Time dependence

So far, we have dealt with electric and magnetic fields that are static. Even though magnetic fields are generated by moving charges, we assumed the charges move at a constant velocity and there is a ‘steady state’ i.e. if a charge moves out of a region, another charge moves in to replace it.

There are a few ways to see that this is not the whole story.

Special relativity tells us that no information can propagate faster than the speed of light.
- If a charge suddenly starts moving, the electric and magnetic fields at a distance from the charge cannot instantly change.
- Yet, with e.g. Coulomb’s law, this ‘instant change’ is exactly what you’d predict.
- So, Coulomb’s law cannot be correct. In fact Coulomb’s law is simply not part of Maxwell’s equations. In the modern view it’s a derived law, valid only for the static case.
There are logical inconsistencies in what we have so far.
- For example, in Ampère’s law, suppose we insert a capacitor with high capacitance into a long wire.
- Depending on where exactly we draw the loop, we would reach different conclusions about the magnetic field. (We’ll draw this in the lecture.)
- This is a paradox, and it is resolved by recognising that the magnetic field is not static.
- In other words, Ampère’s law cannot be correct. A corrected version of Ampère’s law appears in Maxwell’s equations; we will see in the next lecture.
Experimentally, a changing magnetic field can generate an electric field.
- Nothing like this yet appears in any of our equations.
- Conclusion: there is a missing equation, which turns out to be Faraday’s law. We will see that in the current lecture.
We also know experimentally that the electric and magnetic fields can generate each other in a self-sustaining way, leading to electromagnetic waves.

Historically, (3) and then (2) led (after decades of confusion) to Maxwell’s equations. Then (4) and (1) were deduced from Maxwell’s equations, and the experimental demonstration that electromagnetic waves really exist (by Heinrich Hertz) slowly convinced people the equations were correct.

Note that Gauss’s law for electric and magnetic fields is not contradicted by any of the above. Despite originally being ‘derived’ Gauss’s law is in fact fundamental, and will form part of the four Maxwell equations.

Coulomb’s law vs Maxwell’s equations

Here is a demonstration of how Maxwell’s four equations (though we haven’t yet written them down) lead to very different behaviour compared to the Coulomb law. Try moving one of the two charges and see what happens. Compare the behaviour while you are moving the charges, and once the charges have been stationary for a while.

Coulomb’s Law:

Maxwell’s equations:

N.B. for illustrative purposes, the speed of light is hugely reduced compared to its real value. Note also that the walls of the ‘box’ in which the computer simulation is taking place are slightly reflective.

Electromotive force (emf)

We need to generalise the electrostatic potential to be ready to write down time-dependent equations.
The emf $\mathcal{E}$ is defined to be the work done per unit charge moving a charged particle along a given path, i.e.

\[\mathcal{E} \equiv \frac{1}{q} \int \vec{F} \cdot \dd\vec{l}.\]

If the force is just generated by an electrostatic potential this boils down to the potential difference between the two endpoints $A$ and $B$, because
\[\mathcal{E} = \frac{1}{q} \int_{\rm A}^{\rm B} q \vec{E} \cdot \dd\vec{l} = \int_{\rm A}^{\rm B} \vec{E} \cdot \dd\vec{l} = -\int_{\rm A}^{\rm B} \nabla V \cdot \dd\vec{l} = V|^A_B = V_A-V_B\,.\]
(See similar maths in Lecture 4.)
More generally, the emf can be generated by an electric field that is not the gradient of a potential, or by a magnetic field. In this case the emf is given by the Lorentz force law:

\[\mathcal{E} = \int (\vec{E} + \vec{v} \times \vec{B}) \cdot \dd\vec{l}.\]

We will be interested in the emf around a closed loop (i.e. where we end up where we started, and the integral is written as $\oint$ instead of $\int$). Any contribution from electrostatic forces is zero in this case since $V_{\rm A} = V_{\rm B}$ for a closed loop (i.e. A and B are the same point in space).
Beware: electromotive force is not a force! It’s an energy per unit charge, with SI units of ${\rm J\,C^{-1}}$ or equivalently ${\rm V}$. Sorry, it’s just yet another example of bad names in physics.

Faraday’s law

As summarised in YF 29.1, a series of experiments primarily by Michael Faraday showed that a changing magnetic field can generate an electric field.

This experimental fact is summarised in a law named after Faraday.
The most famous of his experiments is the one where Faraday moved a magnet in and out of a coil of wire, and observed a current in the coil. This is the basis of how most electricity is generated today.
Expressed in the language of emf, Faraday’s law is stated as:
\[\mathcal{E} = - \frac{d\Phi_B}{dt}.\]
Here $\mathcal{E}$ is evaluated around a closed loop, ending where it started, otherwise there would also be a term from the potential.
From a modern perspective, Faraday’s law must be true because:
- A loop of wire moving through a magnetic field will have an emf in it arising from the Lorentz force on the moving electrons
- By relativity, we can also analyse this in a frame where the loop is stationary but the magnetic field instead moves past the loop. The same emf must be produced.
- Since $\vec{v}=0$ in this alternative frame, there must be an electric field accelerating the electrons!
- In the lecture, we will use this to justify Faraday’s law starting from the Lorentz force law. A similar (though in my opinion more complicated and less convincing) justification is made in Y&F 29.4.
While Faraday’s law is true for a moving loop, it is actually most commonly used with a static loop.
In this case, $\vec{v}=0$ and only the $\vec{E}$ term arises in the emf. Faraday’s law in this case says:
\[\oint \vec{E} \cdot \dd\vec{l} = - \frac{\dd\Phi_B}{\dd t} \quad \textrm{(static loops only)}.\]
This form, for static loops only, is the one that appears in Maxwell’s equations. It states that an electric field is generated by a changing magnetic field.
The field generated is known as the induced electric field (Y&F 29.5).

Lenz’s law

The term on the right hand side of Faraday’s law is minus the rate of change of the magnetic flux through the loop.

The minus sign is necessary to stop perpetual motion machines:

if the magnetic field is changing, it induces an $\vec{E}$ field
this makes a current flow around loops
the current flowing around the loop makes its own magnetic field
the minus sign ensures that the new magnetic field opposes the old one
if it didn’t, perpetual motion would be possible: the induced electric field would drive a current, that would ramp up the magnetic field, that would induce a stronger electric field, that would drive a stronger current, etc.

N.B. Lenz’s law is not really a separate law. It’s already contained in Faraday’s law. It’s just a helpful way of remembering the direction of the induced electric field.

Circulating electric fields

The electric field generated by a changing magnetic field is not conservative, unlike the electric field generated by static charges:

Lecture 4 pointed out the electric field from static charges is ‘conservative’; it cannot do net work around a loop because the potential returns to its original value.
Lecture 9 showed that magnetic fields cannot do net work around a loop because the force is always perpendicular to the velocity.
But non-conservative electric fields generated by induction can do work on electrons flowing around a loop.
If a conservative electric field is like a landscape with hills and valleys, a non-conservative electric field as generated by induction is like an impossible Escher lithograph where a charge can circulate around and extract net energy.
In reality, this energy is being extracted from the magnetic field (this is another way to state Lenz’s law), so there is no actual paradox.

Lecture 14

Reading

Content