Question

Point \(P\) is midway between two long, straight, parallel wires that run north- south in a horizontal plane. The distance between the wires is $1.0 \mathrm{cm} .\( Each wire carrics a current of \)1.0 \mathrm{A}$ toward the north. Find the magnitude and direction of the magnetic field at point \(P\)

Short answer

Answer: The magnitude of the magnetic field at point P is approximately \(2.83 \times 10^{-5} \text{T}\), and the direction is upward at a 45-degree angle with respect to a horizontal line going through point P.

Step-by-step solution

Identify the distance between point P and each wire

Since point P is midway between the two wires, the distance between point P and each wire is equal to half the distance between the wires. The distance between the wires is given as 1.0 cm, so the distance between point P and each wire is 0.5 cm or \(0.005 \text{m}\) (in SI units).

Calculate the magnetic field at point P due to each wire using the Biot-Savart law

The magnetic field produced by a long, straight wire carrying a current I at a distance d is given by the Biot-Savart law: \(B = \frac{\mu_0 I}{2 \pi d}\) where \(B\) is the magnetic field, \(\mu_0\) is the permeability of free space (\(4 \pi \times 10^{-7} \text{T}\cdot\text{m/A}\)), \(I\) is the current in the wire, and \(d\) is the distance from the wire to the point where we're calculating the magnetic field. We know that each wire carries a current of 1.0 A and the distance from each wire to point P is 0.005 m. Therefore, the magnetic field at point P due to each wire is: \(B_{\text{each}} = \frac{4 \pi \times 10^{-7} \text{T}\cdot \text{m/A} \times 1.0 \text{A}}{2 \pi \times 0.005 \text{m}} = 2 \times 10^{-5} \text{T}\)

Find the total magnetic field at point P by adding the magnetic fields due to each wire

Total magnetic field at point P can be found by adding the magnetic fields produced by both wires. However, since these magnetic fields are not in the same direction (they are perpendicular to each other), they cannot be directly added. Instead, we need to find the vector sum of the magnetic fields. Let's denote the magnetic field produced by the left wire as \(B_{\text{left}}\) and the magnetic field produced by the right wire as \(B_{\text{right}}\). The magnetic field produced by the left wire points into the page, while the magnetic field produced by the right wire points out of the page. To find the total magnetic field at point P, we can compute the vector sum of these magnetic fields, which will be at a 90-degree angle: \(B_{\text{total}} = \sqrt{B_{\text{left}}^2 + B_{\text{right}}^2}\) Since the magnitudes of both magnetic fields are equal, \(B_{\text{left}} = B_{\text{right}} = B_{\text{each}} = 2 \times 10^{-5} \text{T}\), plugging the values into the above formula: \(B_{\text{total}} = \sqrt{(2 \times 10^{-5} \text{T})^2 + (2 \times 10^{-5} \text{T})^2} = 2\sqrt{2} \times 10^{-5} \text{T} \approx 2.83 \times 10^{-5} \text{T}\)

Determine the direction of the magnetic field at point P

The magnetic field at point P is the vector sum of the magnetic field due to each wire. Since both magnetic fields are equal and perpendicular to each other, they form a 45-degree angle with the horizontal line connecting the wires. In other words, the magnetic field at point P is going upward at a 45-degree angle with respect to a horizontal line going through point P. In conclusion, the magnitude of the magnetic field at point P is approximately \(2.83 \times 10^{-5} \text{T}\), and the direction is upward at a 45-degree angle with respect to a horizontal line going through point P.