Potential Difference Path Independence

The potential difference [math]\displaystyle{ \Delta V = V_B - V_A }[/math] between two locations A and B does not depend on the path taken between the locations.

Claimed alanghauser3

The Main Idea

The potential difference between two locations A and B does not depend on the path taken between the locations. A round trip potential difference is always zero.

A Mathematical Model

In a uniform electric field the potential difference is equal to

[math]\displaystyle{ \Delta V }[/math] = -[math]\displaystyle{ \vec{E} }[/math]●[math]\displaystyle{ \Delta \vec{l} }[/math] = = -(Ex●[math]\displaystyle{ \Delta x }[/math] + Ey●[math]\displaystyle{ \Delta y }[/math] + Ez●[math]\displaystyle{ \Delta z }[/math]), units = Volts (V)

In a nonuniform electric field the potential difference is equal to [math]\displaystyle{ \textstyle\int\limits_{i}^{f}-Edl }[/math]

Examples

Simple Example of Two Different Paths

Calculate the potential difference going from A to C: [math]\displaystyle{ \Delta V = V_C - V_A =  ? }[/math]

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Angled Path.

Since the electric field inside the capacitor is uniform all along the path we can use the equation for a uniform electric field [math]\displaystyle{ \Delta V }[/math] = -[math]\displaystyle{ \vec{E} }[/math]●[math]\displaystyle{ \Delta \vec{l} }[/math] = = -(Ex●[math]\displaystyle{ \Delta x }[/math] + Ey●[math]\displaystyle{ \Delta y }[/math] + Ez●[math]\displaystyle{ \Delta z }[/math]), units = Volts (V)

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