Deriving torque from a B field on a magnetic Loop

[Escher360]

Homework Statement

A wire ring lies in the xy-plane with its center at the origin. The ring carries a counter clockwise current I. A uniform magnetic fireld B is the the +x direction.
There are a few parts to the question (Note: Everything except for I is a vector):
1. show that dL = R*d(theta)*(-sin(theta)i + cos(theta)j) and from this Find dF = I*dL x B
2. From part 1, find d(tau) = r x dF. Where r = R(cos(theta)i + sin(theta)j) is the vector from the center of the loop of the element to dl. (dl is perpendicular to r). Show that the result can be written as: tau = mu x B.

Homework Equations

dL = R*d(theta)*(-sin(theta)i + cos(theta)j)
dF = I*dL x B
d(tau) = r x dF
r = R(cos(theta)i + sin(theta)j)
tau = mu x B
i is the x-direction
and j is in the y.

The Attempt at a Solution

Probably not worth writing

 

[jbsteele]

out the attempt because I got stuck. 1. dL = R*d(theta)*(-sin(theta)i + cos(theta)j)2. dF = I*dL x B = I*R*d(theta)*(-sin(theta)i + cos(theta)j) x B 3. d(tau) = r x dF = R(cos(theta)i + sin(theta)j) x I*R*d(theta)*(-sin(theta)i + cos(theta)j) x B 4. tau = mu x B Not sure what to do here.

 

[tomcue]

it all out, but essentially you are deriving the torque equation for a magnetic loop in a uniform magnetic field. By using the right hand rule and cross product, you can show that the torque is given by the product of the magnetic moment (mu) and the magnetic field (B). This is a fundamental equation in electromagnetism and is used in many applications, such as electric motors and generators. The derivation also shows that the torque is dependent on the angle between the magnetic moment and the magnetic field, which is an important concept to understand in electromagnetism. Overall, this exercise is a good way to solidify your understanding of torque and the relationship between magnetic fields and currents.