A 1.73 m long wire carrying a current of 1.43 A forms a 2-turn loop in the shape

Question

A 1.73 m long wire carrying a current of 1.43 A forms a 2-turn loop in the shape of an equilateral triangle. If the loop is placed in a constant uniform magnetic field with a magnitude of 0.406 T, determine the maximum torque that acts on it. A 5-turn coil encloses an elliptical area having a major axis of L = 40.1 cm and a minor axis of W = 28.7 cm, as seen in the figure. The coil lies in the plane of the page and has an I = 5.32 A current flowing clockwise in it. If the coil is in a uniform magnetic field of B = 1.67x10^-4 T, directed toward the left of the page, what is the magnitude of the torque on the coil? Hints: What is the relationship between magnetic field B, magnetic moment p and torque T? (Answer: T = mu x B.) What kind of product is this? (Answer: vector or cross product.) Which hand do you use to determine the direction of a vector product? What is the magnetic moment p of a coil with turns N current I, and area A? The area of an ellipse is A = nab, where a and b are the semi major and semi minor axes of the ellipse.

Explanation / Answer

The torque on any shaped plane current loop with current "i", area "A" and "N" turns is;

T = N(iA) X B

The direction of the first vector is perpendicular to the plane of the loop found by curling fingers of right hand around the perimeter in the direction of "i". Your thumb points in the direction of vector "iA" (also called the magnetic moment "u".
In your case "B" is perpendicular to vector (iA) and the magnitude of the torque is simply;

T = NiAB
= Ni(pi)abB , a&b are semi major/minor axis of the ellipse.
= (5)(5.32)(pi)(.401/2)(.287/2)(1.67x10^-4)
= 4.01x 10^-4 N-m

part B

(torque on current loop) = (magnetic moment of loop) x (external magnetic field)

(magnetic moment of current loop) = (number of turns)(current)(area)

Area of equilateral triangle = s^2 sqrt(3) / 4

The maximum torque will occur when the plane of the loop and the external magnetic field are parallel to each other (direction of vectors A and ? is perpendicular to the loop plane).

?max = ?Bsin90

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