A rectangular loop 3.2 cm wide and 5.1 cm long is placed in a magnetic field. The angle betwee

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A rectangular loop 3.2 cm wide and 5.1 cm long is placed...

A rectangular loop $3.2 \mathrm{~cm}$ wide and $5.1 \mathrm{~cm}$ long is placed in a magnetic field. The angle between the normal to the loop and the magnetic field is $32^{\circ}$, and the magnetic flux through the loop is $2.2 \times 10^{-3} \mathrm{~T} \cdot \mathrm{m}^{2}$. What is the magnitude of the magnetic field?

Key Concepts

Magnetic Flux

Magnetic flux is a measure of the total magnetic field passing through a given area. It is defined as the dot product of the magnetic field vector and the area vector, which inherently includes the cosine of the angle between them. This concept is central in electromagnetism and is expressed as ? = B · A · cos(?), highlighting how both the strength of the field and its orientation affect the net flux.

Vector Dot Product and Angle

The dot product in the context of magnetic flux accounts for the orientation between the magnetic field and the surface normal. The cosine term, cos(?), in the flux equation shows that only the component of the magnetic field perpendicular to the surface contributes to the flux. Understanding this relation is crucial when the magnetic field is not aligned perpendicularly with the surface.

Area Calculation

Calculating the area of the loop is an essential step since the magnetic flux depends directly on the area exposed to the magnetic field. For a rectangular loop, the area is obtained by multiplying its length by its width. This concept is a fundamental component of geometry as applied in physics.

Unit Conversion

Accurate calculation requires consistent units, often necessitating conversion such as centimeters to meters, especially because the SI unit for area in the magnetic flux formula is square meters. Ensuring proper unit conversion is vital to avoid errors in the final computation of physical quantities.

Transcript

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