Four long, parallel conductors carry equal currents of I = 9.50 A. The current direction of the current is into the page at points A and B and out of the page at C and D. Calculate the magnitude and direction of the magnetic field at point P, located at the center of the square of edge length 0.200 m.
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To calculate the magnetic field at point P, we can apply the right-hand rule for the magnetic field due to a long straight conductor.
The magnetic field created by each conductor can be found using the formula:
B = (μ₀ * I) / (2π * r),
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 conductor, and r is the distance from the conductor to the point where we want to calculate the magnetic field.
Since there are four conductors, we need to find the magnetic field created by each individual conductor and then combine them vectorially to get the total magnetic field at point P.
Let's label the four conductors as A, B, C, and D, and assume they are arranged in a clockwise direction.
First, consider conductor A, which carries current into the page. To find the magnetic field at point P due to conductor A, we need to find the perpendicular distance from P to conductor A. Since P is at the center of the square, the perpendicular distance from P to conductor A is half the length of the square's edge. Therefore, the distance from P to conductor A is (0.200 m) / 2 = 0.100 m.
Using the formula mentioned above and plugging in the values, the magnetic field created by conductor A at point P is:
B_A = (μ₀ * I) / (2π * r_A),
= (4π × 10^-7 T·m/A) * (9.50 A) / (2π * 0.100 m),
= 6.0 × 10^-6 T.
The direction of the magnetic field due to conductor A can be found using the right-hand rule. If you point your thumb in the direction of the current flow (into the page at point A), your fingers will curl in the direction of the magnetic field. At point P, the magnetic field created by conductor A points towards conductor B.
Similarly, we can calculate the magnetic fields at point P due to conductors B, C, and D. Given that these conductors are parallel to A and equidistant from point P, they will all have the same magnetic field magnitude and direction.
Therefore, the total magnetic field at point P will be the vector sum of the magnetic fields due to each individual conductor. Since the magnetic fields due to A, B, C, and D are all equal and point in the same direction, their vector sum will yield:
B_total = 4 * B_A,
= 4 * 6.0 × 10^-6 T,
= 2.4 × 10^-5 T.
So, the magnitude of the magnetic field at point P is 2.4 × 10^-5 T, and its direction is towards conductor B.