Solving for Equivalent Resistance in Series Circuits

This question asks us to find the single equivalent resistance value that can replace a group of resistors connected in series within a circuit. The given resistors have resistances of $50 \ \Omega$, $50 \ \Omega$, and $100 \ \Omega$. They are all connected in series.

Understanding Series Circuits

When resistors are connected in series, they are linked end-to-end, forming a single path for the electric current to flow. This means the same current passes through each resistor.

Calculating Equivalent Resistance in Series

To find the total or equivalent resistance ($R_{total}$) of resistors connected in series, you simply add their individual resistances together. The formula is:

$ R_{total} = R_1 + R_2 + R_3 + \dots $

In this specific problem, we have three resistors:

• $R_1 = 50 \ \Omega$
• $R_2 = 50 \ \Omega$
• $R_3 = 100 \ \Omega$

Using the formula for series resistance, we can calculate the equivalent resistance:

$ R_{total} = 50 \ \Omega + 50 \ \Omega + 100 \ \Omega $

Adding these values:

$ R_{total} = 100 \ \Omega + 100 \ \Omega $

$ R_{total} = 200 \ \Omega $

Therefore, a single resistor with a value of $200 \ \Omega$ can replace the three series resistors ($50 \ \Omega$, $50 \ \Omega$, and $100 \ \Omega$) without changing the overall current flow or voltage distribution in the circuit, assuming the voltage source remains the same.