A LevelPhysicsCambridge (CIE)Revision Notes20. Magnetic FieldsForce on a Moving ChargeHall Voltage

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

Exam code: 9702

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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:

V subscript straight H space equals space fraction numerator B I over denominator n t q end fraction

  • 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 greater the magnitude of the Hall voltage V subscript straight H

    • This is why a semiconducting material is often used for a Hall probe

  • Note: if the electrons were replaced by positive charge carriers, the direction of the Hall voltage would reverse

    • This happens because positive charges experience a magnetic force in the opposite direction to that experienced by electrons (provided the direction of the current is unchanged)

Hall voltage 

Hall Voltage, downloadable AS & A Level Physics revision notes

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 subscript straight 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:

E space equals space fraction numerator V subscript straight H space over denominator d end fraction

  • Where:

    • V subscript straight H = Hall voltage (V)

    • d = width of the conductor slice (m)

  • A single electron has a drift velocity v within the conductor

  • The magnetic field is into the plane of the page, so the magnetic force F subscript B on the electron acts to the right:

F subscript B space equals space B q v

  • This is equal to the electric force F subscript E acting to the left:

F subscript E space equals space q E

q E thin space equals space B q v

  • Substituting E and cancelling the charge q

V subscript H over d space equals space B v

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

I space equals space n A v q

  • 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:

V subscript H over d space equals space fraction numerator B I space over denominator n A q end fraction

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

A space equals space d t

  • Substituting A and rearranging for the Hall voltage V subscript straight H leads to the equation:

V subscript H over d equals fraction numerator B I over denominator n open parentheses d t close parentheses q end fraction

V subscript H space end subscript equals fraction numerator B I over denominator n t q end fraction

Electric and magnetic forces creating a Hall voltage

Hall voltage derivation diagram, downloadable AS & A Level Physics revision notes

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

Examiner Tips and Tricks

Remember to use Fleming’s left-hand rule to obtain the direction the electrons move due to the magnetic force created by the magnetic field.

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Ashika

Author: Ashika

Expertise: Physics Content Creator

Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.