## Why we have been wrong about ‘magnetic field strength’

The British Institute of Radiology recently published three information sheets on MRI safety. **Myths and maths** was one of them. In this series I’d like to expand on themes of that sheet. More details can be sourced from my book Essentials of MRI Safety.

Myth:the tesla is the SI unit of magnetic field strengthBIR Information Sheet

Maths:the tesla is the SI unit of magnetic flux density. Magnetic field strength or intensity is measured in amperes per metre. We have been using the wrong terminology for decades!

We’ve all said it: “What field strength is your scanner?”

Incorrect answer: 1.5 tesla

Correct answer: approximately 1.2 million amperes/metre (1,193,662 A/m).

Why is this? Because ‘magnetic field strength’ or intensity (denoted **H**) is a different physical quantity. What we incorrectly refer to as ‘field strength’ is actually **magnetic flux density** (**B**).

### Does it matter?

In a vacuum or in air – *no*. B and H are related through a constant μ_{0} (= 4π x 10^{-7} henrys per metre – H/m).

In a magnetic medium- *yes* it matters very much. You can think of it in this way. The H-field is produced when you have a current (amps) flowing through a coil. So the scanner produces an H-field. The B-field is what you get within a medium. In the case of air B =μ_{0}H, but in another medium the flux density is the sum of the field arising from induction (the H-field) plus the field arising from **magnetisation** of the material.

Considering ferromagnetic metals, we talk about the ‘saturation B-field.’ This is incorrect and misleading. B does not saturate. It is the magnetisation **M** (also measured in A/m) that saturates. Above saturation, B will continue to increase as H is increases, e.g. by moving the object closer to the scanner (although this will probably result in a projectile accident- so don’t try it).

It’s misleading to think of a ‘saturation B value’ or B_{sat} because the simplistic and wrong implication is that the object will magnetically saturate in an external field equal to B_{sat}, whereas the truth is that it will saturate in a much smaller external field- further away from the bore opening – and highly dependent upon the shape of the object.

Another cause of confusion that arises from our misuse of ‘field strength’ lies with the expressions we use to calculate the translational (projectile) force on objects. We see in the literature (including *Essentials of MRI Safety*) complicated equations involving the B-field and its spatial gradient but differing for para/dia magnetic and saturated/unsaturated ferromagnetic objects. These are not wrong, but the *direct* cause of the translational force is not magnetic flux density B, but the **magnetisation** M_{z} of the object as well as the spatial gradient of the B-field. More simply the force is *always* (for all materials):

V is the volume of the object, dB/dz the spatial gradient of the external B-field. Of course the value of M does depend upon B (and H) but it’s the interaction of M with dB/dz that propels the object. As explained in *Myths and Maths 1*, it is *magnetisation* which is the key factor for the translational force.

### So what is B?

To answer that let’s turn to the definition of its unit, the tesla (T). 1 T is defined as being the magnetic flux density required to produce a force of 1 newton (N) on an electrical charge of 1 coulomb (C) moving perpendicular to the B-field with a velocity of 1 metre per second (m/s).

Magnetic flux density B is defined in terms of the **Lorentz force** – not the mechanical force. This is why another name for the B-field is ‘magnetic induction’.

If B is flux *density*, then what is magnetic flux? Flux is the sum of B passing though a surface. You can think of it as the total number of ‘field lines’. Faraday’s law states that the voltage induced around a circuit is equal to the rate of change of flux passing though the circuit.

^{2}= T)

So once again we see that magnetic flux and its density (i.e. B-related quantities) are fundamentally related to induction. We can think of the excitation of MR signal as being another form of induction, so why not use the proper name for our B-field? Let’s call it by its true name(s): magnetic flux density or magnetic induction.

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