• Anything mentioned within the lectures is examinable, unless specifically flagged otherwise.
  • I would suggest working through each lecture, starting with either your own notes or the online notes
  • Then, work through past examinations that are available to you with solutions (you should have already been given instructions for accessing these.)
  • Exams are not intended to be detailed memory tests, but technically you could be tested on anything and there are some key results that you might reasonably be expected to know.
  • I am providing the below summary of what I think are the most important points in case it proves helpful. But please be aware that this is not a definitive guide, as technically the entire course is examinable, except where otherwise stated.
  • Remember in an exam, if you have forgotten some specific detail, don’t panic:
    • The exams are designed such that you can do later bits of the question even if you can’t complete earlier bits.
    • So, if you can’t complete a subsection or it comes out wrong, just leave a gap (in case you can come back to it later) and move onto the next subsection.
    • For example, say you make a mistake or can’t complete part (a), and then part (b) builds on part (a). You can still get full marks for part (b), even if its answer comes out wrong, as long as you have done the right steps within part (b) itself.

Some key things worth revising

  • Make sure you understand the basic notation, as per lecture 1

Electrostatics

  • Coulomb’s law expressed for the electric field of an unmoving (or slowly moving) charge

    \[\vec{E}(\vec{r}) = \frac{1}{4\pi\epsilon_0} \frac{q}{\vert \vec{r} \vert^2} \hat{\vec{r}}\]

  • The principle of superposition (i.e. you can add up the field from different charges)

  • For static charges, $\vec{E} = -\nabla V$ where $V$ is the electrostatic potential

  • For a single charge, $V=q/(4 \pi \epsilon_0 r)$. The potential for multiple charges superposes.

Electric dipoles

  • The dipole moment, $\vec{p} = q\vec{d}$ for two charges $q$ and $-q$ separated by vector distance $\vec{d}$

  • Torque on a dipole moment is $\vec{\tau} = \vec{p} \times \vec{E}$

  • Energy of a dipole in an electric field $\vec{E}$ is $U=-\vec{p} \cdot \vec{E}$

Batteries

  • Use a chemical reaction in two parts to force electrons through an external circuit.

  • The chemistry determines the potential difference between the electrodes.

  • Are limited by the rate at which the chemical reaction can proceed, which in turn is limited by how fast positive ions diffuse within the battery

Capacitors

  • Store energy, a little like a battery, but without using any chemistry.

  • Parallel plate capacitor is the standard example in which two plates become oppositely charged.

  • There is no net charge stored in the capacitor; the two plates balance each other. As a consequence there is also no electric field outside the capacitor itself.

  • The electric field magnitude between the plates is given by $E = \sigma / \epsilon_0$ where $\sigma = Q/A$ is the charge density on a single plate, of surface area $A$.

  • Capacitance $C$ relates the charge $Q$ on one plate with the potential difference $\Delta V$ between the plates, following $\Delta V = Q/C$.

  • Store energy $U=\frac{1}{2} CV^2 = \frac{1}{2} QV$.

  • This energy is stored in the electric field between the plates.

  • Can provide high power temporarily when discharging, but store little energy compared with a battery

Conductors/insulators/dielectrics

  • Due to the atomic/crystalline structure, some of the electrons in a conductor are highly mobile (or ‘free’)

  • They redistribute themselves until the internal electric field is zero (exception: if there is a current flowing and a magnetic field present, they redistribute themselves until the electric/magnetic forces cancel each other, as per Lorentz force below)

  • Consequently, the potential difference between any two points in a perfect conductor is zero (except in the case a magnetic field is present, as above)

  • If a conductor with a uniform surface (e.g. flat plane, or uniform sphere) is charged, the excess charge will distribute itself evenly over the surface.

  • Insulators don’t allow electrons to move through them

  • Dielectrics also don’t allow electrons to move through them, but do allow for a small displacement of the positive and negative charges within the material

  • This displacement suppresses the internal electric field, without cancelling it out entirely

  • Dielectric constant / relative permittivity $\kappa$ is the ratio of suppression of the electric field

  • Electric field calculations in presence of a dielectric: replace $\epsilon_0$ with the permittivity $\epsilon=\kappa\epsilon_0$.

Currents

  • If a current $I$ is flowing through a conductor, this corresponds to a mean net drift of the free electrons. The speed is typically very small compared to the thermal motions.

  • $I$ can be related to the net drift speed by working out the total charge crossing through the cross-sectional area of the wire per unit time.

  • Energy flows far faster than the electron drift speed, in the same way that energy can flow in a wave through water even though the water itself doesn’t move at all.

