Deriving the Force on an Infinitessimal current-loop in a Magnetic Field

AI Thread Summary
The discussion focuses on deriving the force on an infinitesimal current loop in a magnetic field using the equation F = ∇(m·B). The user evaluates the force along a square loop and applies a Taylor expansion to simplify expressions for the magnetic field. They derive terms involving partial derivatives of the magnetic field with respect to y and z, but seek clarification on a specific approximation in the solution manual regarding the integral of the magnetic field's derivative. Additionally, there is a request for assistance with LaTeX coding issues encountered in the post. The conversation emphasizes the mathematical derivation and understanding of the underlying physics principles.
OGrowli
Messages
12
Reaction score
0
We're asked to derive the following equation:

F=\triangledown(\vec{m}\cdot \vec{B})

by evaluating F=\int I(d\vec{l}\times \vec{B}) along a sqaure loop with sides of length ε, parralel to the yz plane. The square's bottom left corner is situated at the origin.

so far I have,
d\vec{l} \times \vec{B}= -dz \times \hat{z} \times \vec{B}(0,0,z) + dy \hat{y} \times \vec{B}(0,y,0)+dz\hat{z} \times \vec{B}(0,\epsilon ,z)-dy \hat{y} \times \vec{B}(0,y,\epsilon)

then using taylor expansion to turn B(0,y,ε) and B(0,ε,z) into expressions of B(0,y,0) and B(0,0,z) respectively and combining like terms I've got:

<br /> - \hat{y} \times \left \epsilon dy\frac{\partial \vec{B}} {\partial z} \right |_{0,y,0} + \hat {z} \times \left \epsilon dz \frac{\partial \vec{B}} {\partial y} \right |_{0,0,z} <br />

the next step would be to evaluate ∫dF and the solutions manual says that,

<br /> \left \int dz\frac{\partial\vec{B}}{\partial y} \right |_{0,0,z}\approx \left \epsilon \frac{\partial\vec{B}}{\partial y} \right |_{0,0,0}<br />

from this point the solution comes pretty easily, but I still don't understand why the above is true. Can anyone explain this to me?
 
Physics news on Phys.org
OGrowli said:
<br /> - \hat{y} \times \left \epsilon dy\frac{\partial \vec{B}} {\partial z} \right |_{0,y,0} + \hat {z} \times \left \epsilon dz \frac{\partial \vec{B}} {\partial y} \right |_{0,0,z} <br />

<br /> \left \int dz\frac{\partial\vec{B}}{\partial y} \right |_{0,0,z}\approx \left \epsilon \frac{\partial\vec{B}}{\partial y} \right |_{0,0,0}<br />

Does anyone know what is wrong with the LaTex coding here?
 
Thread 'Voltmeter readings for this circuit with switches'
TL;DR Summary: I would like to know the voltmeter readings on the two resistors separately in the picture in the following cases , When one of the keys is closed When both of them are opened (Knowing that the battery has negligible internal resistance) My thoughts for the first case , one of them must be 12 volt while the other is 0 The second case we'll I think both voltmeter readings should be 12 volt since they are both parallel to the battery and they involve the key within what the...
Thread 'Struggling to make relation between elastic force and height'
Hello guys this is what I tried so far. I used the UTS to calculate the force it needs when the rope tears. My idea was to make a relationship/ function that would give me the force depending on height. Yeah i couldnt find a way to solve it. I also thought about how I could use hooks law (how it was given to me in my script) with the thought of instead of having two part of a rope id have one singular rope from the middle to the top where I could find the difference in height. But the...
Back
Top