Hall Voltage (Cambridge (CIE) A Level Physics): Revision Note

Hall voltage

• When an external magnetic field is applied perpendicular to the direction of current through a conductor, the electrons experience a magnetic force

• As a result, the electrons drift to one side of the conductor, causing it to become more negatively charged

  • This causes the opposite side to become more positively charged

• As a result of the separation of charge, a potential difference is set up across the conductor

• This is called the Hall voltage, which 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

• It is described by the expression:

• 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 replaced 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

Hall voltage 

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

Derivation of the Hall voltage equation

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

  • The voltage arises from the electrons accumulating on one side of the conductor slice

  • As a result of the charge separation, an electric field is set up between the two opposite sides of the conductor

• The two sides can be treated as 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:

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

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

Electric and magnetic forces creating a Hall voltage

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