- #1
Gertrude
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Homework Statement
I want to determine the net force and torqe on a moving electric dipole in non-uniform magnetic field.
I suspect I should take some kind of a limit, but I'm not sure how to do so.
Please help, I'd really like to understand this.
Homework Equations
##\mathbf{F} = q \mathbf{v} \times \mathbf{B}, \quad \mathbf{\tau} = \mathbf{r} \times \mathbf{F}, \quad \mathbf{p} = q \mathbf{d}##
The Attempt at a Solution
I wrote the velocities as the average velocity ##\mathbf{v} = (\mathbf{v}_1+\mathbf{v}_2)/2## plus the difference (index 1 indicates the positive charge and 2 the negative charge):
$$\mathbf{F} = q\mathbf{v}_1 \times \mathbf{B}(\mathbf{r} + \mathbf{p}/q) - q\mathbf{v}_2 \times \mathbf{B}(\mathbf{r})$$
So what I got in the end was:
$$\mathbf{F} = q \mathbf{v} \times (\mathbf{B}(\mathbf{r} + \mathbf{p}/q) - \mathbf{B}(\mathbf{r})) + \frac{q}{2} \dot{\mathbf{p}} \times (\mathbf{B}(\mathbf{r} + \mathbf{p}/q) + \mathbf{B}(\mathbf{r}))$$
I'm stuck here and I'm not sure how to get a 'nicer' form that would only contain ##\mathbf{B}(\mathbf{r})##.
Regarding the torque:
$$\mathbf{\tau} = \frac{\mathbf{d}}{2} \times \mathbf{F}_1 - \frac{\mathbf{d}}{2} \times \mathbf{F}_2$$
Rearraging a bit I got:
$$\mathbf{\tau} = \frac{\mathbf{p}}{2} \times (\mathbf{v}_1 \times \mathbf{B}(\mathbf{r} + \mathbf{p}/q) + \mathbf{v}_2 \times \mathbf{B}(\mathbf{r}))$$
From here on I have the same problem as before.
If you can help me with a hint or two I'll be really grateful.