To find the current $i$ in the circuit, we need to analyze the given components. The circuit consists of the following:

1. A voltage source of $10 \text{V}$.
2. Two resistors: $4 \Omega$ and $6 \Omega$.

Step-by-Step Solution:

1. Identify the Configuration: The resistors appear to be arranged in parallel, with the $10 \text{V}$ source across them.

2. Calculate the Equivalent Resistance: For parallel resistors, the equivalent resistance $R_{\text{eq}}$ is given by: $\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2}$ Substituting $R_1 = 4 \Omega$ and $R_2 = 6 \Omega$: $\frac{1}{R_{\text{eq}}} = \frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$ Thus, $R_{\text{eq}} = \frac{12}{5} = 2.4 \Omega$

3. Calculate the Total Current: Using Ohm's Law, $V = IR$, we can find the total current supplied by the voltage source: $I = \frac{V}{R_{\text{eq}}} = \frac{10}{2.4} \approx 4.17 \text{A}$

4. Find the Current Through Each Resistor: The current through each resistor in a parallel circuit can be found using $I_i = \frac{V}{R_i}$.

   • For the $4 \Omega$ resistor: $I_1 = \frac{10}{4} = 2.5 \text{A}$
   • For the $6 \Omega$ resistor: $I_2 = \frac{10}{6} \approx 1.67 \text{A}$

Therefore, the currents through the $4 \Omega$ and $6 \Omega$ resistors are approximately $2.5 \text{A}$ and $1.67 \text{A}$, respectively.

Do you need further details or have any questions?

Related Questions:

1. How would the total current change if the voltage source was doubled to $20 \text{V}$?
2. What would happen to the current if an additional resistor was added in series?
3. How does the power dissipation change for each resistor in this circuit?
4. Can you find the power delivered by the voltage source in this configuration?
5. What would be the effect on the equivalent resistance if both resistors had the same resistance value?

Tip: In parallel circuits, the voltage across each resistor is always the same as the source voltage.