A rectangular loop of wire of height H = 20.0 cm and width W = 60.0 cm carries a current I_(2) = 0.170 A that flows counterclockwise about the loop. An infinite straight wire, lying in the plane of…
a) The magnetic field produced by the loop is…
b) The direction of the magnetic field is…
c) The magnetic force acting on the straight wire is…
d) The magnetic field depends on…
The Correct Answer and Explanation is :
To analyze the magnetic field and forces acting in this system, we can use Ampère’s law and the Biot-Savart law. Here’s a structured approach to answer the questions posed:
a) The magnetic field produced by the loop is:
The magnetic field (\mathbf{B}) at a point in space due to a current-carrying rectangular loop can be computed using the Biot-Savart law. The field varies at different points in space but is typically strongest at the center of the loop. For a rectangular loop, the magnetic field at the center can be approximated as:
[
B = \frac{\mu_0 I_2}{2R}
]
where (\mu_0) is the permeability of free space ((4\pi \times 10^{-7} \, \text{T m/A})), (I_2) is the current flowing through the loop, and (R) is the distance from the wire to the center of the loop. In this case, (R) would depend on the position of the straight wire in relation to the loop.
b) The direction of the magnetic field is:
Using the right-hand rule, if the current flows counterclockwise in the loop, the magnetic field inside the loop points upwards (out of the plane of the loop), while outside it points downwards (into the plane of the loop).
c) The magnetic force acting on the straight wire is:
The magnetic force (\mathbf{F}) on a straight wire carrying a current (I) in a magnetic field (\mathbf{B}) is given by:
[
F = I \cdot L \cdot B \cdot \sin(\theta)
]
where (L) is the length of the wire in the magnetic field, and (\theta) is the angle between the wire’s direction and the magnetic field. If the straight wire is parallel to the field, (\theta = 90^\circ) and (\sin(90^\circ) = 1), resulting in maximum force.
d) The magnetic field depends on:
The magnetic field produced by the loop depends on several factors:
- Current ((I_2)): The strength of the magnetic field is directly proportional to the current flowing through the loop.
- Geometry of the loop: The shape and dimensions of the loop affect how the magnetic field lines are distributed in space.
- Distance from the loop: The magnetic field strength diminishes with distance from the loop, following an inverse square or other relevant dependence based on the specific geometry.
In summary, the rectangular loop produces a magnetic field with direction determined by the right-hand rule and strength influenced by current, geometry, and distance. The interaction of this magnetic field with another current-carrying wire results in a force whose magnitude depends on the wire’s length, current, and the angle relative to the field.