The figure below shows two current carrying wires and the directions of the currents they carry. The wires carry the following currents: I1 = 12.5 A and I2 = 9.8 A. Wire 1 is at (0, 4.0) mm; wire 2 is at (13.3, 0) mm; point P is at (13.3, 4.0) mm.
webassign. net/userimages/twoCurrents. JPG?db=v4net&id=133103
What are the magnitude and direction (measured with respect to the +x direction) of the total magnetic field due to the two wires at point P?
(b) direction: °
To determine the magnitude and direction of the total magnetic field at point P due to the two wires, we can use the principle of superposition. According to this principle, the magnetic field produced by each wire can be calculated separately, and then the individual contributions can be added up vectorially to obtain the total magnetic field.
The magnitude of the magnetic field produced by a current-carrying wire at a point can be obtained using Ampere's Law. Ampere's Law states that the magnetic field around a long, straight wire is proportional to the current and inversely proportional to the distance from the wire.
Let's denote the magnetic field produced by wire 1 at point P as B1, and the magnetic field produced by wire 2 at point P as B2.
To calculate B1, we can use the formula:
B1 = μ₀ * I1 / (2πr₁)
where μ₀ is the permeability of free space (4π × 10⁻⁷ T*m/A), I1 is the current in wire 1 (12.5 A), and r₁ is the distance between wire 1 and point P (the y-coordinate difference between them, which is 4.0 mm = 0.004 m).
Substituting the given values, we have:
B1 = (4π × 10⁻⁷ T*m/A) * (12.5 A) / (2π * 0.004 m)
Simplifying, we get:
B1 = 10⁻⁴ T
Similarly, to calculate B2, we can use the same formula, using the values for wire 2:
B2 = μ₀ * I2 / (2πr₂)
where I2 is the current in wire 2 (9.8 A), and r₂ is the distance between wire 2 and point P (the x-coordinate difference between them, which is 13.3 mm = 0.0133 m).
Substituting the given values, we have:
B2 = (4π × 10⁻⁷ T*m/A) * (9.8 A) / (2π * 0.0133 m)
Simplifying, we get:
B2 = 7.36 × 10⁻⁵ T
Now that we have obtained the individual magnetic fields B1 and B2, we can add them vectorially to obtain the total magnetic field at point P. Since B2 is along the x-axis (from wire 2), and B1 is along the y-axis (from wire 1), the resulting magnetic field can be found using the vector addition rule.
The magnitude of the total magnetic field (Bt) can be calculated using the Pythagorean theorem:
Bt = √(B1² + B2²)
Substituting the obtained values for B1 and B2, we have:
Bt = √((10⁻⁴ T)² + (7.36 × 10⁻⁵ T)²)
Simplifying, we get:
Bt ≈ 1.1 × 10⁻⁴ T
So, the magnitude of the total magnetic field at point P is approximately 1.1 × 10⁻⁴ Tesla.
To determine the direction of the total magnetic field, we can use trigonometry. Since wire 1 is in the positive y-direction and wire 2 is in the positive x-direction, the total magnetic field will be at an angle relative to the positive x-axis.
To calculate this angle, we can use the arctan function:
θ = arctan(B1 / B2)
Substituting the calculated values for B1 and B2, we have:
θ = arctan((10⁻⁴ T) / (7.36 × 10⁻⁵ T))
Using a calculator, we find:
θ ≈ 53.1°
So, the direction of the total magnetic field at point P, measured with respect to the positive x-direction, is approximately 53.1 degrees.