# Is force perpendicular to magnetic field in a bar magnet?

Recently I started this topic. My lecturer used

$$F_{B}=q_{n}B$$

I have two questions in this topic.

By this equation both the magnetic field and Force are in same direction. But aren't they supposed to be perpendicular to eachother, just like in the case of current carrying conductor? Are $$F$$ and $$B$$ in the same direction?

Is the force equation even correct? Because when asked about this, my lecturer replied it is due to coloumb's magnetic force law. But how much reliable is that law as monopoles do not exist?

(Sorry there are lots of questions and i have so many confusions about this topic. Kindly, if i made any mistake correct me.)

Thank you!

If you look at extremely old high school books (say a high school textbook from the US in the 1920's, like Millikan), it's fairly common to see this style of presentation. However, it's extremely problematic for a number of reasons. First of all, many magnetic objects, such as a toroid, do not have any identifiable poles, and even for something like a bar magnet the notion of a pole is an approximate one. And fundamentally, magnets are not made out of magnetic monopoles. In classical electromagnetism, they're made out of currents.

A sufficiently small magnetic dipole can be modeled as two opposite monopoles, of equal magnitude, connected by a rigid stick. In this situation, a uniform magnetic field generates only a net torque, but zero net force. In a nonuniform field, the dipole experiences a net force as well, but that's a more complex topic and I'm pretty sure that's not what your instructor is referring to if this is a basic intro to magnetism.

Because when asked about this, my lecturer replied it is due to coloumb's magnetic force law. But how much reliable is that law as monopoles do not exist?

It's hard to comment too much without understanding more about the context being assumed in the presentation. However, there is not really any uncertainty that this gives correct answers when applied to real-world objects that can be appropriately modeled as a collection of poles. There is only one reasonable way of extending Maxwell's equations to include magnetic monopoles. When you do that, you get a form of Gauss's law for the magnetic field that looks exactly like Gauss's law for the electric field. Just as Coulomb's law follows from Gauss's law for the electric field (with symmetry), so does a Coulomb's law for the magnetic case.

• Thanks... But also what do you think about my force and magnetic field argument? Are they always in the same direction in a bar magnet?? Sep 7, 2022 at 15:20

$$\mathbf F$$ and $$\mathbf B$$ are indeed in the same direction. Remember that in the equation you quote the (imaginary) thing on which the force acts is not a moving charge but a (North) monopole.

Think about a compass needle (pivoted magnet). It points with its North end in the direction of the external magnetic field, and its South end in the opposite direction. An early theoretical model of a magnet was a separated pair of equal and opposite monopoles, one near each end. $$\mathbf F=m\mathbf B$$, with $$m$$ as pole strength, would account for the equal and opposite forces experienced by the ends of the needle, and the direction in which the needle points when it has stopped oscillating. What's more, if each monopole gives rise to an inverse square law field (one inwards, one outwards), it also accounts quite nicely (using vector addition of fields) for the shape of a magnet's field outside the magnet. But it gives exactly the wrong direction for the field inside the magnet! And it's a bit disconcerting that no-one has yet found a monopole by itself.

The theory has long since been replaced by the idea that a magnet's magnetic properties arise electrically from spinning electrons. A current-carrying solenoid is a better model of a magnet than a pair of monopoles. Nonetheless, as mentioned earlier, there is a tempting partial fit between the monopoles model and what we observe, and at least until the mid twentieth century many courses in magnetism and electromagnetism made use of the idea of monopoles, at least in the early stages. Messy patches were used to compensate for the wrong field direction inside magnets and magnetic materials when treated as collections of true dipoles (as opposed to small current loops).

You ask about the accuracy of the inverse square law for magnetic poles. I believe that Coulomb used a torsion balance with ball-ended magnets, which behave approximately as if their poles were situated at the centres of the spheres. This method would not be capable of great accuracy. Because poles always come in pairs, there's no magnetic analogue of Maxwell's very accurate hollow sphere method for demonstrating the inverse square law for charges. But... If we use magnetic poles as a theoretical construct, we can obtain some results that are as accurate as we like. In particular, that a long way from it, the magnetic field pattern of a current loop is exactly the same as that of a dipole consisting of equal North and South monopoles on the loop axis, separated by a small distance. The Wiki article on 'Magnetic Dipoles' gives more detail.