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To solve for the current \( i \) and voltage \( v \) in this circuit, we need to analyze the arrangement of the conductances (given in siemens, \( S \)) and apply Kirchhoff's laws or Ohm's Law for conductance.

### Step 1: Simplify the Parallel Conductances
We can observe that there are several parallel and series connections of conductances:

- The three conductances labeled \( 1S \), \( 2S \), and \( 4S \) on the left form a parallel combination.

The total conductance of resistors in parallel is given by the sum of their conductances:
\[
G_{\text{parallel}} = G_1 + G_2 + G_3
\]
where \( G_1 = 1S \), \( G_2 = 2S \), and \( G_3 = 4S \). So:
\[
G_{\text{parallel}} = 1S + 2S + 4S = 7S
\]

This parallel combination has an equivalent conductance of \( 7S \).

### Step 2: Series Combination with Other Conductances
Now, the \( 7S \) equivalent conductance is in series with the \( 4S \) conductance (above it in the circuit diagram). In series, the total conductance is the sum of the reciprocal of individual conductances:
\[
\frac{1}{G_{\text{total}}} = \frac{1}{G_{\text{series 1}}} + \frac{1}{G_{\text{series 2}}}
\]
Where \( G_{\text{series 1}} = 7S \) and \( G_{\text{series 2}} = 4S \).

Let me calculate the total equivalent conductance for the series combination.The total equivalent conductance \( G_{\text{total}} \) of the circuit is approximately \( 2.545S \).

### Step 3: Calculate Voltage \( v \)
Since the current source provides a current of \( 12A \), we can calculate the voltage \( v \) across the total conductance using Ohm's Law for conductance, which states:
\[
v = \frac{I}{G}
\]
Where \( I = 12A \) and \( G_{\text{total}} = 2.545S \). Let's calculate the voltage \( v \).The voltage \( v \) across the circuit is approximately \( 4.71 \, \text{V} \).

### Step 4: Calculate Current \( i \)
To find the current \( i \) through the \( 4S \) conductance (the resistor on the top branch), we can use Ohm's Law. The current through a conductance is given by:
\[
i = G \cdot v
\]
Where \( G = 4S \) and \( v \approx 4.71 \, \text{V} \). Let's calculate the current \( i \).The current \( i \) through the \( 4S \) conductance is approximately \( 18.86 \, \text{A} \).

### Summary:
- Voltage \( v \) across the total circuit: \( 4.71 \, \text{V} \)
- Current \( i \) through the \( 4S \) conductance: \( 18.86 \, \text{A} \)

Would you like further details or have any questions?

Here are 5 related questions:
1. How does the total conductance affect the overall circuit behavior?
2. Can we calculate the power dissipated in the 4S resistor?
3. How would the circuit change if the current source was replaced by a voltage source?
4. What happens to the current through the resistors if we add another resistor in parallel?
5. How do we apply Kirchhoff's laws to verify these results?

**Tip:** When analyzing circuits with conductances, remember that higher conductance means lower resistance, which leads to higher current through that path.

This problem demonstrates how to calculate the current and voltage in an electrical circuit with a 12A current source. By applying Ohm's Law for conductance and Kirchhoff's Laws, we simplify parallel and series connections of conductances and solve for voltage (v = 4.71 V) and current (i = 18.86 A). Suitable for high school physics students.

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