Hall Voltage (CIE A Level Physics)

• The Hall voltage is a product of the Hall effect
• Hall voltage is defined as:

The potential difference produced across an electrical conductor when an external magnetic field is applied perpendicular to the current through the conductor

• When an external magnetic field is applied perpendicular to the direction of current through a conductor, the electrons experience a magnetic force
• This makes them drift to one side of the conductor, where they all gather and becomes more negatively charged
• This leaves the opposite side deficient of electrons, or positively charged
• There is now a potential difference across the conductor
  • This is called the Hall Voltage, V_H

Hall voltage 

The positive and negative charges drift to opposite ends of the conductor producing a hall voltage when a magnetic field is applied

• An equation for the Hall voltage V_H is derived from the electric and magnetic forces on the charges

Electric and magnetic forces creating the Hall voltage

The electric and magnetic forces on the electrons are equal and opposite

• The voltage arises from the electrons accumulating on one side of the conductor slice
• As a result, an electric field is set up between the two opposite sides
• The two sides can be treated like oppositely charged parallel plates, where the electric field strength E is equal to:

• Where:
  • V_H = Hall voltage (V)
  • d = width of the conductor slice (m)

• A single electron has a drift velocity of v within the conductor. The magnetic field is into the plane of the page, therefore the electron has a magnetic force F_B to the right:

• This is equal to the electric force F_E to the left:

• Substituting E and cancelling the charge q

• Recall that current I is related to the drift velocity v by the equation:

• Where:
  • A = cross-sectional area of the conductor (m²)
  • n = number density of electrons (m^-3)

• Rearranging this for v and substituting it into the equation gives:

$\frac{V_H}{d} = \frac{BI }{nAq}$

• The cross-sectional area A of the slice is the product of the width d and thickness t:

$A = dt$

• Substituting A and rearranging for the Hall voltage V_H leads to the equation:

$\frac{V_H}{d} =\frac{BI}{I\left(dt\right)q}$

$V_{H }=\frac{ BI}{ntq}$ 

• Where:
  • B = magnetic flux density (T)
  • q = charge of the electron (C)
  • I = current (A)
  • n = number density of electrons (m^-3)
  • t = thickness of the conductor (m)

• This equation shows that the smaller the electron density n of a material, the larger the magnitude of the Hall voltage
  • This is why a semiconducting material is often used for a Hall probe

• Note: if the electrons were placed by positive charge carriers, the negative and positive charges would still deflect in opposite directions
  • This means there would be no change in the polarity (direction) of the Hall voltage