Consider following circuit parallel 10 15 ohm

General
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consider following circuit parallel 10 15 ohm

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It seems like you’re discussing a parallel circuit with two resistors of 10 ohms and 15 ohms. Let’s break this down step by step with the necessary explanation and formulas.

Parallel Circuit with Resistors: Key Idea

In a parallel circuit, the voltage across all resistors is the same, but the total or equivalent resistance (R_\text{eq}) is different. The formula for calculating the total resistance in a parallel circuit is given by:

\frac{1}{R_\text{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots

Here, R_1, R_2, R_3, etc., represent the resistances of the individual resistors connected in parallel.

Solution for Two Resistors: 10 Ω and 15 Ω

We are calculating the equivalent resistance of two resistors, R_1 = 10 \, \Omega and R_2 = 15 \, \Omega, in parallel:

Step 1: Use the Parallel Resistance Formula

Substitute the values of R_1 and R_2 into the formula:

\frac{1}{R_\text{eq}} = \frac{1}{10} + \frac{1}{15}

Step 2: Find a Common Denominator

The least common denominator (LCD) of 10 and 15 is 30. Rewrite the fractions:

\frac{1}{10} = \frac{3}{30}, \quad \frac{1}{15} = \frac{2}{30}

Add the fractions:

\frac{1}{R_\text{eq}} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30}

Step 3: Simplify and Invert

Simplify the fraction \frac{5}{30}:

\frac{1}{R_\text{eq}} = \frac{1}{6}

Inverting gives the equivalent resistance:

R_\text{eq} = 6 \, \Omega

Final Answer

The equivalent resistance of the parallel circuit with two resistors (10 Ω and 15 Ω) is:

6 Ω

Additional Notes for Reference

  1. Why Does R_\text{eq} Decrease in a Parallel Circuit?

    • In a parallel circuit, more current paths are available as you add resistors, which decreases the overall resistance compared to the individual resistors.

  2. General Observation for Two Resistors in Parallel:

    • If you have exactly two resistors in parallel, you can use the shortcut formula:
R_\text{eq} = \frac{R_1 \cdot R_2}{R_1 + R_2}

For our case:

R_\text{eq} = \frac{10 \cdot 15}{10 + 15} = \frac{150}{25} = 6 \, \Omega

This gives the same result.

  1. Power Dissipation in Each Resistor:
    • If you want to calculate power dissipation, you’d need additional information like the voltage applied across the circuit or the total current.

If you have further questions about this problem or wish to see power calculations, voltage distribution, etc., feel free to ask! :blush: @username