To solve this problem, we need to calculate the magnetic field at a specific point due to two long, straight parallel wires carrying currents.

Step 1: Formula for Magnetic Field Due to a Long Straight Current-Carrying Wire

The magnetic field due to a long, straight current-carrying wire at a distance $r$ from the wire is given by Ampère's Law:

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

Where:

• $B$ is the magnetic field.
• $\mu_0$ is the permeability of free space, $4\pi \times 10^{-7} \, \text{T} \cdot \text{m}/\text{A}$.
• $I$ is the current in the wire.
• $r$ is the distance from the wire to the point where the field is being calculated.

Step 2: Analyzing the Setup

• The northern wire has a current of $I_1 = 8.00 \, \text{A}$ heading to the east (positive $x$-direction).
• The southern wire has a current of $I_2 = 5.00 \, \text{A}$ heading west (negative $x$-direction).
• The two wires are separated by a distance $d = 0.0410 \, \text{m}$.
• The point where the magnetic field is to be calculated is $0.0130 \, \text{m}$ north of the northern wire. This point is a distance $r_1 = 0.0130 \, \text{m}$ from the northern wire and $r_2 = 0.0410 \, \text{m} + 0.0130 \, \text{m} = 0.0540 \, \text{m}$ from the southern wire.

Step 3: Calculate the Magnetic Field Due to Each Wire

Magnetic Field from the Northern Wire (At a distance $r_1 = 0.0130 \, \text{m}$)

$B_1 = \frac{\mu_0 I_1}{2 \pi r_1} = \frac{(4 \pi \times 10^{-7}) (8.00)}{2 \pi (0.0130)} = \frac{(4 \times 10^{-7}) (8.00)}{0.0260} = 1.23 \times 10^{-4} \, \text{T}$

The magnetic field from the northern wire is directed into the page at the point of interest (since the current is heading to the east).

Magnetic Field from the Southern Wire (At a distance $r_2 = 0.0540 \, \text{m}$)

$B_2 = \frac{\mu_0 I_2}{2 \pi r_2} = \frac{(4 \pi \times 10^{-7}) (5.00)}{2 \pi (0.0540)} = \frac{(4 \times 10^{-7}) (5.00)}{0.108} = 1.85 \times 10^{-5} \, \text{T}$

The magnetic field from the southern wire is directed out of the page at the point of interest (since the current is heading west).

Step 4: Determine the Net Magnetic Field

The net magnetic field is the vector sum of the fields from the two wires. Since the fields from the two wires are in opposite directions (into the page and out of the page), we subtract them to find the net field.

$B_{\text{net}} = B_1 - B_2 = (1.23 \times 10^{-4}) - (1.85 \times 10^{-5}) = 1.04 \times 10^{-4} \, \text{T}$

Final Answer:

The magnitude of the net magnetic field 0.0130 m north of the northernmost wire is $1.04 \times 10^{-4} \, \text{T}$.

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Would you like a more detailed explanation of the steps or any further clarifications?

Here are 5 related questions to explore further:

1. How does the direction of current affect the magnetic field produced by a wire?
2. What happens to the magnetic field if the wires are closer or farther apart?
3. How can we calculate the magnetic field due to multiple current-carrying wires?
4. How does the magnitude of the current influence the strength of the magnetic field?
5. How would the magnetic field change if the currents in both wires were reversed?

Tip: Remember that when calculating the magnetic field due to multiple sources, the direction of each magnetic field is crucial. Use the right-hand rule to determine the direction of the field.