Let's start by recalling Ampere's law which states that the magnetic field by a long straight wire is given by B = μ₀I / (2πr), where B is the magnetic field, μ₀ is the permeability of free space which is equal to 4π × 10^-7 Tm/A, I is the current in the wire, and r is the distance from the wire.
Now, let's use this law to calculate the magnetic field due to each wire at the point of interest which is 5cm to the left from the left wire.
Starting with the left wire:
The current I1 in the left wire is 117A and the distance r1 from the point to the left wire is 0.05m. Substituting these values into Ampere's law we get:
B1 = (4π × 10^-7 Tm/A * 117A) / (2π * 0.05m) ≈ 0.000468 T
Now let's consider the right wire:
The current I2 in the right wire is 191A. However, since it's directed out of the page, which is the opposite direction to the current in the left wire, the sign of the current should be opposite. The distance r2 from the point to the right wire is 0.20m. Substituting these values into Ampere's law gives:
B2 = -(4π × 10^-7 Tm/A * 191A) / (2π * 0.20m) ≈ -0.0001528 T
The total magnetic field at the point is the vector sum of the magnetic fields due to the left and the right wires. Adding the values we got:
B_total = B1 + B2 = 0.000468 T - 0.0001528 T = 0.0003152 T
To conclude, the magnitude of the magnetic field at the point that is 5cm to the left from the left wire is approximately equal to 0.0003152 T. The direction of the magnetic field is into the page since the contribution from the left wire, which points into the page, is stronger than the contribution from the right wire.