A square loop of wire has a perimeter of 4.00 mm and is oriented such that two of its parallel sides form a 13.0 ∘∘ angle with the xx-axis and other two sides are parallel to the yy-axis. A uniform horizontal magnetic field parallel to the xx-axis has a magnitude of 0.250 TT . passes though the loop
a) Calculate the magnetic flux through the loop.
b) Another loop lies in the same plane but has an irregular shape, resembling a starfish. Its area is three times greater than that of the square loop. Calculate the flux through this loop.

a) (2.436 × 10⁻⁷) Wb

b) (7.308 × 10⁻⁷) Wb

Explanation:

Magnetic flux is the dot product of the magnetic field vector and the Area vector.

Mathematically, it is given as

Φ = BA cos θ

where B = magnetic field strength

A = Cross sectional Area of the loop enclosed

θ = angle between the magnetic field and the plane of the area.

a) B = 0.250 T

To find A, the perimeter of the loop is given as 4.00 mm.

Perimeter of a square = 4L

4L = 4.00

L = 1.00 mm = 0.001 m

The area is given as L²

A = (0.001)² = 0.000001 m²

θ = 13°

Φ = BA cos θ

Φ = 0.25 × 0.000001 × cos 13°

Φ = 0.0000002436 Wb = (2.436 × 10⁻⁷) Wb

b) Another loop lies in the same plane but has an irregular shape, resembling a starfish. Its area is three times greater than that of the square loop

Φ = BA cos θ

B = 0.25 T

A = 3 × 0.000001 = 0.000003 m²

θ = 13°

Φ = 0.25 × 0.000003 × cos 13°

Φ = 0.0000007308 Wb = (7.308 × 10⁻⁷) Wb

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