 : I_2 : I_3 = 6 \times 1 : 6 \times \frac{1}{2} : 6 \times \frac{1}{3}\)

\(I_1 : I_2 : I_3 = 6 : 3 : 2\)

Final Current Ratio

Therefore, the ratio of the currents flowing through the three resistors connected in parallel is 6 : 3 : 2. This demonstrates how current divides in a parallel circuit based on the resistance; lower resistance allows more current to flow.