(a) Show for a 2D Hall bar of length \(L\) and width \(w\) that \(j=\sigma E\) (where \(j\) is the current density, \(E\) is the electric field, and \(\sigma\) is the conductivity) is equivalent to the simple form of Ohm's law, \(V=I R\). The resistance is \(R=ho L / w\), where \(ho=1 / \sigma\) is the resistivity.

(b) Show that for Fig. 28.1 the resistance components \(R_{i j}=V_{i} / I_{j}\) and resistivity components \(ho_{i j}=E_{i} / j_{j}\) are related by \(R_{y x}=ho_{y x}\) and \(R_{x x}=(L / w) ho_{x x}\).

(c) Verify the final forms of the \(ho\) and \(\sigma\) tensors in Box 28.1 .

Data from Fig. 28.1

Data from Box 28.1

Transcribed Image Text:

(a) Ex (b) Magnetic field Bz jx jx Z 2 Lov. + + + + H + (c) Ex Ey jx (VH (d) L (VL) W jx