To solve this circuit problem using the Node Voltage Method, we need to follow these steps:

1. Label the Nodes: Identify the nodes and assign voltage labels to each unknown node. For simplicity, let's call the voltage at the node between the two $500 \, \Omega$ resistors and the $1 \, k\Omega$ resistor as $V_A$.

2. Identify Known Currents: The currents $I_1$ and $I_2$ are given as $I_1 = 3 \, \text{mA}$ and $I_2 = 5 \, \text{mA}$. These currents are sourced by current sources on each side of the circuit.

3. Set Up Node Equations:

   • For the node at $V_A$, apply Kirchhoff’s Current Law (KCL). The sum of currents entering and leaving the node must equal zero.

Let's write out the current terms:

• The current flowing through the $500 \, \Omega$ resistor from $V_A$ to ground is $\frac{V_A}{500}$.
• The current flowing through the $500 \, \Omega$ resistor to the left of $V_A$ (towards $I_1$) is $\frac{V_A}{500}$.
• The current flowing through the $1 \, k\Omega$ resistor (1000 $\Omega$) towards $I_2$ is $\frac{V_A}{1000}$.

With these currents, the KCL equation at $V_A$ becomes:

$\frac{V_A}{500} + \frac{V_A}{500} + \frac{V_A}{1000} = I_1 + I_2$

Substitute the values for $I_1$ and $I_2$ and resistances:

$\frac{V_A}{500} + \frac{V_A}{500} + \frac{V_A}{1000} = 3 \, \text{mA} + 5 \, \text{mA}$

Convert mA to A (so 8 mA becomes 0.008 A):

$\frac{V_A}{500} + \frac{V_A}{500} + \frac{V_A}{1000} = 0.008$

4. Combine Terms:

$\frac{2V_A}{500} + \frac{V_A}{1000} = 0.008$

Convert this into a common denominator:

$\frac{4V_A + V_A}{1000} = 0.008$

$\frac{5V_A}{1000} = 0.008$

5. Solve for $V_A$:

$V_A = \frac{0.008 \times 1000}{5} = 1.6 \, \text{V}$

6. Calculate Voltages Across Components:
   • Voltage across each $500 \, \Omega$ resistor is $V_{500} = V_A = 1.6 \, \text{V}$.
   • Voltage across the $1 \, k\Omega$ resistor is also $V_{1k} = V_A = 1.6 \, \text{V}$.

The resulting voltages are:

• Voltage across each $500 \, \Omega$ resistor: $1.6 \, \text{V}$
• Voltage across the $1 \, k\Omega$ resistor: $1.6 \, \text{V}$

Would you like further details on the calculations or have questions?

Related Questions:

1. How do you apply the Mesh Analysis method to this circuit instead of Node Voltage Method?
2. What is the power dissipated in each resistor in the circuit?
3. How would you solve this circuit if the current sources were replaced with voltage sources?
4. How does the total equivalent resistance change if another resistor is added in parallel to one of the $500 \, \Omega$ resistors?
5. What happens to the node voltage if the value of $I_1$ or $I_2$ changes?

Tip:

Always double-check units when working with milliamps and ohms; ensure consistency for accurate calculations.