The problem asks us to determine the net magnetic field at the center of a copper wire coil. To solve this problem, we need to consider the resistance of the wire, the EMF of the battery, and the lengths of the two segments of the wire.

1. Calculate the total resistance of the circuit:
- The resistance of the wire is given as 10 ohms.
- The internal resistance of the battery is given as 0.5 ohms.
- Since the wire and battery are connected in series, the total resistance is the sum of the individual resistances:
Total resistance = wire resistance + internal resistance = 10 ohms + 0.5 ohms = 10.5 ohms.

2. Determine the current flowing through the circuit:
- The EMF of the battery is given as 5 volts.
- Using Ohm's Law (V = IR), we can find the current (I) flowing through the circuit:
I = V / R = 5 V / 10.5 ohms = 0.476 A.

3. Calculate the magnetic field due to each segment of the wire:
- The magnetic field due to a current-carrying wire can be determined using Ampere's Law, which states that the magnetic field (B) is proportional to the current (I) and inversely proportional to the distance (r) from the wire:
B = (μ₀ * I) / (2πr), where μ₀ is the permeability of free space.

- Let's consider the two segments of the wire: segment 1 with length L1 and segment 2 with length L2, where L1/L2 = 2/3.

- The magnetic field at the center of the coil due to segment 1 can be calculated as:
B1 = (μ₀ * I) / (2π * (L1/2)) = (μ₀ * I) / (πL1).

- Similarly, the magnetic field at the center of the coil due to segment 2 can be calculated as:
B2 = (μ₀ * I) / (2π * (L2/2)) = (μ₀ * I) / (πL2).

4. Determine the net magnetic field at the center of the coil:
- The net magnetic field is the vector sum of the magnetic fields due to each segment.
- Since the currents are in the same direction, we can simply add the magnitudes of the magnetic fields:
Bnet = B1 + B2 = (μ₀ * I) / (πL1) + (μ₀ * I) / (πL2) = (μ₀ * I) / π * (1/L1 + 1/L2).

In conclusion, the net magnetic field at the center of the coil can be calculated using the formula Bnet = (μ₀ * I) / π * (1/L1 + 1/L2), where L1 and L2 are the lengths of the two segments of the wire.

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