Quick question about magnetic fields

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To find the total force on a rectangular loop in the z = 0 plane with the magnetic field B = (6x, -9y, 3z) and current of 5 A in the counterclockwise direction, the torque equation T = ∫(dl × B) should be applied. Integration must be performed for all four sides of the rectangle due to the non-constant nature of the magnetic field. Each side's contribution needs to be calculated using the electromagnetic force expression involving the cross product. After computing the integrals for each side, the resulting vectors should be summed to obtain the total force. This approach ensures an accurate calculation of the force experienced by the loop.
formulajoe
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given that B = (6x, -9y, 3z) find the total force experienced by a rectangular loop in the z = 0 plane.
1<x<3
1<y<2

i used the torque equation, but my question is if i have to do the integration for all 4 sides of the rectangle.
T = int(dl cross B)
 
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I don't get it,what's the curreent in the loop?

Daniel.
 
oh yea, there's 5 A in the counterclockwise direction
 
Okay,then.U know the expression for the electromagnetic force (the one containing the cross product) in diff.form.Apply it for the 4 sides of the rectangle and do those integrations (u need them,because the field is not constant) .Then simply add the 4 resulting vectors...

Daniel.
 
The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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