The problem involves solving for the currents in a given electrical circuit, specifically using Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL).

Let's go step by step.

Given Values:

• $E_1 = 10 \, V$
• $E_2 = 15 \, V$
• $R_1 = 100 \, \Omega$
• $R_2 = 200 \, \Omega$
• $R_3 = 100 \, \Omega$
• $R_4 = 300 \, \Omega$
• $R_5 = 50 \, \Omega$

The circuit diagram includes:

• A voltage source $E_1$ connected in series with resistors $R_1$ and $R_3$.
• Another voltage source $E_2$ connected in series with resistors $R_2$ and $R_4$.
• The resistor $R_5$ is connected between two points in the circuit.

Steps to Solve:

1. Label the Currents:

   • Let's denote the current in the loop with $E_1$ as $I_1$.
   • The current in the loop with $E_2$ as $I_2$.
   • The current through $R_5$ is $I_3$.

2. Apply KCL at the Junction:

   • At the junction, where $R_1$, $R_2$, and $R_5$ meet, we have: $I_3 = I_1 - I_2$

3. Apply KVL to the Loops:

   • For the left loop (with $E_1$): $E_1 - I_1 R_1 - I_3 R_3 = 0$ Substituting $I_3 = I_1 - I_2$: $10 - 100 I_1 - 100 (I_1 - I_2) = 0$ Simplify to: $200 I_1 - 100 I_2 = 10$ $2 I_1 - I_2 = 0.1 \quad \text{(Equation 1)}$

   • For the right loop (with $E_2$): $E_2 - I_2 R_2 - I_3 R_4 = 0$ Substituting $I_3 = I_1 - I_2$: $15 - 200 I_2 - 300 (I_1 - I_2) = 0$ Simplify to: $300 I_2 - 300 I_1 = 15$ $I_2 - I_1 = 0.05 \quad \text{(Equation 2)}$

4. Solve the Simultaneous Equations: From Equation 2: $I_2 = I_1 + 0.05$

   Substitute into Equation 1: $2 I_1 - (I_1 + 0.05) = 0.1$ Simplify: $I_1 - 0.05 = 0.1$ $I_1 = 0.15 \, A$

   Substituting $I_1$ back into Equation 2: $I_2 = 0.15 + 0.05 = 0.2 \, A$

   Now, $I_3 = I_1 - I_2 = 0.15 - 0.2 = -0.05 \, A$ (negative sign indicates direction opposite to assumed direction).

Conclusion:

• $I_1 = 0.15 \, A$
• $I_2 = 0.2 \, A$
• $I_3 = -0.05 \, A$

These currents can be used to further analyze the voltage drops across each resistor or to find power dissipated if needed.

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Would you like further details on any step or help with a different problem?

Follow-Up Questions:

1. What are the voltage drops across each resistor using these current values?
2. How would the analysis change if one of the resistors had a different value?
3. What if the resistors were connected in parallel instead of series?
4. How would the current distribution change if $E_1$ and $E_2$ had different polarities?
5. How do you calculate the power dissipated in each resistor?

Tip:

Always double-check the signs of the currents and voltages, especially when using KVL and KCL, to ensure they match the assumed directions.