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Nernst Effect: How a Temperature Difference Generates an Electric Field

Important Formula

Formula: Nernst Effect

$$\class{purple}{E_{\text y}} ~=~ C_{\text N} \, \frac{\text{d} T}{\text{d} x} \, \class{violet}{B_{\text z}}$$ $$\class{purple}{E_{\text y}} ~=~ C_{\text N} \, \frac{\text{d} T}{\text{d} x} \, \class{violet}{B_{\text z}}$$ $$\frac{\text{d} T}{\text{d} x} ~=~ \frac{ 1 }{ C_{\text N} } \, \frac{ \class{purple}{E_{\text y}} }{ \class{violet}{B_{\text z}} }$$ $$\class{violet}{B_{\text z}} ~=~ \frac{ 1 }{ C_{\text N} } \, \left( \frac{\text{d} T}{\text{d} x} \right)^{-1} \, \class{purple}{E_{\text y}}$$ $$C_{\text N} ~=~ \frac{ \class{purple}{E_{\text y}} }{ \class{violet}{B_{\text z}} } \, \left( \frac{\text{d} T}{\text{d} x} \right)^{-1}$$

Rearrange formula

What do the formula symbols mean?

Electric field (E field)

$$ \class{purple}{E_{\text y}} $$ Unit $$ \frac{\mathrm{V}}{\mathrm{m}} = \frac{\mathrm{N}}{\mathrm{C}} = \frac{\mathrm{kg} \, \mathrm{m}}{\mathrm{A} \, \mathrm{s}^3} $$

Due to the temperature difference $ \frac{\text{d} T}{\text{d} x} $ in the conductor, which is in a magnetic field $B_{\text z}$, the electrons move towards the hot conductor side. Since this motion happens in a magnetic field, the electrons are deflected by the Lorentz force, so in this case an electric field $ E_{\text y} $ is formed in the $y$ direction.

Since this effect is practically similar to the Hall effect; with the only difference that instead of the E-field a temperature gradient is the cause for the electron motion, this effect is also called thermal Hall effect.

Temperature gradient

$$ \frac{\text{d} T}{\text{d} x} $$

Temperature difference in a current-carrying conductor which is in a magnetic field. The temperature difference is caused by the deflected electrons in the magnetic field due to the Lorentz force. Because slow electrons are deflected more than fast electrons, one side of the conductor becomes cooler than the other.

The temperature gradient forms in the $x$ direction in this case.

Magnetic flux density (B-field)

$$ \class{violet}{B_{\text z}} $$ Unit $$ \mathrm{T} = \frac{\mathrm{kg}}{\mathrm{A} \, \mathrm{s}^2} $$

The magnetic field penetrates the current-carrying conductor perpendicularly and points in the $z$ direction in this formula.

Nernst coefficient

$$ C_{\text N} $$

The Nernst coefficient is material-specific and tells how well the electric field can form in the respective material due to the temperature gradient.

The Nernst Effect describes the occurrence of an electric field $ \class{purple}{\boldsymbol{E}_{\text y}} $ perpendicular to a temperature gradient in a material situated in a magnetic field $ \class{violet}{B_{\text z}} $. Since this effect is practically similar to the Hall Effect, with the only distinction being that, instead of the electric field, a temperature gradient is the cause of electron motion, this effect is also called the thermal Hall effect.

$$ \begin{align} \class{purple}{E_{\text y}} ~=~ C_{\text N} \, \frac{\text{d} T}{\text{d} x} \, \class{violet}{B_{\text z}} \end{align} $$

Due to the temperature difference $ \frac{\text{d} T}{\text{d} x} $ in the material, which is located in a magnetic field $B_{\text z}$, the electrons move towards the hot side of the conductor. Since this movement takes place in a magnetic field, the electrons are deflected by the Lorentz force, so that in this case an electric field $ E_{\text y} $ forms in the $y$ direction.