(II) Two long thin parallel wires 13.0 cm apart carry 25-A currents in the same direction. Determine the magnetic field vector at a point 10.0 cm from one wire and 6.0 cm from the other (Fig. 28–37). [Hint: You could try using the law of cosines, Appendix A.]

Verified step by step guidance

1. Step 1: Understand the problem. Two parallel wires carry currents of 25 A in the same direction. The goal is to find the net magnetic field vector at a point that is 10.0 cm from one wire and 6.0 cm from the other. The magnetic field due to a current-carrying wire is given by the Biot-Savart law or Ampere's law, specifically: B = (μ₀ * I) / (2π * r), where μ₀ is the permeability of free space, I is the current, and r is the distance from the wire.
2. Step 2: Calculate the magnetic field magnitude due to each wire at the given point. For the first wire, use the formula B₁ = (μ₀ * I) / (2π * r₁), where r₁ = 10.0 cm = 0.1 m. For the second wire, use B₂ = (μ₀ * I) / (2π * r₂), where r₂ = 6.0 cm = 0.06 m. These fields will have different magnitudes because the distances r₁ and r₂ are different.
3. Step 3: Determine the direction of the magnetic fields. Use the right-hand rule: point your thumb in the direction of the current, and your curled fingers show the direction of the magnetic field. Since the currents are in the same direction, the magnetic fields at the point of interest will form circular patterns around each wire. Analyze the geometry to determine the angle between the two magnetic field vectors.
4. Step 4: Use vector addition to find the net magnetic field. The two magnetic fields, B₁ and B₂, are not aligned, so you need to resolve them into components. Use trigonometry to find the x- and y-components of each field. For example, if θ is the angle between the wires and the point, the components of B₁ are B₁x = B₁ * cos(θ₁) and B₁y = B₁ * sin(θ₁). Similarly, resolve B₂ into its components.
5. Step 5: Combine the components to find the net magnetic field. Add the x-components and y-components separately: B_net_x = B₁x + B₂x and B_net_y = B₁y + B₂y. Then, calculate the magnitude of the net field using the Pythagorean theorem: B_net = √(B_net_x² + B_net_y²). Finally, determine the direction of the net field using the arctangent function: θ_net = arctan(B_net_y / B_net_x).

Key Concepts

Magnetic Field Due to a Current-Carrying Wire

A long straight wire carrying an electric current generates a magnetic field around it. The strength and direction of this magnetic field can be determined using the right-hand rule and the formula B = (μ₀/4π) * (2I/r), where B is the magnetic field, I is the current, r is the distance from the wire, and μ₀ is the permeability of free space.

Superposition Principle

The superposition principle states that the total magnetic field at a point due to multiple sources is the vector sum of the magnetic fields produced by each source individually. In this case, the magnetic fields from both wires must be calculated separately and then combined to find the resultant magnetic field at the specified point.

Law of Cosines

The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is useful in determining distances and angles in problems involving multiple vectors, such as calculating the resultant magnetic field from two wires at a specific point. The formula is c² = a² + b² - 2ab * cos(θ), where c is the side opposite the angle θ.