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### Number and algebra

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### Working mathematically

### For younger learners

### Advanced mathematics

# Hold Still Please

### Why do this problem?

This problem will develop students' intuition and skill with
vectors and force laws in a context which encourages exploration
and does not require any calculation to get started. By trying to
construct stable configurations students will be led towards the
notion of static points only occurring at places of zero
electrostatic potential.
### Possible approach

### Key questions

### Possible extension

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Age 16 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

This problem is very open and well suited to discussion to get
things started. The main requirement is the encouragement that
students start to try out specific configurations and then use the
force law to determine whether or not the configuration will remain
static.

More general statements are, of course, more subtle and
difficult to analyse than particular examples. Clear, systematic
thinking will be needed, as will a clear
representation system (diagrammatically or
algebraically).

After doing this problem, you might refer students to
'Earnshaw's theorem' http://en.wikipedia.org/wiki/Earnshaw%27s_theorem

Are there any obvious simple cases to look at?

Can you make any symmetrical configurations to look at?

What is special, if anything, about configurations in which
$1$ or more particles don't move?

How might you attempt to fix down a configuration which will
move by adding more fixed particles?

Take an entirely stable configuration. Consider the question:
what happens if you nudge one of the particles slightly?

Concentrate on intuitive attempts at the problem.