understand the origin of the Hall voltage and derive and use the expression VH = BI / (ntq), where t = thickness

Force on a Current‑Carrying Conductor

When a straight conductor of length \$L\$ carrying a current \$I\$ is placed in a magnetic field \$\mathbf{B}\$, the charge carriers experience a magnetic force. The resultant force on the conductor is given by the Lorentz force summed over all carriers, leading to the macroscopic expression

\$\mathbf{F}=I\,\mathbf{L}\times\mathbf{B}\$

where \$\mathbf{L}\$ is a vector of magnitude \$L\$ directed along the conventional current.

Derivation from the Lorentz Force on Charge Carriers

1. Consider \$n\$ charge carriers per unit volume, each of charge \$q\$, moving with drift velocity \$\mathbf{v}_d\$.
2. The current density is \$\mathbf{J}=nq\mathbf{v}_d\$ and the total current \$I = J A\$, where \$A\$ is the cross‑sectional area.
3. Each carrier feels a magnetic force \$q\,\mathbf{v}_d\times\mathbf{B}\$.
4. Summing over all carriers in the volume \$V = A L\$ gives

   \$\mathbf{F}= (nV) q\,\mathbf{v}d\times\mathbf{B}= (nA L) q\,\mathbf{v}d\times\mathbf{B}.\$

5. Substituting \$\mathbf{v}_d = \mathbf{J}/(nq)\$ and \$I = J A\$ yields

   \$\mathbf{F}= I\,\mathbf{L}\times\mathbf{B}.\$

Hall Effect and Hall \cdot oltage

The magnetic force on moving charge carriers pushes them toward one side of the conductor, creating a transverse electric field. This field balances the magnetic force at equilibrium, giving rise to the Hall voltage \$V_H\$ measured across the thickness \$t\$ of the conductor.

Origin of the Hall \cdot oltage

• Magnetic force on a carrier: \$q\,\mathbf{v}_d\times\mathbf{B}\$.
• Charges accumulate on the side, producing an electric field \$\mathbf{E}_H\$.
• Equilibrium condition: \$q\mathbf{E}H = q\,\mathbf{v}d\times\mathbf{B}\$, so \$\mathbf{E}H = \mathbf{v}d\times\mathbf{B}\$.
• The Hall voltage is the potential difference across the thickness \$t\$: \$VH = EH\,t\$.

Derivation of \$V_H = \dfrac{B I}{n t q}\$

1. From the equilibrium condition, \$EH = vd B\$ (for \$\mathbf{v}_d\perp\mathbf{B}\$).
2. Current \$I = n q A v_d\$, where \$A = w t\$ (width \$w\$, thickness \$t\$).
3. Eliminate \$vd\$: \$vd = \dfrac{I}{n q w t}\$.
4. Substitute into \$EH\$: \$EH = \dfrac{I B}{n q w t}\$.
5. Since \$VH = EH w\$, the width \$w\$ cancels, giving

   \$V_H = \frac{B I}{n t q}.\$

Using the Hall \cdot oltage Formula

The expression can be rearranged to determine any unknown quantity, for example the carrier density \$n\$:

\$n = \frac{B I}{t q V_H}.\$

Typical \cdot alues and Units

Worked Example

Problem: A copper strip of thickness \$t = 2.0\times10^{-4}\,\text{m}\$ carries a current \$I = 2.5\,\$A. It is placed in a uniform magnetic field \$B = 0.5\,\$T perpendicular to the current. The measured Hall voltage is \$V_H = 3.2\times10^{-5}\,\$V. Determine the carrier density \$n\$.

1. Write the formula for \$n\$: \$n = \dfrac{B I}{t q V_H}\$.
2. Insert the known values (use \$q = e = 1.60\times10^{-19}\,\$C):

   \$n = \frac{0.5\times 2.5}{(2.0\times10^{-4})(1.60\times10^{-19})(3.2\times10^{-5})}.\$

3. Calculate:

   \$\$n \approx \frac{1.25}{(2.0\times10^{-4})(5.12\times10^{-24})}

   = \frac{1.25}{1.024\times10^{-27}}

   \approx 1.22\times10^{27}\,\text{m}^{-3}.\$\$

4. Interpretation: This value is of the same order as the known electron density in copper (\$\sim 8.5\times10^{28}\,\text{m}^{-3}\$), confirming the method.

Common Pitfalls

• Forgetting that \$V_H\$ is measured across the thickness \$t\$, not the width.
• Mixing up the sign of \$q\$; the Hall voltage polarity indicates the type of dominant carriers (positive holes vs. electrons).
• Using \$B\$ in gauss instead of tesla without conversion (1 T = 10⁴ G).

Summary

• The magnetic force on moving charges in a conductor gives a net force \$ \mathbf{F}=I\mathbf{L}\times\mathbf{B}\$.
• Charge separation creates a transverse electric field, leading to the Hall voltage.
• For a thin plate of thickness \$t\$, the Hall voltage is \$V_H = \dfrac{B I}{n t q}\$.
• The formula can be rearranged to find \$n\$, \$B\$, \$I\$, \$t\$, or \$V_H\$ as required.
• Experimental measurement of \$V_H\$ provides a direct method to determine carrier density and sign.