(II) Two long parallel wires 8.20 cm apart carry 19.5-A dc currents in the same direction. Determine the magnetic field vector at a point P, 12.0 cm from one wire and 13.0 cm from the other. See Fig. 28–43. [Hint: Use the law of cosines. See Appendix A or inside rear cover.]

Verified step by step guidance

1. Step 1: Understand the problem. Two parallel wires carry currents in the same direction, and we need to find the net magnetic field at a point P located at specific distances from each wire. The magnetic field due to a current-carrying wire is given by the Biot-Savart law, simplified for a long straight wire as: $B=\frac{\mu ₀I}{2\pi r}$, where $\mu ₀$ 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 contributions from each wire at point P. For the first wire, use the formula $B$₁ = μ₀I2πr₁, where $r$₁ is the distance from the first wire (12.0 cm). Similarly, for the second wire, use $B$₂ = μ₀I2πr₂, where $r$₂ is the distance from the second wire (13.0 cm).
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 fingers curl in the direction of the magnetic field. Since the currents are in the same direction, the magnetic fields at point P will have different orientations. Represent these fields as vectors and note their directions.
4. Step 4: Use vector addition to find the net magnetic field at point P. The angle between the two magnetic field vectors can be determined using the law of cosines. The distance between the two wires is 8.20 cm, so the angle $\theta$ can be calculated as: $\cos \theta =\frac{r}{}$₁² + r₂² - d²2r₁r₂, where $d$ is the distance between the wires (8.20 cm).
5. Step 5: Combine the magnetic field vectors. Use the formula for the resultant of two vectors: $B$_net = B₁² + B₂² + 2B₁B₂cosθ. This will give the magnitude of the net magnetic field at point P. The direction can be found using trigonometry if needed.

Key Concepts

Magnetic Field Due to a Current-Carrying Wire

The magnetic field generated by a long straight wire carrying a current can be described by Ampère's Law. The magnetic field (B) at a distance (r) from a long straight wire is given by the formula B = (μ₀I)/(2πr), where μ₀ is the permeability of free space and I is the current. The direction of the magnetic field follows the right-hand rule, curling around the wire in a circular pattern.

Superposition of Magnetic Fields

When multiple current-carrying wires are present, the total magnetic field at a point is the vector sum of the magnetic fields produced by each wire. This principle of superposition allows us to calculate the resultant magnetic field by considering both the magnitude and direction of the fields from each wire at the point of interest.

Law of Cosines

The Law of Cosines is a mathematical formula used to relate the lengths of the sides of a triangle to the cosine of one of its angles. It is particularly useful in physics for determining distances and angles in problems involving vectors. In this context, it helps to find the resultant distance from the point of interest to the wires, which is essential for calculating the magnetic field.