09.1 – Gradienten von Feldern

 

Vektorfelder (z.B. Kraftfelder, elektrische Felder) lassen sich oft als Gradient des Potentials ausdrücken. Es gilt:

F = -{\text{grad}}\left( \varphi \right) = -\left( {\begin{array}{*{20}{c}} \frac{{\partial \varphi }} {{\partial x}} \\ \frac{{\partial \varphi }} {{\partial y}} \\ \frac{{\partial \varphi }} {{\partial z}} \\ \end{array} } \right)

Berechnen Sie die Felder für folgende Potentiale:

  1. \varphi = C \cdot z
  2. \varphi = -\frac{C} {r}
  3. \varphi = -C \cdot \frac{z} {r}

Lösung

a )

F = -{\text{grad}}\left( \varphi \right) = -C \cdot \left( {\begin{array}{*{20}{c}} 0 \\ 0 \\ 1 \\ \end{array} } \right)

b )

\varphi = -\frac{C} {r}

\varphi = -C \cdot \frac{1} {r} = -C \cdot \frac{1} {{\sqrt {x^2 +y^2 +z^2 } }} = -C \cdot \left( {x^2 +y^2 +z^2 } \right)^{-\frac{1} {2}}

\frac{{\partial \varphi }} {{\partial x}} = -C \cdot \left( {-\frac{1} {2}} \right)\left( {x^2 +y^2 +z^2 } \right)^{-\frac{3} {2}} \cdot 2x

\frac{{\partial \varphi }} {{\partial y}} = -C \cdot \left( {-\frac{1} {2}} \right)\left( {x^2 +y^2 +z^2 } \right)^{-\frac{3} {2}} \cdot 2y

\frac{{\partial \varphi }} {{\partial z}} = -C \cdot \left( {-\frac{1} {2}} \right)\left( {x^2 +y^2 +z^2 } \right)^{-\frac{3} {2}} \cdot 2z

F = -{\text{grad}}\left( \varphi \right) = -C \cdot \left( {x^2 +y^2 +z^2 } \right)^{-\frac{3} {2}} \cdot \left( {\begin{array}{*{20}{c}} x \\ y \\ z \\ \end{array} } \right) = -C \cdot r^{-3} \cdot \left( {\begin{array}{*{20}{c}} x \\ y \\ z \\ \end{array} } \right)

c )

\varphi = -C \cdot \frac{z} {r}

\varphi = -C \cdot \frac{z} {r} = -C \cdot z \cdot \frac{1} {{\sqrt {x^2 +y^2 +z^2 } }} = -C \cdot z \cdot \left( {x^2 +y^2 +z^2 } \right)^{-\frac{1} {2}}

\frac{{\partial \varphi }} {{\partial x}} = -C \cdot z\left( {-\frac{1} {2}} \right)\left( {x^2 +y^2 +z^2 } \right)^{-\frac{3} {2}} \cdot 2x

\frac{{\partial \varphi }} {{\partial y}} = -C \cdot z\left( {-\frac{1} {2}} \right)\left( {x^2 +y^2 +z^2 } \right)^{-\frac{3} {2}} \cdot 2y

\frac{{\partial \varphi }} {{\partial z}} = -C \cdot \left( {1 \cdot \left( {x^2 +y^2 +z^2 } \right)^{-\frac{1} {2}} +z \cdot \left( {-\frac{1} {2}} \right)\left( {x^2 +y^2 +z^2 } \right)^{-\frac{3} {2}} \cdot 2z} \right)

vereinfacht:

\frac{{\partial \varphi }} {{\partial x}} = C \cdot zr^{-3} \cdot x

\frac{{\partial \varphi }} {{\partial y}} = C \cdot zr^{-3} \cdot y

\frac{{\partial \varphi }} {{\partial z}} = -C \cdot \left( {1 \cdot r^{-1} +z \cdot \left( {-\frac{1} {2}} \right)r^{-3} \cdot 2z} \right) = -C \cdot \left( {r^{-1} -z^2 \cdot r^{-3} } \right)

= C \cdot \left( {-r^{-1} +z^2 \cdot r^{-3} } \right)

= C \cdot z \cdot r^{-3} \cdot \left( {-\frac{{r^2 }} {z}+z} \right) = C \cdot zr^{-3} \cdot \left( {z-\frac{{r^2 }} {z}} \right)

eingesetzt:

F = -{\text{grad}}\left( \varphi \right) = -C \cdot zr^{-3} \left( {\begin{array}{*{20}{c}} x \\ y \\ {z-\frac{{r^2 }} {z}} \\ \end{array} } \right)