[NOTES/EM-01006]--Hall Effect

If a current carrying conductor is placed in a magnetic field that is perpendicular to the current, a potential difference develops in Hall effect is production of potential difference in across the conductor. The direction in which the potential difference develops is perpendicular to both the current and the magnetic field. 

The figure shows a slab \(ABCD\) with a current flowing in the \(X\)- direction. A magnetic field is applied in the \(Y\)-direction. The force on the current due to the magnetic field will be \(Iv(dl)\) and is easily seen to be along the \(Z\)- axis. The current consists of electrons which will move towards the side \(AB\) and start accumulating there giving rise to a potential difference between the side \(AB\) and \(CD\). Thus an electric field will be set up along the \(Z\) axis. In the equilibrium situation the force on the electrons due to the electric and magnetic fields will be equal and opposite. Thus we have \begin{equation} e E = e v B \end{equation} where \(V\) is the drift velocity of the electrons. If the width \(AD\) is denoted by \(w\)then \(E= V_H /w,\) where \(V_H\) is the Hall voltage developed across \(AD\) and . This gives an expression for the Hall voltage \begin{equation} V_H = \frac{vB}{w} \end{equation} If the number of electrons per unit volume is \(n\), \(t\) is the thickness of the slab then \( I=e v ne (w t)\) and we get \begin{equation} V_H =\frac{IB}{net} \end{equation} The Hall resistance \(R_H\) is defined to be the ratio of the Hall voltage to the current is given by \begin{equation} R_H = \frac{B}{net} \end{equation}

The Hall effect continues to be very important way of measuring the fundamental constant \(\hbar\).Hall probes are often used as magnetometers, i.e. to measure magnetic fields, or inspect materials (such as tubing or pipelines) using the principles of magnetic flux leakage.