# Altitude Analogy

Electric shock?

When I first took the beginnings of these notes in class, I found it extremely difficult to gain an intuitive grasp of what all these terms and concepts meant - they all sounded vaguely similar and I had no clear picture of what they represented. During a discussion online (at 2:30 in the morning 😶) with a university physics student, they introduced me to a wonderful analogy which represents electric potential as elevation on a map - and suddenly, the topic became much less abstract and started making sense in my head!

To share some of this intuition via this analogy, I'll try my best to re-explain below the altitude analogy which was explained to me. This does not have any direct relevance to the SQA Advanced Higher course, however in my opinion I consider this well worth a read.

This analogy applies to gravitational fields as well, however I'm lazy applying it yourself would be a good exercise to show your understanding. Just substitute the terms and concepts relevant to electric fields with their gravitational field counterparts.

Massive thank you to chroma for their help with this topic!

### Electric Potential

Imagine that we sampled every point within a space, measured the electric potential at that point, and extruded into another dimension based on the value of that electrical potential (like a heightmap, for those familiar with computer graphics). In simple terms, we convert electrical potential into elevation - the higher the altitude, the higher the electrical potential, and vice versa.

When we add charges to this map, they influence the electric potential. A positive charge raises a peak of high electric potential in the map, and a negative charge sinks down and creates a well of low electric potential.

If we were considering electric potential in a 2D plane, the resulting map would look something like this in 3D:

To consider electrical potential along a 1D line, we can simply take a cross-section of this map, to get what would simply look like a graph of electrical potential (y-axis) against distance (x-axis). However, to consider electrical potential within a 3D space, that would require extrusion into a fourth spacial dimension - which would be a mathematically accurate model, however it would be very difficult to visualise. For the purposes of this explanation, let's stick to 3D maps representing 2D planes.

### Potential Difference

Now that we understand electric potential, potential difference (voltage) is simply a difference in altitude. It's the distance you have to climb up or down on the map - the greater the climb, the greater the potential difference. Climbing up would correspond to a positive potential difference, whereas descending down would be negative.

Just as we can measure both altitude and difference in altitude using metres, we can measure both electric potential and potential difference using volts - despite the term of voltage only referring specifically to a potential difference. This is a source of confusion for many!

### Field Strength

We can think of electric field strength as being the gradient at a point - therefore electric field strength is a sort of multidimensional derivative of electric potential. To put it simply, the electric field strength at a point is a vector starting from that point and pointing down the slope. The steeper the slope, the greater the magnitude of the vector - and conversely, the field strength at a flat point would have a magnitude of zero.

If we drew many of these field strength vectors and connected them up in a chain from tip to tail, the lines would represent the path a positive test charge would take, which wants to "roll down" to get to as low a potential it can. If this sounds familiar, good - because that is what field lines are! A birds-eye view of the map with the vectors drawn would look exactly like the 2-dimensional electric field diagrams you'll be familiar with.

## Electrostatic Force

The electrostatic force is simply the "gravity" in the analogy - the force which pulls positive charges downwards towards a lower electric potential, and negative charges upwards towards a greater electric potential. The strength of this force can depend on factors such as permittivity.