A very long wire carries an I1 = 7.80 A current along the x axis and another long wire carries an I2 = 5.13 A current along the y axis.
(I2 is directed upward on the y axis, and I1 is directed to the right on the x axis)
What is the magnetic field at point P located at x = 4.42 m and y = 2.78 m? Use out of the page as the positive direction.
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To find the magnetic field at point P, we can use the Biot-Savart law. The Biot-Savart law states that the magnetic field at a given point in space due to a current-carrying wire is directly proportional to the current flowing through the wire and inversely proportional to the distance between the point and the wire.
The formula for the magnetic field due to a current-carrying wire is given by:
B = (μ₀ / 4π) * (I * dl × r) / r³
Where:
- B is the magnetic field
- μ₀ is the permeability of free space (4π × 10^(-7) T m/A)
- I is the current in the wire
- dl is an infinitesimally small length of wire
- r is the distance between the point and the wire
In this case, we have two wires, one carrying a current I1 along the x-axis and the other carrying a current I2 along the y-axis. We can find the magnetic field at point P by summing the magnetic fields due to each wire.
Let's calculate the magnetic field due to each wire separately and then combine the results:
1. Magnetic field due to I1 along the x-axis:
To find the magnetic field at point P due to the wire carrying I1, we will use the Biot-Savart law. The vector dl will be in the positive x-direction since the current flows to the right along the x-axis.
B1 = (μ₀ / 4π) * (I1 * dl × r1) / r1³
Here r1 is the distance between the wire carrying I1 and point P, and dl will be a small length along the wire. Since the wire is along the x-axis, dl can be considered as a small element dx.
The distance r1 can be calculated using the Pythagorean theorem:
r1 = √(x² + y²)
Substituting the values, we get:
r1 = √(4.42² + 2.78²) = √(19.5364 + 7.7284) = √27.2648 = 5.218 m
Using this value of r1, we can calculate the magnetic field due to I1 along the x-axis.
B1 = (μ₀ / 4π) * (I1 * dx) / r1²
2. Magnetic field due to I2 along the y-axis:
To find the magnetic field at point P due to the wire carrying I2, we can follow the same procedure. The vector dl will be in the positive y-direction, and the distance r2 can be calculated using the Pythagorean theorem:
r2 = √(x² + y²)
Substituting the values, we get:
r2 = √(4.42² + 2.78²) = √(19.5364 + 7.7284) = √27.2648 = 5.218 m
Using this value of r2, we can calculate the magnetic field due to I2 along the y-axis.
B2 = (μ₀ / 4π) * (I2 * dy) / r2²
3. Combine the magnetic fields:
To find the total magnetic field at point P, we can sum the magnetic fields due to each wire:
B_total = B1 + B2
Substituting the values, we get:
B_total = (μ₀ / 4π) * (I1 * dx) / r1² + (μ₀ / 4π) * (I2 * dy) / r2²
After substituting the known values for I1, I2, dx, dy, r1, and r2, and evaluating the expression, you will get the magnetic field at point P.
Note: The direction of the magnetic field can be determined using the right-hand rule.