James Clerk Maxwell
James Clerk Maxwell

Start this free course now. Just create an account and sign in. Enrol and complete the course for a free statement of participation or digital badge if available.

Free course

James Clerk Maxwell

3 The Ampère–Maxwell law

3.1 Limitations of Ampère's law

In order to analyse the limitations of Ampère's law, and suggest ways of overcoming them, we need to use some properties of divergence. For ease of reference, these properties are given below:

Some properties of divergence

  1. The divergence of any curl is equal to zero:

  2. A constant k can be taken outside a divergence:

  3. A time derivative can be taken outside a divergence:

You can take these properties on trust if you wish, but it is easy enough to prove them by expanding both sides in Cartesian coordinates.

SAQ 2

Prove Equation 7.5.

Answer

Expanding the left-hand side of Equation 7.5 gives

which vanishes because mixed partial derivatives do not depend on the order of partial differentiation.

Now let's examine the differential version of Ampère's law, which states that

The limitations of this law are revealed by taking the divergence of both sides. This gives

The divergence of any curl is equal to zero so, using Equation 7.6 and the equation of continuity, we have

We therefore see that Ampère's law requires the charge density to remain constant. Put another way:

Ampère's law fails when the charge density changes in time.

SMT359_2

Take your learning further371

Making the decision to study can be a big step, which is why you'll want a trusted University. The Open University has 50 years’ experience delivering flexible learning and 170,000 students are studying with us right now. Take a look at all Open University courses372.

If you are new to university level study, we offer two introductory routes to our qualifications. Find out Where to take your learning next?373 You could either choose to start with an Access courses374or an open box module, which allows you to count your previous learning towards an Open University qualification.

Not ready for University study then browse over 1000 free courses on OpenLearn375 and sign up to our newsletter376 to hear about new free courses as they are released.

Every year, thousands of students decide to study with The Open University. With over 120 qualifications, we’ve got the right course for you.

Request an Open University prospectus371