Electricity & Magnetism

I. Electric Charge

A way to define electric charge is to say that it is a quantity that a body must have in order to experience a force due to an electric field, analogous to mass for the gravitational field. Charge is a scalar quantity, with units of Coulombs (C). A coulomb is a large unit of charge, for examples electrons have a charge in the order of magnitude $10^{-19}C$.

Coulomb’s Law

An important law in electrostatics is Coulomb’s law. This gives the force between two charges of charge $q_{1}$ and $q_{2}$ respectively, and is often stated as the following, wherer is the distance between the two charges.

    $$F= \frac{q_{1}q_{2}}{4 \pi \varepsilon_{o}r^{2}}$$

Here \varepsilon_{0} is the permittivity of free space, a fundamental constant with magnitude of approximately 8.85 \times 10^{-12}Fm^{-1}.

The Principle of Superposition

Forces in electrostatics add linearly. This means that they follow linear superposition when being combined. This means that it is straightforward to extend from two particles to many. The force on any one particle in a system of charged particles is simply the vector sum of the force on that one particle from each of the other particles in the system.

Point Charges

Coulomb’s law applies to point charges, and only point charges. This means a the ideal situation where there exists a charge with no spatial extent. This ends up being a good model because on a macroscopic scale the effect of the charge acts as if it were localised to a single point.

Conservation of Charge

The principle of conservation of charge, similar to other conservation laws, states that for a given isolated system the total charge must be constant. There are no known exceptions to this principle. This can be observed in particle physics via particle interactions. Take for example the decay of a proton via beta plus decay.

    $$p \rightarrow n+ \nu_{e} + \beta^{+}$$

The total charge before the decay is +1, and that is also true after the decay.

II. Movement of Charge & Conduction

The movement of charge relative to some reference is described as an electric current. The electric current, Ithrough a surface is the rate of flow of charge through this surface. This means it can be expressed as the following.

    $$I= \frac{dQ}{dt}$$

This is a scalar quantity (can verify this by considering that both charge and time are scalar quantities), and has units of Ampères (A). By convention the positive sense in which current flows is the direction in which a positive charge moves in a system.

Current Density

Current density is however a vector quantity. Current density is the current per unit area, where the area is defined by a vector plane. The current can vary over this area so the rigorous definition of the current density (\mathbf{J}) is the following.

    $$\mathbf{J} = \lim_{A \rightarrow 0} \frac{I}{A}$$

This means that the current density is defined at a point. The equation above can be written in it’s integral form.

    $$ I= \iint_{s} (\mathbf{J} \cdot \mathbf{\hat{n}}) \ dA $$

This tells us that the surface integral of the current density over a given area gives the current through that area.

The Continuity Equation

The continuity equation is an mathematical expression of the principle of conservation of charge, thus is often useful. Here the derivation for the continuity equation is shown.

Let us consider a volume V bounded by a surface S and the current though the volume is I. The current into the volume can be expressed as the following, where the minus sign represents the inwards nature of the current we are considering.

    $$ I= -\iint_{s} (\mathbf{J} \cdot \mathbf{\hat{n}}) \ dA =-\iint_{s} (\mathbf{J} \cdot d\mathbf{S}) $$

Using the divergence theorem this can be written in the following way.

    $$ I= -\iiint_{V} (\nabla \cdot \mathbf{J}) \ dV = \frac{dq}{dt}$$

If \rho is the charge per unit volume, the total charge for our volume must be given by the following volume integral.

    $$ q= \iiint_{V} \rho \cdot dV$$

    $$\therefore \frac{dq}{dt}= \iiint_{V} \frac{d\rho}{dt} \cdot dV$$

We can now equate the two equations found for the rate of flow of charge.

    $$ \iiint_{V} \frac{d\rho}{dt} \cdot dV =-\iiint_{V} (\nabla \cdot \mathbf{J}) \ dV$$

    $$\iiint_{V} \bigg (\frac{d\rho}{dt} +\nabla \cdot \mathbf{J} \bigg ) \ dV = 0$$

This has to be true for all volumes, thus the following can be stated. This is the continuity equation.

    $$\frac{d\rho}{dt} + (\nabla \cdot\mathbf{J})=0$$

III. Electric Fields

Electric Field Strength

The electric field strength, \mathbf{E}, is defined as the force per unit positive test charge at a point in the field. This means that the field strength of the electric field is a vector quantity with the same direction as the force at the point in the field.

    $$\mathbf{E}= \frac{\mathbf{F}}{q}$$

This also means that if the field strength at a point in the field is known, the force on an arbitrary charge can be calculated.

