For the circuit below: Vs = 10 V, v2 = 2 V, and R1 = 5 Ohms.  Find the resistor, R2, in Ohms.

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.

For the circuit below: Vs = 10 V, v2 = 2 V, and R1 = 5 Ohms. Find the resistor, R2, in Ohms.

Learn how to find the value of resistor R2 in a series circuit where Vs = 10 V, v2 = 2 V, and R1 = 5 Ohms. This problem covers essential concepts in circuit analysis, including Ohm's Law and Kirchhoff's Voltage Law (KVL). Suitable for students in Grades 9-12.

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