Can Calculus of Variations Prove the Path of Least Resistance for Current Flow?

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The discussion centers on using calculus of variations to demonstrate that current follows the path of least resistance. The original poster seeks to formulate a functional to represent this concept mathematically. It is noted that while current does not exclusively take the path of least resistance, it tends to prefer this route, particularly when maximizing current at junctions. Key equations provided include current density and resistance formulas, emphasizing the relationship between current, electric field, and conductivity. The conversation highlights the complexity of current flow, suggesting that while paths may vary, the principle of least resistance remains significant in certain contexts.
sodaboy7
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Current follows the path of least resistance or shortest path. I just want to prove this or rather reproduce it using calculus of variations. I just want to show it in a fancy way. I want help to form the FUNCTIONAL for it.
Useful equations:
I=dq/dt=nqvA
R=rho*l/A
Where v is drift velocity

Any suggestion (may be using different equations and parameters)?
 
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The current does not actually take the path of least resistance. It takes all available paths. In general:

J = \sigma E

Where J is current density, E is electric field, and σ is the electrical conductivity.
 
It prefers path of least resistance. Or it maximum current follows the path of least resistance upon division at a point.
 

Thread 'How to picture the vector potential?'

Susskind (in The Theoretical Minimum, volume 1, pages 203-205) writes the Lagrangian for the magnetic field as ##L=\frac m 2(\dot x^2+\dot y^2 + \dot z^2)+ \frac e c (\dot x A_x +\dot y A_y +\dot z A_z)## and then calculates ##\dot p_x =ma_x + \frac e c \frac d {dt} A_x=ma_x + \frac e c(\frac {\partial A_x} {\partial x}\dot x + \frac {\partial A_x} {\partial y}\dot y + \frac {\partial A_x} {\partial z}\dot z)##. I have problems with the last step. I might have written ##\frac {dA_x} {dt}...
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