To determine the maximum number of strands $N$ in the aluminum wiring bundle, we need to ensure that the current $i = 12 \, \text{A}$ remains below the specified limit when the wiring is connected to a battery with voltage $V = 3.0 \, \text{V}$. We'll use Ohm's Law and the formula for resistance of a cylindrical conductor.

Step 1: Resistance of a single strand

The resistance of a single strand of aluminum wire is given by the formula:

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

where:

• $\rho = 2.65 \times 10^{-8} \, \Omega \cdot \text{m}$ is the resistivity of aluminum,
• $L = 15 \, \text{m}$ is the length of each strand,
• $A$ is the cross-sectional area of the strand, which can be found using the formula for the area of a circle: $A = \pi r^2$,
• $r = 0.22 \, \text{mm} = 0.22 \times 10^{-3} \, \text{m}$ is the radius of each strand.

So the resistance of a single strand is:

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

Step 2: Total resistance of the bundle

If there are $N$ strands in the bundle, the total resistance $R_{\text{total}}$ is given by the formula for resistors in parallel, since each strand is electrically connected in parallel:

$\frac{1}{R_{\text{total}}} = N \times \frac{1}{R}$

Thus, the total resistance of the bundle is:

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

Step 3: Applying Ohm's Law

Ohm's Law relates the total resistance $R_{\text{total}}$, the current $i$, and the voltage $V$:

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

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

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

Solving for $N$:

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

Step 4: Substitution of known values

Now, we can substitute the given values:

• $\rho = 2.65 \times 10^{-8} \, \Omega \cdot \text{m}$,
• $L = 15 \, \text{m}$,
• $i = 12 \, \text{A}$,
• $V = 3.0 \, \text{V}$,
• $r = 0.22 \, \text{mm} = 0.22 \times 10^{-3} \, \text{m}$.

Thus:

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

Let me calculate that for you.The maximum number of strands $N$ that you can use while ensuring the current remains below 12 A is approximately $N \approx 10$.

Would you like any further details or have any questions?

Here are 5 related questions to explore further:

1. How does increasing the voltage affect the number of strands required?
2. What happens to the total resistance if you halve the radius of the strands?
3. How does the length of the wire impact the overall current capacity?
4. What is the effect of using a material with higher resistivity for the wiring?
5. Can we calculate the total power dissipated in the wiring bundle?

Tip: When resistors are connected in parallel, the total resistance decreases as the number of resistors increases.