two perpendicular, long, straight wires, both of which lie in the plane of the paper. The current in each of the wires is I = 4.5 A. In the drawing dH = 0.13 m and dV = 0.48 m. Find the magnitudes of the net magnetic fields at points A and B.
The drawing shows one horizontal wire with the current going up and one vertical wire with the current going to the right. Point A is dH to the right and dV above the intersection. Point B is dH to the left and dV below the intersection.
To find the magnitudes of the net magnetic fields at points A and B, we can use the Biot-Savart Law, which states that the magnetic field at a point due to a current-carrying wire is directly proportional to the current and inversely proportional to the distance from the wire.
First, let's calculate the magnetic field at point A due to the horizontal wire. Since the wire is straight and current is flowing upwards, the magnetic field at point A will be into the plane of the paper. The magnitude of the magnetic field at point A due to the horizontal wire can be calculated using the formula:
dB_horizontal = (μ₀ * I_horizontal * d_horizontal) / (2π * r_horizontal)
Where:
- μ₀ is the permeability of free space (4π * 10⁻⁷ T·m/A)
- I_horizontal is the current in the horizontal wire (4.5 A)
- d_horizontal is the distance from the horizontal wire to point A (dH)
- r_horizontal is the distance from the wire to point A (the distance of the wire from point A)
Given the wire lies in the plane of the paper, the distance of the wire from point A is dV. So, r_horizontal = dV.
Substituting the given values, we have:
dB_horizontal = (4π * 10⁻⁷ T·m/A) * (4.5 A) * (0.13 m) / (2π * 0.48 m)
Simplifying the expression, canceling out π and simplifying the units, we get:
dB_horizontal = 4.5 * 10⁻⁷ T
Therefore, the magnitude of the magnetic field at point A due to the horizontal wire is 4.5 * 10⁻⁷ T.
Next, let's calculate the magnetic field at point A due to the vertical wire. Since the wire is straight and the current is flowing to the right, the magnetic field at point A will be out of the plane of the paper. The magnitude of the magnetic field at point A due to the vertical wire can be calculated using the same formula:
dB_vertical = (μ₀ * I_vertical * d_vertical) / (2π * r_vertical)
Where:
- I_vertical is the current in the vertical wire (4.5 A)
- d_vertical is the distance from the vertical wire to point A (dV)
- r_vertical is the distance from the wire to point A (the distance of the wire from point A)
Given the wire lies in the plane of the paper, the distance of the wire from point A is dH. So, r_vertical = dH.
Substituting the given values, we have:
dB_vertical = (4π * 10⁻⁷ T·m/A) * (4.5 A) * (0.48 m) / (2π * 0.13 m)
Simplifying the expression, canceling out π and simplifying the units, we get:
dB_vertical = 6.92 * 10⁻⁷ T
Therefore, the magnitude of the magnetic field at point A due to the vertical wire is 6.92 * 10⁻⁷ T.
To find the net magnetic field at point A, we can use the principle of vector addition since the horizontal and vertical magnetic fields are perpendicular to each other:
B_net_A = √(B_horizontal² + B_vertical²)
Substituting the calculated values, we have:
B_net_A = √((4.5 * 10⁻⁷ T)² + (6.92 * 10⁻⁷ T)²)
Simplifying the expression, we get:
B_net_A ≈ 8.29 * 10⁻⁷ T
Therefore, the magnitude of the net magnetic field at point A is approximately 8.29 * 10⁻⁷ T.
Similarly, we can repeat the above steps to find the magnitude of the net magnetic field at point B by using the distances dH and dV in the appropriate places and then applying vector addition.
Note: The direction of the net magnetic field at points A and B can also be determined using the right-hand rule, and it will be perpendicular to the plane of the paper.