To find the current necessary to produce a magnetic field of 5.7 mT at a distance of 13.1 mm from an infinitely long wire, we use Ampere's Law in the form of the magnetic field around a long straight wire:

$B = \frac{\mu_0 I}{2 \pi r}$

Where:

• $B$ is the magnetic field ($5.7 \, \text{mT} = 5.7 \times 10^{-3} \, \text{T}$),
• $\mu_0$ is the permeability of free space ($4\pi \times 10^{-7} \, \text{T} \cdot \text{m/A}$),
• $I$ is the current (in amperes),
• $r$ is the distance from the wire ($13.1 \, \text{mm} = 0.0131 \, \text{m}$).

Step 1: Rearrange for $I$:

$I = \frac{2 \pi r B}{\mu_0}$

Step 2: Plug in the values:

$I = \frac{2 \pi (0.0131) (5.7 \times 10^{-3})}{4 \pi \times 10^{-7}}$

Step 3: Simplify the calculation:

$I = \frac{2 (0.0131) (5.7 \times 10^{-3})}{4 \times 10^{-7}}$

$I = \frac{0.00014934}{4 \times 10^{-7}}$

$I = 373.4 \, \text{A}$

Final Answer:

The current required is 373.4 A.

Would you like further details or have any questions? Here are 5 related questions to consider:

1. How does the distance from the wire affect the magnetic field strength?
2. What assumptions are made when using this formula for an "infinitely long wire"?
3. What is the role of the permeability of free space ($\mu_0$) in this calculation?
4. How would the result change if the wire was not straight but coiled?
5. How can you measure the magnetic field around a wire in practice?

Tip: Always check the units when solving physics problems to ensure consistency and avoid mistakes!