Force of an MRI on a paperclip

Problem

An MRI magnet produces $B = 1{,}5$ T at its centre. Estimate the attractive force on a $1$ g paperclip at $50$ cm from the centre, assuming a dipolar decay $B \propto 1/r^3$, and compare it with the weight.

Solution

The force on a ferromagnetic object is proportional to $B$ and to the gradient of the field: $F\propto B\,\frac{\dd B}{\dd r}$

With $B\propto1/r^3$ we have $\dfrac{\dd B}{\dd r}\propto-\dfrac{3B}{r}$, so $F\propto\dfrac{B^2}{r}$ decays very rapidly with distance from the centre. At $50$ cm the field is already much reduced with respect to the central value.

An order-of-magnitude estimate gives: $F \sim 2\cdot10^{-4}\;\text{N}$

To be compared with the weight of the paperclip: $P=mg\approx10^{-3}\cdot9{,}8\approx1\cdot10^{-2}\;\text{N}$

The magnetic force is much smaller than the weight: at this distance the paperclip stays on the ground. Getting closer to the centre (where $B$ and its gradient grow by orders of magnitude) the force would exceed the weight and the paperclip would be sucked in. This is a rough estimate, valid only as an order of magnitude.

$\ev{F \sim 2\cdot 10^{-4}\;\text{N}\ \ll\ P\approx1\cdot10^{-2}\;\text{N}}$

Links

Topics: Magnetismo Concepts: Campo magnetico Skills: Stime di Fermi