To solve this problem, we'll use Ohm's Law and Kirchhoff's Voltage Law (KVL). Here's how we can approach it:

Step-by-step Solution:

1. Identify the Given Values:

   • Voltage source $V_s = 10 \, V$
   • Voltage across $R_2 = v_2 = 2 \, V$
   • Resistor $R_1 = 5 \, \Omega$

2. Analyze the Circuit:

   • If $V_s = 10 \, V$ is the total voltage and $v_2 = 2 \, V$ is the voltage drop across $R_2$, the remaining voltage drop must be across $R_1$.
   • By Kirchhoff's Voltage Law (KVL), the sum of voltage drops across $R_1$ and $R_2$ must equal the source voltage $V_s$.

   $V_s = V_{R_1} + V_{R_2}$

3. Calculate the Voltage Drop Across $R_1$: $V_{R_1} = V_s - V_{R_2} = 10 \, V - 2 \, V = 8 \, V$

4. Apply Ohm's Law to Find the Current Through $R_1$:

   • Using Ohm's Law $V = IR$, the current $I$ through the circuit is given by:

   $I = \frac{V_{R_1}}{R_1} = \frac{8 \, V}{5 \, \Omega} = 1.6 \, A$

5. Find the Value of $R_2$:

   • The current through $R_2$ is the same as the current through $R_1$ since they are in series. Using Ohm's Law for $R_2$:

   $R_2 = \frac{V_{R_2}}{I} = \frac{2 \, V}{1.6 \, A} = 1.25 \, \Omega$

Conclusion:

The value of the resistor $R_2$ is 1.25 Ohms.

Would you like more details on any of these steps, or do you have any questions?

Related Questions:

1. What would happen to the current if $R_2$ were doubled?
2. How do you find the total resistance in a series circuit?
3. What is the power dissipated across $R_1$ and $R_2$?
4. How does Kirchhoff's Voltage Law apply to parallel circuits?
5. How can we solve a circuit problem with more than two resistors in series?

Tip: When analyzing circuits, always double-check if components are in series or parallel, as this affects how you apply Ohm's Law and Kirchhoff's Laws.