Biot-Savart Law

  • There are no ‘magnetic charges’; instead magnetic fields arise from moving electric charges.

  • Biot-Savart law gives magnetic field arising from a steady current $I$ flowing through a wire $L$ as

\[\vec{B} = \frac{\mu_0 I}{4 \pi} \int_L \frac{ {\rm d} \vec{L} \times \hat{\vec{r}}}{\vert \vec{r} \vert^2}\]

Lorentz force / emf

  • The force experienced by a charge $q$ in an electric and/or magnetic field is
\[\vec{F} = q(\vec{E} + \vec{v}\times\vec{B})\]
  • Electromotive force, emf, is the work done per unit charge along a path (often a closed path):
\[\mathcal{E} = \int (\vec{E} + \vec{v} \times \vec{B}) \cdot {\rm d}\vec{l}\]

  • Remember emf is not a force! It is energy per unit charge, like the electrostatic potential $V$

  • In fact, in an electrostatic problem, emf and potential difference are the same thing. But emf is more general.

Magnetic (dipole) moment

  • In many ways, a magnetic moment has similar physics to an electric dipole moment. However their underlying physical causes are different.

  • If a current $I$ circulates around a small loop with area $A$ and the vector normal to the loop is $\hat{\vec{n}}$, the magnetic moment is $\vec{\mu} = I \vec{A}$ where $\vec{A} = A \hat{\vec{n}}$.

  • Torque on a magnetic moment in a magnetic field is $\vec{\tau} = \vec{\mu} \times \vec{B}$

  • Associated potential energy is $U = -\vec{\mu} \cdot \vec{B}$

  • There is no net force on a magnetic moment in a uniform magnetic field. “Magnetic attraction” occurs due to magnetic fields not being uniform.

Magnetic materials

  • Due to quantum mechanics, matter is full of current loops from orbiting electrons (and electron spin).

  • However many of them cancel out either at the atomic scale (electron pairing) or because the orbits are randomly aligned in a material

  • Paramagnetic materials align their magnetic moments in an external magnetic field and therefore are roughly analagous to a dielectric

  • Ferromagnetic materials have interactions between their magnetic moments that makes them try to spontaneously align so can be made into permanent magnets

  • $K_m$, the relative permeability, is the ratio between the magnetic field in a material and that expected in a vacuum

  • Calculations when working with a paramagnetic/ferromagnetic materials replace $\mu_0 \to \mu \equiv K_m \mu_0$.

Coils/Solenoids

  • Stacking up lots of current-carrying wire in loops (sometimes known as ‘turns’) creates much stronger magnetic fields than possible from a single loop

  • The interior magnetic field strength depends on the number of turns (for a ‘short’ coil where the loops are all basically on top of each other) or the number of turns per unit length (for a ‘long’ solenoid)

  • Due to Faraday’s law (covered under Maxwell’s equations below), coils/solenoids are inductors
    • mutual inductance: a changing current in one coil induces an emf in another coil
    • self inductance: a single coil/solenoid flowing through it
    • Self-inductance is given symbol $L$, and the self-emf is then $\mathcal{E} = -L {\rm d} I / {\rm d} t$.
  • Inductors can store energy, rather like capacitors can. $U = LI^2/2$.
  • The energy is stored in the magnetic field.

Maxwell’s equations

  • The definition of flux for electric fields, $\Phi_E = \oiint \vec{E} \cdot \mathrm{d} \vec{A}$ and similarly for magnetic fields $B$.

  • Maxwell’s equations.

    Name Description Equation
    Gauss’ Law Electric field lines start and end on charges $\oiint \vec{E} \cdot d\vec{A} = \frac{q_{\rm enc}}{\epsilon_0}$
    Gauss’ Law for magnetism There are no magnetic charges, so magnetic field lines are closed loops $\oiint \vec{B} \cdot d\vec{A} = 0$
    Faraday’s Law A changing magnetic field generates an electric field around loops $\oint \vec{E} \cdot d\vec{l} = - \frac{d\Phi_B}{dt}$
    Ampère-Maxwell Law A current or a changing electric field generates a magnetic field $\oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I_{\rm enc} + \epsilon_0 \frac{d\Phi_E}{dt} \right)$

  • Ampère’s law, $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\rm enc}$ (this is the Ampère-Maxwell Law but without the displacement current term).

  • Maxwell’s equations + the Lorentz force law imply all of classical electromagnetism, though other equations are obviously still useful.

  • Maxwell’s equations in a vacuum can be combined to predict the existence of electromagnetic radiation (i.e. light).