# What is equation to find force of magnetic attraction

I was wondering as a student, when I was playing with magnets I thought like Gravitational force equation of Newton's Gravitation Theory, can we also calculate force of an magnetic attraction on an para-magnetic object\material as it can help me learn more about how big an magnet needs to be to create this much pull or something along lines of Engineering or understanding how objects work.

I could not find any formulas pertaining the force of attraction on an para-magnetic body when induced to an magnetic field, therefore I want to ask an equation to calculate force of an magnetic attraction on an para-magnetic object\material? Or does one even exist?

It would be kind, if someone explains the equation.

## 1 Answer

Equations for magnetic interactions between objects tend to be a lot more complicated than for electrostatic forces. This is because while electric fields are produced by and exert forces on charge, a scalar, magnetic fields interact with electric currents, the flow of charge in a particular direction, which is a vector quantity. This makes the equations for calculating a magnetic field inherently more complicated, as they depend on the direction as well as the magnitude of the current. In addition to this a current is always a flow of charge from somewhere to somewhere else, so you never really encounter a "point current" in the same way as a point charge.

The equivalent to Newtons Law or Coulombs Law for magnetism is the Boit-Savart Law. If you have a current $I$ flowing through an infinitesimal length $\mathrm{d}\vec{l}$ then the field at a point $\vec{r}$ is given by $$\mathrm{d}\vec{B} = \frac{\mu_0}{4\pi} \frac{I \;\mathrm{d}\vec{l}\times\vec{r}}{|\vec{r}|^3}$$ The field does still drop of with the inverse square of the distance, but the vector nature of the current makes the formula rather more complicated. Generally this expression has to be integrated over the length of a wire carrying the current. This can get quite complicated and so this method is generally only used for simple situations such as the field around a magnetic dipole. Having found the field, the force on a infinitesimal length of current can be found from. $$\vec{F} = I\;\mathrm{d}\vec{l}\times\vec{B}$$ This is again analogous to the electrostatic case but with added complication due to current being a vector.

Magnetic materials can have a wide variety of properties and the shape of the object also has a significant effect on the fields in and around it, and so on the forces it experiences. There are general methods for finding the force on an object due to an applied magnetic field, but not simple formulae.