Question

To understand how to use the principle of superposition in conjunction with the Biot-Savart (or Ampere's) law.From the Biot-Savart law, it can be calculated that the magnitude of the magnetic field due to a long straight wire is given byBwire=?0I/2?d ,where ?0 (=4?×(10^?7T)?m/A) is the permeability constant, I is the current in the wire, and d is the distance from the wire to the location at which the magnitude of the magnetic field is being calculated.The same result can be obtained from Ampere's law as well.The direction of vector B? can be found using the right-hand rule. (Take care in applying the right-hand rule. Many students mistakenly use their left hand while applying the right-hand rule since those who use their right hand for writing sometimes automatically use their "pencil-free hand" to determine the direction of B? .)A) Which of the vectors best represents the direction of the magnetic field created at point K (see the diagram in the problem introduction) by wire 1 alone?B) Which of the vectors best represents the direction of the magnetic field created at point K by wire 2 alone?C) Which of these vectors best represents the direction of the net magnetic field created at point K by both wires?D) Find the magnitude of the magnetic field Bsub1K created at point K by wire 1.E) Find the magnitude of the net magnetic field BsubK created at point K by both wires

Answer 1

A) The direction of the magnetic field created at point K by wire 1 alone is represented by a vector pointing into the page.

B) The direction of the magnetic field created at point K by wire 2 alone is represented by a vector pointing out of the page.

C) The direction of the net magnetic field created at point K by both wires is the vector sum of the individual fields, resulting in a resultant vector pointing slightly into the page.

D) The magnitude of the magnetic field B₁ᴷ created at point K by wire 1 is B₁ᴷ = μ₀I₁ / 2πd₁.

E) The magnitude of the net magnetic field Bᴷ created at point K by both wires is Bᴷ = B₁ᴷ + B₂ᴷ = μ₀I₁ / 2πd₁ + μ₀I₂ / 2πd₂.

Step-by-step explanation:

A) For wire 1, applying the right-hand rule indicates that the magnetic field curls in a clockwise direction around the wire, suggesting a field directed into the page at point K.

B) Conversely, for wire 2, the magnetic field curls counterclockwise around the wire, implying a field directed out of the page at point K.

C) When considering both wires, the net magnetic field at point K results from the vector sum of the individual fields. Due to the proximity and alignment of the wires, the resultant field is slightly into the page.

D) The magnitude of the magnetic field created by wire 1 at point K, B₁ᴷ, is determined using the Biot-Savart law formula, where μ₀ is the permeability constant, I₁ is the current in wire 1, and d₁ is the distance between wire 1 and point K.

E) To find the net magnetic field at point K due to both wires, the individual magnetic field magnitudes (B₁ᴷ and B₂ᴷ) are summed since these fields act concurrently. This sum gives the overall magnetic field strength at point K caused by the combined effect of both wires.

Answer 2

To determine the magnetic field at a point due to current in wires, use the Biot-Savart law for magnitude and the right-hand rule for direction. Apply the principle of superposition through vector addition to find the net magnetic field. Ampère's law can simplify calculations in symmetrical or infinite straight wire situations.

Step-by-step explanation:

The principles of superposition and the Biot-Savart or Ampère's law are central to understanding magnetic fields around conductors. The Biot-Savart law is particularly useful for calculating the magnetic field at any point in space due to current flow through a conductor.

Applying the Biot-Savart Law

1. Identify the current element length and the position vector relative to the point where the magnetic field is calculated.
2. Use the cross product to find the direction of the resultant magnetic field from individual segments of the current-carrying wire.

Applying the Right-Hand Rule

The right-hand rule (RHR-2) assists in determining the direction of the magnetic field around a straight conductor. To apply RHR-2, one should use their right hand and point the thumb in the direction of the current. The fingers will then curl in the direction of the magnetic field lines.

To find the total magnetic field due to multiple wires, you calculate the individual magnetic fields using the Biot-Savart law and then apply the principle of superposition, which involves vector addition of the individual fields to find the net magnetic field.

Calculating Magnetic Field Magnitudes

• To calculate the magnetic field magnitude at a point due to a single wire, use the Biot-Savart law and plug in the known values of the current (I), the permeability constant (μ0), and the distance from the wire (d).
• For determining the combined effect of both wires on the magnetic field, use vector addition to sum up the magnetic fields created by each wire at the given point.

In practical scenarios, if the problem involves symmetrical situations or long straight wires, Ampère's law can also be used, which sometimes simplifies the calculations.