Electric Field around a point charge

Using Coulomb’s law, and the definition of the Electric field strength it is possible to express the electric field around a point charge.

    $$\mathbf{E}= \frac{Q}{4\pi\varepsilon_{0}r^{2}}\mathbf{\hat{r}}$$

Electric Dipoles

An electric dipole is an arrangement of any two point charges at any distance from each other. However, the case in which the charges are opposite in sign but the same in magnitude at a fixed difference from each other is often of significant interest. When referring to dipoles it is often the case that this specific case is being referenced. Dipoles in nature, often popping up all over atomic and molecular physics, can either be permanent or induced by the presence of a non uniform field.

**Diagram

The dipole moment,  \mathbf{p}, is defined to be the magnitude of the charges in a dipole arrangement, multiplied by the displacement vector between the negative and positive charge.

    $$\mathbf{p}=q\mathbf{d}$$

Electric Fields inside Conductors

Let us consider an ideal conductor in an electric field. At any point in time the electric field inside an ideal conductor is the sum of the external field and the field inside the conductor due to the charge distribution.

**Diagram

Initially (t=0) the random distribution of charges means that the internal field is 0. Thus the net field in the region is equal to the external field. However the charges inside start moving as soon as the external field is applied, moving to opposite ends of the conductor. This creates an internal field which opposes the external field. At electrostatic equilibrium the internal field exactly opposes the internal field, thus the net field in the conductor is 0. For the charges to stop moving the distorted external field at the surface of the conductor must be perpendicular to the surface at each point. Any parallel component would cause a charge to move, which doesn’t happen at electrostatic equilibrium.

Faraday Cages

A Faraday cage describes a system where a conductor totally surrounds a non-conductor. The field inside the cage, as mentioned above is 0, so interior of the cage is shielded from external fields.

IV. Gauss’s Law

Electric Flux

The electric Flux, \Phi, through an area is defined to be the dot product between the Electric field and the area. This means that electric flux is a scalar quantity.

    $$\Phi = \mathbf{E} \cdot\mathbf{A}$$

For non uniform fields and non-planar areas this has to be generalised. The infinitesimal flux, \delta\Phi, using the definition above will be equal to \mathbf{E} \cdot \delta\mathbf{A}. This is because as the area tends to zero a plane will be a good approximation to any surface. In the infinitesimal limit \delta\mathbf{A} \rightarrow 0 this becomes a surface integral.

    $$\Phi = \iint_{s} \mathbf{E} \cdot d\mathbf{A}$$

This allows for electric flux to be calculated for any field through any surface. Note that often numerical solutions are required to compute the integral.

Flux due to a point charge through a sphere

Consider a point charge at the centre of a sphere. Let’s find the flux through the sphere. Let’s take the charge of the point charge to be q and the radius of the sphere to be R. We know the field strength of the field at any point a distance r from the charge from Coulomb’s law and the definition of field strength. Here \mathbf{\hat{r}} is a unit vector in the direction of the field.

    $$\Phi=\varoiint_{sphere}\bigg (\frac{q}{4\pi\varepsilon_{0}R^{2}}\mathbf{\hat{r}}\bigg)\cdot (d\mathbf{A})$$

We can factor out the constants then evaluate the surface integral, as it is simply the surface integral over the whole sphere.

    $$ \Phi=\frac{q}{4\pi\varepsilon_{0}R^{2}} \varoiint \mathbf{\hat{r}} \cdot d\mathbf{A} =\bigg (\frac{q}{4\pi\varepsilon_{0}R^{2}} \bigg )(4\pi R^{2}) = \frac{q}{\varepsilon_{0}}$$

This ends up being the flux through any closed surface, according to Gauss’s law!