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## Vectors in 3 dimensions

In this blog, I explained some of the basics of vectors. I only dealt with 2-dimensional vectors, but of course we live in a 3-dimensional world. So, to fully specify a vector in 3-dimensions we need to add a third component. Usually, but not always, we can use the x, y and z-components of the vector, and so can write any vector in terms of its x, y and z-components.

For example, let us suppose we have some force $\vec{F}$ which is in 3 dimensions, then we can write it in terms of its Cartesian $(x,y,z)$ components as $\vec{F} = F_{x} \; \hat{x} + F_{y} \; \hat{y} + F_{z} \; \hat{z}$, where $\hat{x}, \hat{y}, \hat{z}$ are the unit vectors in the (x,y,z) directions respectively, and $F_{x}, F_{y}$ and $F_{z}$ are the x, y and z-components respectively (in this figure the unit vectors $\hat{x}, \hat{y}$ and $\hat{z}$ are labelled $\hat{i}, \hat{j}$ and $\hat{k}$, which is an alternative nomenclature ).

The 3-D force F can be broken down into its 3 components in the x, y and z-directions.

Sometimes, we may be interested in how much the vector is changing with position. A good example of this might be the force on a charged particle produced by a bar magnet. We have all seen the so-called “field pattern” produced by the magnetic field of a bar magnet. We sprinkle iron filings on a piece of card, put the bar magnet under the card, and the iron filings align along the field lines.

The field lines of a bar magnet.

We might be interested in how much the force due to this magnetic field varies with position. To make things easier, if we only consider the change in the strength of the force on the surface of the card, then we can ignore any change in the vertical dimension, which we will call the z-dimension.

How much a quantity changes with position can be determined by taking the derivative with respect to x, with respect to y and with respect to z, our three spatial directions.

## The vector differential operator

The vector differential operator $\nabla$ (often called del) is the mathematical operator which determines how much vectors change in each of their 3 spatial components. Mathematically it can be written as the partial derivative in each of the 3 spatial dimesions, so usually

$\nabla = \hat{x} \frac{\partial }{\partial x} + \hat{y} \frac{\partial }{\partial y} + \hat{z} \frac{\partial }{\partial z}$

if we are using Cartesian coordinates. The partial derivative $\partial$ is the derivative with respect to one of the variables when the other variables are kept constant. For example, suppose we had a force field which could be described by the following (completely arbitrary) equation

$\vec{F} = (2x) \hat{x} + (3yz) \hat{y} + (4xy) \hat{z}$

What would be the partial derivative with respect to $x$ of this force?

$\frac{\partial}{\partial x}\vec{F} = \frac{\partial}{\partial x}(2x + 3yz + 4xy)= 2 +4y$

The partial derivate with respect to $y$ would be

$\frac{\partial}{\partial y}\vec{F} = \frac{\partial}{\partial y}(2x + 3yz +4xy) = 3z + 4x$

and the partial derivate with respect to $z$ would be

$\frac{\partial}{\partial z}\vec{F} = \frac{\partial}{\partial z}(2x + 3yz + 4xy) = 3y$.

So, finally, we can write $\boxed{ \nabla \vec{F} = (2+4y)\hat{x} + (3z + 4x)\hat{y} + (3y)\hat{z} }$.

The term we use for a quantity like $\nabla \vec{F}$ is the gradient (or grad) of the vector. This word derives from the fact that, in two dimensions, if we have $y=f(x)$, then the gradient of the function (the slope of the function) is just given by $\frac{d}{dx}f(x)$. So the grad is just the three (or more) dimensional equivalent of taking the derivative of a two dimensional function like $y=f(x)$.

As I will explain in a future blog, the vector differential operator $\nabla$ is a very powerful and important operator in mathematics and physics. It forms the basis of vector calculus and can be combined in two other ways with a vector to derive the divergence of the vector and its curl.

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## Galilean Relativity and Electrodynamics

Quite a few months ago now I derived the so-called Galilean transformations, which allow us to relate one frame of reference to another in the case of Galilean Relativity.

