- #1

tim_lou

- 682

- 1

I have searched various websites... but all they have about magnetic moments is:

[tex]\vec\mu=I\vec{A}[/tex]

and

[tex]\vec\tau=\vec\mu\times\vec{B}[/tex]

but all the the sources only showed that the equation is true for rectangular loop of current.

but I need a more satisfactory answer. So I try to prove this in a general case:

I want to prove that the torque is indeed [tex]\vec\mu\times\vec{B}[/tex], when magnetic moment is defined as:

[tex]\vec\mu=I\int d\vec{A}[/tex]

(after proving that, i will try to prove the energy relation: [tex]E=-\vec\mu\cdot{\vec{B}}[/tex])

So, I forge ahead, let a closed current loop be the curve r, and a constant magnetic field B exists (I is current in the loop, tau is torque):

[tex]d\vec\tau=\vec{r}\times d\vec{F}=I \vec{r}\times (d\vec{r}\times\vec{B})[/tex]

where I is current in the wire, assume that it is constant.

[tex]\vec\tau=I\oint\vec{r}\times(d\vec{r}\times\vec{B})[/tex]

I used the triple product formula and some manipulations, then I got:

[tex]\vec\tau=I\oint\vec{r}\times{d\vec{r}}\times{\vec{B}}+I\oint\vec{r}(\vec{B}\cdot{d\vec{p})[/tex]

the first term looks like [tex]\vec\mu\times\vec{B}[/tex] but then I can't tell if the second term is zero...well, i checked the manipulation numerous times... i basically used:

[tex]\vec{r}\times{d\vec{r}}\times\vec{B}=d\vec{r}(\vec{B}\cdot\vec{r})-\vec{r}(\vec{B}\cdot{d\vec{r}})[/tex]

and

[tex]\vec{r}\times{}(d\vec{r}\times\vec{B})=d\vec{r}(\vec{r}\cdot\vec{B})-\vec{B}(\vec{r}\cdot{d\vec{r}})[/tex]

when integrating, B(r * dr) goes to zero since it is a closed loop.

Hence,

[tex]\vec\tau=I\oint\vec{r}\times{d\vec{r}}\times{\vec{B}}+I\oint\vec{r}(\vec{B}\cdot{d\vec{p})=\vec\mu\times\vec{B}+I\oint\vec{r}(\vec{B}\cdot{d\vec{p})[/tex].

I just changed the integrand.

now I'm stuck...supposedly, torque=mu cross magnetic field. but what is the second term of the integral doing there? perhaps the formula for the torque on any closed current loop is not simply mu cross B?

Questions: what frame should I use to calculate the torque? does it matter? should I choose the center of mass frame? Is the torque always the same regardless of what reference frame one is in?

edited: clarified my intentions.

[tex]\vec\mu=I\vec{A}[/tex]

and

[tex]\vec\tau=\vec\mu\times\vec{B}[/tex]

but all the the sources only showed that the equation is true for rectangular loop of current.

but I need a more satisfactory answer. So I try to prove this in a general case:

I want to prove that the torque is indeed [tex]\vec\mu\times\vec{B}[/tex], when magnetic moment is defined as:

[tex]\vec\mu=I\int d\vec{A}[/tex]

(after proving that, i will try to prove the energy relation: [tex]E=-\vec\mu\cdot{\vec{B}}[/tex])

So, I forge ahead, let a closed current loop be the curve r, and a constant magnetic field B exists (I is current in the loop, tau is torque):

[tex]d\vec\tau=\vec{r}\times d\vec{F}=I \vec{r}\times (d\vec{r}\times\vec{B})[/tex]

where I is current in the wire, assume that it is constant.

[tex]\vec\tau=I\oint\vec{r}\times(d\vec{r}\times\vec{B})[/tex]

I used the triple product formula and some manipulations, then I got:

[tex]\vec\tau=I\oint\vec{r}\times{d\vec{r}}\times{\vec{B}}+I\oint\vec{r}(\vec{B}\cdot{d\vec{p})[/tex]

the first term looks like [tex]\vec\mu\times\vec{B}[/tex] but then I can't tell if the second term is zero...well, i checked the manipulation numerous times... i basically used:

[tex]\vec{r}\times{d\vec{r}}\times\vec{B}=d\vec{r}(\vec{B}\cdot\vec{r})-\vec{r}(\vec{B}\cdot{d\vec{r}})[/tex]

and

[tex]\vec{r}\times{}(d\vec{r}\times\vec{B})=d\vec{r}(\vec{r}\cdot\vec{B})-\vec{B}(\vec{r}\cdot{d\vec{r}})[/tex]

when integrating, B(r * dr) goes to zero since it is a closed loop.

Hence,

[tex]\vec\tau=I\oint\vec{r}\times{d\vec{r}}\times{\vec{B}}+I\oint\vec{r}(\vec{B}\cdot{d\vec{p})=\vec\mu\times\vec{B}+I\oint\vec{r}(\vec{B}\cdot{d\vec{p})[/tex].

I just changed the integrand.

now I'm stuck...supposedly, torque=mu cross magnetic field. but what is the second term of the integral doing there? perhaps the formula for the torque on any closed current loop is not simply mu cross B?

Questions: what frame should I use to calculate the torque? does it matter? should I choose the center of mass frame? Is the torque always the same regardless of what reference frame one is in?

edited: clarified my intentions.

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