How Does a Square Wire Frame React in a Constant Magnetic Field?

[Dell]

http://lh6.ggpht.com/_H4Iz7SmBrbk/SikOL1lh7BI/AAAAAAAABCY/_70wWSK_-Vw/s640/DEVAN.jp

given a square wire frame with a side of length L, placed in a constant extarnal magnetic field B, (B=B*$$\hat{k}$$). a current of i flows through the frame anticlockwise(as shown by arrows in diagram)

the frame can turn freely about the x-axis (which passes through its centre)
the frame has a linear mass density of λ

a) what is the force applied on the frame?
b) what is the momentum applied of the frame?
c) what is the angular acceleration of the frame?

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a) since the force applied would be ILxB,
the force on either side parallel to the field B has no contribution since ILB*sinθ=0 (θ=0/180)
for either side perpendicular to the field ILxB=ILB

therefore my total force on the frame would be 0+0+ILB-ILB=0

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b) for the momentum i say that N=RxF=(L/2)*F=(L/2)*ILB=0.5L²IB

and since i have 2 contributors to the momentum and the are both on x-,
N=-L²IB*$$\hat{i}$$
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c) for the angular acceleration α, i need to somehow find the moment of enetria for a frame like this, and then i can say that N=Iα
α=N/I

can i use the moment of a pole 1/12(mL²) and multiply it by 2 to take into account each side? this doesn't seem right to me, what aout the other 2 sides wgich arent revolving about their centre?

 

[jbsteele]

so i found a different formula which is I=λL2/3 so using this i can say that α=N/I= (L2IB)/(λL2/3)= 3IB/λ

 

[nlightner]

I would like to clarify a few things about the concept of magnetic force and fields in this scenario. First, let's define some terms for clarity. The square wire frame in the image is a closed loop of wire, also known as a current-carrying loop. The external magnetic field B is a vector quantity, meaning it has both magnitude and direction. In this case, B is pointing in the z-direction, as denoted by the unit vector ̂k. The current i is the flow of electric charge through the wire loop, and is measured in amperes (A). The linear mass density λ refers to the mass per unit length of the wire loop, and is measured in kilograms per meter (kg/m).

Now, let's answer the questions posed in the content.

a) The force applied on the wire loop can be calculated using the equation F = IL x B, where I is the current, L is the length of the wire loop, and B is the external magnetic field. In this case, since the wire loop is oriented perpendicular to the magnetic field, the force on each side of the loop is equal and opposite, resulting in a net force of 0. This is because the angle between the current and the magnetic field is 90 degrees, making the sine of the angle 1 (sin90=1). Therefore, the force applied on the wire loop is 0.

b) The momentum applied on the wire loop can be calculated using the equation p = I x B, where p is the momentum, I is the current, and B is the external magnetic field. In this case, since the wire loop is oriented perpendicular to the magnetic field, the momentum on each side of the loop is equal and opposite, resulting in a net momentum of 0. Again, this is because the angle between the current and the magnetic field is 90 degrees, making the sine of the angle 1 (sin90=1). Therefore, the momentum applied on the wire loop is 0.

c) The angular acceleration of the wire loop can be calculated using the equation α = N/I, where N is the net torque applied on the loop and I is the moment of inertia. The moment of inertia for a wire loop can be calculated using the equation I = 1/12mL^2, where m is the mass of the wire loop and L is the length of the loop. In