$\boxed {\begin{array}{lcl} x^{\prime} & = & x + vt \\ y^{\prime} & = & y \\ z^{\prime} & = & z \\ t^{\prime} & = & t \end{array} }$

It had been shown that for experiments involving mechanics, the Galilean transformations seemed to be valid. To put it another way, mechanical experiments were invariant under a Galiean transformation. However, with the development of electromagnetism in the 19th Century, it was thought that maybe results in electrodynamics would not be invariant under the Galilean transformation.

## The electrostatic force between two charges

If we have two charges which are stationary, they experience a force between them which is given by Coulomb’s law.

$\vec{F}_{C} = \frac{ Q^{2} }{ 4\pi\epsilon_{0}\vec{r}^{2} }$ where $Q$ is the charge of each charge, $r$ is the distance between their centres, and $\epsilon_{0}$ is the permittivity of free space, which determines the strength of the force between two charges which have a charge of 1 Coulomb and are separated by 1 metre.

Coulomb’s law gives us the force between two charges. If the charges are the same sign the force is repulsive, if the charges are opposite in sign the force is attractive.

## Moving charges produce a magnetic field

If charges are moving we have an electric current. An electric current produces a magnetic field. The strength of this field is given by Ampère’s law

$\oint \vec{B} \cdot d\vec{\l} = \mu_{0}I$ where $d\vec{l}$ is the length of the wire, $\vec{B}$ is the magnetic field, $\mu_{0}$ is the permeability of free space and $I$ is the current. So, if the two charges are moving, each will be surrounded by its own magnetic field.

A wire carrying a current produces a magnetic field as given by Ampère’s law.

## The Lorentz force

If the two charges are moving and hence producing magnetic fields around each of them then there will be an additional force between the two charges due to the magnetic field each is producing. This force is called the Lorentz force and is given by the equation

$\vec{F}_{L} = Q\vec{v}\times\vec{B}$. If $r$ is the distance between the two wires, and they are carrying currents $I_{1}$ and $I_{2}$ respectively, and are separated by a distance $r$, we can write $B=\frac{\mu_{0}I}{2\pi r}$ which then gives us that the Lorentz force $F_{L} = \frac{ I_{1} \Delta L \mu_{0} I_{2} }{2 \pi r }$ and so the Lorentz force per unit length due to the magnetic field in the other wire that each wire feels is given by $\boxed{ \frac{ F_{L} }{\Delta L} = \frac{ \mu_{0} I_{1} I_{2} }{ 2 \pi r} }$. Writing the currents in terms of the rate of motion of the charges, we can write this as

$F_{L} = \frac{ \mu_{0} Q_{1} Q_{2} }{ 4\pi r^{2} } v^{2}$

The Lorentz force is the force on a wire due to the magnetic field produced in the other wire from the current flowing in it.

## Putting it all together

Let us suppose the two charges are sitting on a table in a moving train. This would mean that someone on the train moving with the charges would measure a different force between the two charges (just the electrostatic force) compared to someone who was on the ground as the train went past (the electrostatic force plus the Lorentz force).

The force measured on one of the charges by the person on the train, for whom the charges are stationary, which we shall call $F$ will be

$F = \frac{ Q_{1}Q_{2} }{ 4 \pi \epsilon_{0}r^{2} }$.

The force measured on one of the charges by the person on the ground, for whom the charges are moving with a velocity $v$, which we shall call $F^{\prime}$ will be

$F^{\prime} = \frac{ Q_{1}Q_{2} }{4 \pi \epsilon_{0}r^{2} } + \frac{ \mu_{0} Q_{1} Q_{2} }{ 4\pi r^{2} } v^{2}$.

These two forces are clearly different, and so it would seem that the laws of Electrodymanics are not invariant under a Galilean transformation, or to put it another way that one would be able to measure the force between the two charges to see if one were at rest or moving with uniform motion because the forces differ in the two cases.

As I will explain in a future post, Einstein was not happy with this idea. He believed that no experiment, be it mechanical or electrodynamical, should be able to distinguish between a state of rest or of uniform motion. His solution to this problem, On the Electrodynamics of Moving Bodies, was published in 1905, and led to what we now call his Special Theory of Relativity. This theory revolutionised our whole understanding of space and time.

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