Questions: Two very long currents, both of magnitude 2 A run as shown. One is flowing into the page (-z), while the other is flowing to the right (+x). Three locations (X, Y, and Z) are labeled. i. At which location is the magnitude of the total net magnetic field the lowest? A. Point X B. Point Y C. Point Z D. Points X and Y tied for lowest E. It is equal at all three points ii. In which direction is the total net magnetic field at Point Z? A. Right and Out of the page (+x,+z) B. Right and Into the page (+x,-z) C. Left and Out of the page (-x,+z) D. Left and Into the page (-x,-z)

Solution Steps

• The magnetic field due to a long straight current-carrying wire is given by \( B = \frac{\mu_0 I}{2 \pi r} \), where \( I \) is the current and \( r \) is the distance from the wire.
• For the wire with current into the page (at the origin), use the right-hand rule to determine the direction of the magnetic field at points X, Y, and Z.
• For the wire with current to the right (along the x-axis), use the right-hand rule to determine the direction of the magnetic field at points X, Y, and Z.

• Point X (2 mm above the wire at the origin):

  • Distance from the wire at the origin: \( r = 2 \text{ mm} \)
  • Distance from the wire along the x-axis: \( r = \sqrt{(2 \text{ mm})^2 + (2 \text{ mm})^2} = \sqrt{8} \text{ mm} \)
  • Calculate the magnetic fields and their directions.

• Point Y (4 mm above the wire at the origin):

  • Distance from the wire at the origin: \( r = 4 \text{ mm} \)
  • Distance from the wire along the x-axis: \( r = \sqrt{(2 \text{ mm})^2 + (4 \text{ mm})^2} = \sqrt{20} \text{ mm} \)
  • Calculate the magnetic fields and their directions.

• Point Z (2 mm below the wire at the origin):

  • Distance from the wire at the origin: \( r = 2 \text{ mm} \)
  • Distance from the wire along the x-axis: \( r = 2 \text{ mm} \)
  • Calculate the magnetic fields and their directions.

• Point X:
  • Sum the magnetic field vectors from both wires.
• Point Y:
  • Sum the magnetic field vectors from both wires.
• Point Z:
  • Sum the magnetic field vectors from both wires.

Final Answer