Potential Difference Path Independence: Difference between revisions
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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. | 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. | ||
=== | ===Potential Difference Equations=== | ||
In a uniform electric field the potential difference is equal to | In a uniform electric field the potential difference is equal to | ||
<math>\Delta V = -\vec{E}●\Delta \vec{l} = -(E_x●\Delta x + E_y●\Delta y + E_z●\Delta z</math>). | <math>\Delta V = -\vec{E}●\Delta \vec{l} = -(E_x●\Delta x + E_y●\Delta y + E_z●\Delta z</math>). | ||
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===Simple Example of Two Different Paths=== | ===Simple Example of Two Different Paths=== | ||
Calculate the potential difference going from A to C: <math>\Delta V = V_C - V_A = ?</math> | Calculate the potential difference going from A to C: <math>\Delta V = V_C - V_A = ?</math> | ||
[[File:figure16.30.png|thumb|none|alt=Potential Difference Path Independence.|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>\Delta V = -\vec{E}●\Delta \vec{l} = -(E_x●\Delta x + E_y●\Delta y + E_z●\Delta z</math>). | [[File:figure16.30.png|thumb|none|alt=Potential Difference Path Independence.|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>\Delta V = -\vec{E}●\Delta \vec{l} = -(E_x●\Delta x + E_y●\Delta y + E_z●\Delta z</math>). The displacement vector <math>\Delta l = <\Delta x, \Delta y, \Delta z> = <(x_1 - 0),(-y_1 - 0)> = <x_1,-y_1></math>. The electric field vector is given as <math>\vec{E} = <E_x,0,0></math>. | ||
==Middling=== | ==Middling=== | ||
Revision as of 14:22, 1 December 2015
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.
Potential Difference Equations
In a uniform electric field the potential difference is equal to [math]\displaystyle{ \Delta V = -\vec{E}●\Delta \vec{l} = -(E_x●\Delta x + E_y●\Delta y + E_z●\Delta z }[/math]).
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]
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 = -\vec{E}●\Delta \vec{l} = -(E_x●\Delta x + E_y●\Delta y + E_z●\Delta z }[/math]). The displacement vector [math]\displaystyle{ \Delta l = \lt \Delta x, \Delta y, \Delta z\gt = \lt (x_1 - 0),(-y_1 - 0)\gt = \lt x_1,-y_1\gt }[/math]. The electric field vector is given as [math]\displaystyle{ \vec{E} = \lt E_x,0,0\gt }[/math].
Middling=
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