To solve this problem, we need to calculate the maximum number of strands $N$ such that the current $i$ remains below 12 A when connected to a voltage of $V = 3.0 \, \text{V}$.

Step 1: Formula for Resistance

The resistance of a single strand of aluminum wire can be calculated using the formula for resistance in terms of resistivity $\rho$, length $L$, and cross-sectional area $A$:

$R = \frac{\rho L}{A}$

For a circular wire, the cross-sectional area $A$ is given by:

$A = \pi r^2$

where $r$ is the radius of the wire. Therefore, the resistance of one strand is:

$R = \frac{\rho L}{\pi r^2}$

Step 2: Total Resistance of the Bundle

The total resistance $R_{\text{total}}$ of $N$ strands in parallel can be calculated using the formula for resistors in parallel. If $R$ is the resistance of a single strand, then the total resistance of $N$ strands is:

$R_{\text{total}} = \frac{R}{N} = \frac{\rho L}{N \pi r^2}$

Step 3: Ohm’s Law and Current

By Ohm's Law, the current $i$ is related to the voltage $V$ and the total resistance $R_{\text{total}}$ by the equation:

$i = \frac{V}{R_{\text{total}}}$

Substituting the expression for $R_{\text{total}}$:

$i = \frac{V}{\frac{\rho L}{N \pi r^2}} = \frac{N \pi r^2 V}{\rho L}$

Step 4: Solving for $N$

We are given that the current $i$ must be less than or equal to 12 A. Therefore, we can set up the inequality:

$12 \geq \frac{N \pi r^2 V}{\rho L}$

Solving for $N$:

$N \leq \frac{12 \rho L}{\pi r^2 V}$

Thus, the maximum number of strands $N$ is given by:

$N_{\text{max}} = \frac{12 \rho L}{\pi r^2 V}$

Step 5: Substituting the Given Values

Now, we substitute the known values:

• $\rho = 2.65 \times 10^{-8} \, \Omega \cdot \text{m}$
• $L = 15 \, \text{m}$
• $r = 0.22 \, \text{mm} = 0.22 \times 10^{-3} \, \text{m}$
• $V = 3.0 \, \text{V}$

Substituting these into the equation:

$N_{\text{max}} = \frac{12 \times 2.65 \times 10^{-8} \times 15}{\pi \times (0.22 \times 10^{-3})^2 \times 3.0}$

Let me calculate this value for you.The maximum number of strands $N_{\text{max}}$ that can be used while ensuring the current remains below 12 A is approximately 10.46. Since the number of strands must be a whole number, the maximum number of strands $N$ you can use is 10.

Would you like more details on any of the steps, or have further questions?

Here are five related questions to expand your understanding:

1. How would the result change if the radius of each strand were doubled?
2. What happens to the current if the length of the bundle increases?
3. How does using a material with higher resistivity affect the maximum number of strands?
4. What is the role of parallel wiring in reducing the overall resistance?
5. How would the calculation change if the voltage were increased to 6 V?

Tip: In parallel circuits, adding more paths for current to flow (more strands) decreases the overall resistance.