To solve for the current $I_2$ traveling through resistor $R_2$, we need to break the circuit down into its components and apply Ohm's Law and the rules for resistors in series and parallel.

Step 1: Analyze the circuit

• Resistor $R_1$ is in series with the parallel combination of $R_2$ and $R_3$.
• Resistors $R_2$ and $R_3$ are in parallel, so we first need to calculate their combined resistance.

Step 2: Calculate the equivalent resistance of $R_2$ and $R_3$ in parallel

The formula for the total resistance of two resistors in parallel is:

$\frac{1}{R_{\text{eq, parallel}}} = \frac{1}{R_2} + \frac{1}{R_3}$

Substituting the values of $R_2 = 32 \, \Omega$ and $R_3 = 82 \, \Omega$:

$\frac{1}{R_{\text{eq, parallel}}} = \frac{1}{32} + \frac{1}{82}$

First, calculate the sum of the fractions:

$\frac{1}{32} + \frac{1}{82} = \frac{82 + 32}{32 \times 82} = \frac{114}{2624}$

Now, calculate the reciprocal to find the equivalent resistance:

$R_{\text{eq, parallel}} = \frac{2624}{114} \approx 23.03 \, \Omega$

Step 3: Calculate the total resistance of the circuit

Now, we add the resistance $R_1 = 23 \, \Omega$ in series with the parallel combination:

$R_{\text{total}} = R_1 + R_{\text{eq, parallel}} = 23 + 23.03 = 46.03 \, \Omega$

Step 4: Calculate the total current in the circuit

Next, apply Ohm's Law to find the total current $I_{\text{total}}$ in the circuit. The potential difference $\Delta V = 7.5 \, \text{V}$ is applied across the total resistance:

$I_{\text{total}} = \frac{\Delta V}{R_{\text{total}}} = \frac{7.5}{46.03} \approx 0.163 \, \text{A}$

Step 5: Calculate the current through $R_2$

Since $R_2$ and $R_3$ are in parallel, the voltage across both resistors is the same. We can use the current division rule to find $I_2$, the current through $R_2$:

$I_2 = I_{\text{total}} \times \frac{R_3}{R_2 + R_3}$

Substituting the known values:

$I_2 = 0.163 \times \frac{82}{32 + 82} = 0.163 \times \frac{82}{114} \approx 0.116 \, \text{A}$

Thus, the current through resistor $R_2$ is approximately 0.116 A.

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Would you like further details on any part of the process? Here are 5 related questions to explore further:

1. How does the current division rule work in parallel circuits?
2. What would happen if $R_2$ and $R_3$ were in series instead of parallel?
3. How do you calculate power dissipated by each resistor in this circuit?
4. How would the current through $R_2$ change if $R_3$ were much smaller than $R_2$?
5. How can we verify the current through $R_2$ using Kirchhoff's current law?

Tip: When solving circuits with resistors in series and parallel, always simplify the circuit step by step—first calculate equivalent resistances, then apply Ohm's Law to find currents and voltages.