Force on a magnet in a magnetic field

[TheBigDig]

Homework Statement

Deduce an expression for the force on a magnet in a field gradient dB/dz assuming m || B

Relevant Equations

$$E = -\vec{m}\cdot \vec{B}$$
$$\vec{F} = \nabla (\vec{m}\cdot \vec{B})$$

So I'm kinda stumped. I'm assuming that since $\vec{m}||\vec{B}$, the x and y components of both are zero. But I'm unsure how to take this further.

 

[Orodruin]

Why do you assume that the x and y components are zero? The only thing you have been told is that the field has a non-zero gradient in the z-direction.

 

[TheBigDig]

Why do you assume that the x and y components are zero?

I assumed that because the fields are parallel to one another. But you're right, that isn't strong enough reasoning. If you take the dot product of m and B you'll get $\vec{m}\cdot\vec{B} = mBcos\theta$. But I don't see how that'll help me.

 

[Orodruin]

No, you get $mB$, the angle is zero since $\vec m$ and $\vec B$ are parallel. However, the field depends on the position.

 

[TheBigDig]

Sorry, I don't quite understand what you mean by position

 

[Orodruin]

The field has a gradient and therefore depends on where the magnet is located, i.e., the position. This is what leads to a force.

 

[TheBigDig]

Okay, so if the magnet was located at the origin how would that affect the field?

 

[Orodruin]

Forget the origin. All you need to know is the gradient at the point where it is actually located.

 

[TheBigDig]

Forgive me for being dense but what'll that imply for the force?

 

[Orodruin]

The force is the gradient of $\vec m \cdot \vec B$. If that quantity depends on position, then there will be a force.