physic pls help me

[obtuse spindle]

4. If a rectangular carbon solid has sides with lengths a, 2a, and 3a, how would you connect wires coming from a battery to obtain a) the least resistance and b) the greatest resistance?

[austere plover]

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[forest merlin]

With no further information this is quite hard to solve. What is the circuit diagram? Can I just connect one edge a and call it a day? Do I need to draw wire across the faces and find the minimum and maximum distances from the body diagonals of the cube?

[obtuse spindle]

unfortunately the professor only gave us the problem without giving us any kind of diagram.

it is in Spanish but in the question I translated it to English.

btw, thank you very much for your help

[forest merlin]

Okay. Thank you also for posting the full problem. If this was my problem I would just consider the "draw wire across the faces".

If you have an infinite amount of wire, you would be able to wrap the carbon solid into an infinite length.

However if we consider only straight lines, the maxima would quite trivially be a+2a+3a = 6a, you can verify this by a drawing.

The problem is then to find the minima, without using the body diagonal (going through the solid). We do this by unwrapping the solid. Now we can draw two different "straight lines" using different faces (See picture, not drawn to scale). Can you find the length of the blue line connecting the orange and the pink dots using pythagoras? Are the two blue lines the same length?

[obtuse spindle]

wao now you gave me a key to better analyze the problem, according to your structured analysis I can now get the rest.
my problem was that I didn't know how to start.

thank you very much

and if I can get that by means of Pythagoras

[candid condor]

The key to solving this problem is first understanding how resistance depends on the dimensions of the carbon solid.
The resistance (R) of a materials is given by:
R = pL/A

p is the resistivity of the material, L is the length of the material through which the current flows, and A is the cross sectional area perpendicular to the current flow

To achieve the least resistance, you need to minimize L and maximize A

To achieve the greatest resistance, you need to maximize L and minimize A

[candid condor]

That works, as you can identify the diagonal paths within the rectangular solid, which are the longer paths through the solid.

You're given the dimensions a, 2a, and 3a, then you can calculate the diagonals (lengths) across different faces...

1. Face Diagonal (across (a) and (2a)):
   [
   d_{\text{face1}} = \sqrt{a^2 + (2a)^2} = \sqrt{a^2 + 4a^2} = \sqrt{5a^2} = a\sqrt{5}
   ]

[hollow bay]

Estratos Infinitio

[forest merlin]

So the whole block is used as a resistor? But carbon is a good conductor? I thought this was about maxing and minimizing the internal resistance of the wires 😅

In case of a body diagonal being the greatest length, the area varies and then this becomes an integral to solve for R. The minima would be to fit the broadest face and make the current go through the shortest length, then L = a and A = 6a^2. However, considering a "carbon solid" as a cuboid, the empirical results of R = pL/A do not hold, those hold for small cylinders.

[candid condor]

2. Face Diagonal (across (2a) and (3a)):
   [
   d_{\text{face2}} = \sqrt{(2a)^2 + (3a)^2} = \sqrt{4a^2 + 9a^2} = \sqrt{13a^2} = a\sqrt{13}
   ]

[hollow bay]

Estratos Infinitio

[candid condor]

Yes, in this problem, we are considering the entire carbon block as a resistor. The dimensions of the block and the material -- carbon -- are given and were tasked with finding the configurations that give the least and greatest resistances when current flows through the block.

Also, sure carbon is a decent conductor, but it also has resistive properties.

The problem focuses on how the dimensions and configuration of the carbon block affect it's resistance.

This isn't about the resistance of the wires connecting the carbon block but about resistance of the carbon block itself.

And so using the Pythagorean thereom, you can calculate the length of the body diagonal, but of course for practical purposes and simplicity, you'd often consider straight paths through block dimensions.

I think the integral approach can be complicated (or complex) and generally isn't needed for this type of problem.

Feel free to correct me if you believe I'm interpreting this incorrectly.

[forest merlin]

Yeah exactly straight paths. The integral approach might be too high level but it is the only way to prove the existence of unique maxima. This is definitely open up for interpretation like you said.

[candid condor]

3. Space Diagonal (through the solid):
   [
   d_{\text{space}} = \sqrt{a^2 + (2a)^2 + (3a)^2} = \sqrt{a^2 + 4a^2 + 9a^2} = \sqrt{14a^2} = a\sqrt{14}
   ]

[hollow bay]

Estratos Infinitio