91Ó°ÊÓ

When dealing with magnetic fields, it is essential to understand that they can be described in terms of their components along different axes, typically the x, y, and z axes. In this example, we have three components: \( B_x = 0.10 \)T, \( B_y = 0.15 \)T, and \( B_z = 0.17 \)T. Each of these components reflects the strength of the magnetic field in a particular direction.
To analyze the effect of a magnetic field on a current-carrying wire, identifying the relevant components is crucial.

• The wire is aligned along the z-axis, meaning the current runs in that direction.
• The components that affect the wire’s magnetic force are those perpendicular to its direction: \( B_x \) and \( B_y \).

These perpendicular components interact with the current in the wire, and their combined effect determines the magnetic force experienced by the wire.

The concept of a current-carrying wire is fundamental in physics, where electric currents produce magnetic fields around the wire. In this case, we have a wire that is 25 cm long (0.25 m in meters) and carries a current of 4.3 A along the z-axis.
The interaction between the wire and external magnetic fields results in a force that can be calculated using the cross product of the current and magnetic field vectors. The equation used for finding the force is:

• \( F = I \times L \times B \times \sin(\theta) \)

For our wire, since it is aligned along the z-axis, the angle \( \theta \) between the current direction and the field's perpendicular components \( B_x \) and \( B_y \) is 90 degrees, which simplifies calculations because \( \sin(90) = 1 \).
This setup allows physicists to predict how wires will behave in various magnetic environments, vital for designing electrical devices.

The Pythagorean Theorem is a mathematical principle that applies to right triangles. It states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This theorem helps us find the magnitude of a resultant vector, such as the effective magnetic field component that is perpendicular to the wire.
Given our magnetic field components \( B_x \) and \( B_y \), we use the Pythagorean Theorem to find \( B_\perp \):

• \( B_\perp = \sqrt{B_x^2 + B_y^2} \)

Using the available data:

• \( B_\perp = \sqrt{(0.10)^2 + (0.15)^2} = \sqrt{0.01 + 0.0225} \)
• Which calculates to \( B_\perp = \sqrt{0.0325} \approx 0.18 \text{ Tesla} \)

This effective perpendicular component is then used in the magnetic force formula to determine the force acting on the wire. The Pythagorean Theorem thus plays a key role in dissecting complex problems into simpler, manageable calculations.