# Altitude Analogy

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.