= Solution
The <Hall effect> is a transverse electric response to a magnetic <force> on moving <charge carriers>. Take conventional <current density> $\mathbf J=J_x\widehat{\mathbf x}$ and <magnetic field> $\mathbf B=B_z\widehat{\mathbf z}$ in a conducting slab. For one carrier type of <number density> $n$ and signed <electric charge> $q$, the <drift velocity> is $\mathbf v_d=\mathbf J/(nq)$. Carriers initially deflect sideways until a Hall <electric field> balances the magnetic <Lorentz force>:
$$
q(\mathbf E_H+\mathbf v_d\times\mathbf B)=0,\qquad
E_y=\frac{J_xB_z}{nq}.
$$
The <Hall coefficient> is therefore
$$
\boxed{R_H=\frac{E_y}{J_xB_z}=\frac1{nq}.}
$$
For <Electrons> it is $-1/(ne)$, whereas a simple hole-dominated conductor gives $+1/(pe)$. The sign reveals the effective carrier sign; the magnitude estimates their density in the one-carrier approximation.
For width $w$ along $y$ and thickness $t$ along $z$, $I=J_xwt$. Define the signed <Hall voltage> by $V_H=E_yw$, namely the electric potential at the lower-$y$ edge minus that at the higher-$y$ edge. Then
$$
\boxed{V_H=\frac{IB_z}{nqt},\qquad n=\frac{|IB_z|}{e t|V_H|}\quad(|q|=e).}
$$
This explicit contact convention fixes the voltage sign. A <Hall bar> with longitudinal and transverse contacts measures both <electrical resistivity> and the Hall response. Reverse the field and take the part of the transverse voltage odd in $B_z$ to suppress longitudinal-voltage pickup from misaligned contacts; reversing current provides another useful check.
A Hall measurement is useful for carrier identification, semiconductor doping measurements and magnetic-field sensing. Combining a one-carrier $R_H$ with <electrical conductivity> gives a mobility magnitude $\mu=\sigma|R_H|$. However, mixed <Electron> and hole conduction is not described by a single density: in weak fields,
$$
R_H=\frac{p\mu_h^2-n\mu_e^2}{e(p\mu_h+n\mu_e)^2}.
$$
The response weights mobilities differently from ordinary conductivity. Complicated <energy bands>, anisotropic <Fermi surfaces> and multiple bands can likewise change the simple interpretation in metals; a positive Hall coefficient does not mean that the material contains positively charged fundamental <Electrons>.
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