The circuit provided contains multiple voltage sources and a resistor of 1 kΩ, and we are asked to determine the current labeled $i$ by first combining the four voltage sources into a single equivalent source.

Step-by-Step Solution:

1. Identifying Voltage Sources:

   • We have four voltage sources: two 5V sources, one 6V source, and one 2V source.
   • The two 5V sources are arranged in parallel but facing opposite directions.
   • The 6V and 2V sources are arranged in parallel as well, but again, opposite in polarity.

2. Simplifying the Voltage Sources:

   • Since opposite voltage sources in parallel can cancel each other out, let’s combine them step-by-step.

     • The two 5V sources cancel each other because they are of equal magnitude but opposite polarity:
       $V_{eq1} = 5V - 5V = 0V$
     • The 6V and 2V sources subtract due to their opposite polarity:
       $V_{eq2} = 6V - 2V = 4V$

   • Therefore, the equivalent voltage source is 4V.

3. Applying Ohm's Law: Now, the circuit is simplified to a single 4V source in series with a 1 kΩ resistor.

   • Using Ohm's Law $V = IR$, where $V$ is the voltage, $I$ is the current, and $R$ is the resistance: $I = \frac{V}{R} = \frac{4V}{1k\Omega} = 4 \, \text{mA}$

Conclusion:

The current $i$ is $4 \, \text{mA}$.

Would you like more details or have any questions?

Here are 5 related questions that expand on this topic:

1. How can voltage sources in parallel or series be combined in other configurations?
2. What happens if the voltage sources are not equal but still in opposite polarity?
3. How do we calculate the total resistance if multiple resistors are added to this circuit?
4. How would the current change if one of the voltage sources had a higher value?
5. What if the resistor value was different—how would it affect the current?

Tip: When simplifying circuits, always check if voltage sources are in parallel or series. They can often be combined to make the circuit easier to analyze.