# How does electricity pass through a surface (or more generally, conductive material with complex geometries)?

I was reading this article on how the breaking mechanism of a Sawstop works. For those who might not know, Sawstop is a table saw fitted with at least a conductivity sensor that detects changes in current passing through the blade as it comes into contact with something other than wood (the table saw operator's fingers, for instance).

1. This got me thinking about how electricity flows through a circular plate. If we connect wires to the plate (see attached diagram) and assuming the wires form a conductive closed loop, would the current change if we changed where the 2 wires were connected to that plate? I'm guessing no since Kirchoff's junction rule states that same current drawn through one wire must leave through the other wire. We can just treat the plate as a resistor.

2. If a person touches the plate and assuming he doesn't form another closed-loop path, would he change the current flowing through the wire? I'm guessing no since he'd just be a floating conductor/resistor, in principle.

• The current wouldn't differ between the two wires, but it would change based on the resistance between them. Jul 9, 2016 at 8:41
• I thought this system was more like a capacitative touch sensor, which is based on varying electric fields and does not rely on actually touching the sensor (which is in this case the blade!) Jul 9, 2016 at 8:55
• Physically, the centrifugal force acts on electrons in the disk, too, and this generates a small electric field, or a voltage between the wires. This depends on distance from the axis and speed. However, I think that's not what you're asking for... Aug 8, 2016 at 14:08

Part 1.

The current will distribute through the disk proportionally to the resistance of each of the infinite paths it can take. Essentially the disk is a whole bunch of resistors in parallel. In order to work out the effect, you would have to sum the currents through each possible path by integrating current over the 3D structure.

In terms of whether anything changes when moving the connections, yes it will. If you consider both wires placed on the circle, all of the current paths have to travel at least the distance between those connections through the structure. If both are placed at the centre of the circle, the shortest current path will be only the thickness of the disk. Conversely if you place one at each end of a diameter line, all current paths will be through at least a diameter length of the material.

Clearly if you change the length of the current path through any material, the resistance of that path will change. The longer the path, the higher the resistance. If the total resistance in a circuit is increased without changing the supplied voltage, this will the lower the current.

Part 2.

If a person touches the disk and no closed loop is formed, then no, it will not change anything. However this is not physically possible.

In practice the finger will have some conductivity and be touching more than just an infinitely small point. As a result of the contact point being distributed across an area of the disk, it is equivalent to adding an additional resistive path into the circuit which will cause a small current change by reducing the resistance of the disk.

Additionally, if there is any AC component to the signal, the finger would act as an additional capacitance which will have an effect on the impedance of the disk.

• Ah. So how the conductor is shaped and where the electrical connections (in/out) are physically located determines the where the majority of the current flows. The facts to reconcile are: (1) Current flows through the large/infinite number of paths in parallel (2) The sum of the current going into the conductor across all paths must equal the sum of the current coming out of all paths (3) The larger the conductor and if voltage is kept the same, the smaller the sum of current through all paths. Jul 17, 2016 at 4:17
• @MinhTran The most direct path will be the largest current path (assuming uniform resistivity), though only by a small amount. Increasing the size of the conductor will effect the current, but how depends on in which direction you increase it. In your example, making the disk thicker will decrease current as it increases the length of the current paths (basically resistor in series), whereas increasing the diameter will increase the current as it effectively adds more paths in parallel. Jul 17, 2016 at 13:36