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The Biot-Savart Law is a fundamental principle in electromagnetism that describes how electric currents produce magnetic fields. It is essential when calculating the magnetic field generated by a long, straight current-carrying wire or other geometries. The formula it provides is given by:\[ \vec{B} = \frac{\mu_0 I}{2\pi r} \hat{k} \]where:

• \( \vec{B} \) is the magnetic field produced.
• \( I \) is the current in the wire.
• \( r \) is the perpendicular distance from the wire to the point where the field is being calculated.
• \( \mu_0 \) (the permeability of free space) is a constant \(4\pi \times 10^{-7} \mathrm{Tm} / \mathrm{A}\).

This law applies to any shape of wire, but is most straightforward with long straight wires.
The direction of the magnetic field vector depends on the current's direction as explained by the right-hand rule, and the field lines wrap around the wire in concentric circles.

Current-carrying wires are the basic elements used in the Biot-Savart Law to calculate the magnetic field. When a wire carries a current, it generates a magnetic field around it, forming closed loops.A current-carrying wire can be described in terms of:

• Intensity of the current, \(I\), measured in Amperes (A).
• Its orientation in space, often along the coordinate axes in problems.

Long straight wires are often used in textbook problems because their magnetic field can easily be calculated using symmetric properties. The strength of the magnetic field decreases with distance from the wire.
In our problem context, you see two wires: One carries \(6.00 \mathrm{~A}\) positioned at \(10.0 \mathrm{~cm}\), and the other carries \(10.0 \mathrm{~A}\) at \(5.00 \mathrm{~cm}\). Both currents flow in the same direction initially, affecting the calculation of net magnetic fields at a specific point, such as the origin.

Vector addition is a critical mathematical concept used to calculate the net effect of multiple vectors acting simultaneously in a space. For many physical phenomena, like the magnetic fields from different wires, we add their vector contributions to find the resultant field.When dealing with magnetic fields from each current-carrying wire:

• Each magnetic field is a vector with both magnitude and direction.
• To find the net field at a point, such as the origin, we perform vector addition.
• In the exercise, \( \vec{B}_1 \) and \( \vec{B}_2 \) are summed to find \( \vec{B} \).

The direction is essential, determined through rules like the right-hand rule. The challenges involve ensuring the sum respects both magnitudes and directions, such as two magnetic fields pointing into the page, reinforcing each other at the origin. As solved, the mathematical addition results in:\[ \vec{B} = (-1.20 \times 10^{-5} - 4.00 \times 10^{-5}) \hat{k} = -5.20 \times 10^{-5} \hat{k} \, \mathrm{T} \]

The right-hand rule is an intuitive tool used to determine the direction of the magnetic field generated by a current-carrying wire. This rule provides a quick way to ascertain the vector direction without complex calculations. To apply the right-hand rule:

• Imagine holding the wire in your right hand, with your thumb pointing in the direction of the conventional current (positive direction).
• Your fingers will naturally curl around the wire in the direction of the magnetic field lines.

For the exercise involving wires on the xy-plane:

• The rule indicates whether the magnetic field points into or out of the page.
• For both wires having currents in the positive x-direction, both magnetic fields point into the page (negative z-direction).
• If the current direction reverses, the magnetic field direction also reverses based on this intuitive rule.

By following the right-hand rule, one can correctly visualize and determine the magnetic field directions necessary for accurate vector addition and solving the problem effectively.