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

In summary, the conversation discusses the concept of current and how it follows the path of least resistance or shortest path. The speaker wants to prove this using calculus of variations and is looking for suggestions on how to form the functional for it. They mention useful equations such as I=dq/dt=nqvA and R=rho*l/A, where v is the drift velocity. However, it is noted that the current does not strictly follow the path of least resistance, but rather takes all available paths. The general equation for current density is J = \sigma E, where J is current density, E is electric field, and σ is the electrical conductivity. The speaker is open to suggestions for using different equations and parameters to demonstrate this concept.
  • #1
sodaboy7
81
0
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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  • #2
The current does not actually take the path of least resistance. It takes all available paths. In general:

[tex]J = \sigma E[/tex]

Where J is current density, E is electric field, and σ is the electrical conductivity.
 
  • #3
It prefers path of least resistance. Or it maximum current follows the path of least resistance upon division at a point.
 

1. What is the "path of current functional"?

The "path of current functional" refers to the sequence of steps or processes that an electric current follows through a circuit or system. It is the flow of electrical energy from the source to the load, and is important in understanding the behavior and operation of electrical devices.

2. How does the path of current functional affect the performance of a circuit?

The path of current functional plays a crucial role in determining the performance of a circuit. The efficiency, stability, and reliability of a circuit are all affected by the path of current functional. A well-designed circuit will ensure that the path of current is optimized for maximum performance.

3. Can the path of current functional be altered or controlled?

Yes, the path of current functional can be altered or controlled through the use of different circuit components such as resistors, capacitors, and transistors. By manipulating the flow of current, the path can be redirected or modified to achieve a desired outcome.

4. What factors can affect the path of current functional?

There are several factors that can affect the path of current functional, including the voltage and current levels, the type and arrangement of circuit components, and the presence of any external influences such as temperature or electromagnetic interference.

5. How can knowledge of the path of current functional be applied in real-world situations?

Understanding the path of current functional is crucial in the design, troubleshooting, and maintenance of electrical systems and devices. It is also important in fields such as electronics, telecommunications, and power generation, where the flow of current is a key factor in the operation and performance of equipment.

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