The Hall Probe

The Hall Effect

​AS & A Level

​Derive the expression VH=BI/ntq for the Hall voltage, where t = thickness

Describe and analyse qualitatively the deflection of beams of charged particles by uniform electric and uniform magnetic fields

Explain how electric and magnetic fields can be used in velocity selection

explain the main principles of one method for the determination of v and e/me for electrons

​Learning Goals!

Hall Voltage​

​Hall voltage is 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 deficident of electrons, or positively charged

There is now a potential difference across the conductor. This is called the Hall Voltage, V_H

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

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

​An equation for the Hall voltage V_H is 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, 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:

F_B = Bqv

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

F_E = qE

qE = Bqv

• Substituting E and cancelling the charge q

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

I = nAvq

• 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:

A = dt

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

• 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​

Motion of a Charged Particle in a Magnetic Field​

​A charged particle in uniform magnetic field which is perpendicular to its direction of motion travels in a circular path

• This is because the magnetic force F_B will always be perpendicular to its velocity v

  F_B will always be directed towards the centre of the path

​A charged particle moves travels in a circular path in a magnetic field

​The magnetic force F_B provides the centripetal force on the particle

• Recall the equation for centripetal force:

• m = mass of the particle (kg)

• v = linear velocity of the particle (m s^-1)

• r = radius of the orbit (m)

• Equating this to the force on a moving charged particle gives the equation:

​Rearranging for the radius r obtains the equation for the radius of the orbit of a charged particle in a perpendicular magnetic field:

​Velocity Selection

​A velocity selector is:

A device consisting of perpendicular electric and magnetic fields where charged particles with a specific velocity can be filtered

• Velocity selectors are used in devices, such as mass spectrometers, in order to produce a beam of charged particles all travelling at the same velocity

• The construction of a velocity selector consists of two horizontal oppositely charged plates situated in a vacuum chamber

  • The plates provide a uniform electric field with strength E between them

• There is also a uniform magnetic field with flux density B applied perpendicular to the electric field

​If a beam of charged particles enter between the plates, they may all have the same charge but travel at different speeds v

The electric force does not depend on the velocity: F_E = EQ

• However, the magnetic force does depend on the velocity: F_B = BQv

  The magnetic force will be greater for particles which are travelling faster

• To select particles travelling at exactly the desired the speed v, the electric and magnetic force must therefore be equal, but in opposite directions

F_E = F_B

​The particles travelling at the desired speed v will travel through undeflected due to the equal and opposite electric and magnetic forces on them

​The resultant force on the particles at speed v will be zero, so they will remain undeflected and pass straight through between the plates

By equating the electric and magnetic force equations:

​EQ = BQv

The charge Q will cancel out on both sides to give the selected velocity v equation:

​

​Therefore, the speed v in which a particle will remain undeflected is found by the ratio of the electric and magnetic field strength

• If a particle has a speed greater or less than v, the magnetic force will deflect it and collide with one of the charged plates

• This would remove the particles in the beam that are not exactly at speed v

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

​

​Summary!

​A charged particle in uniform magnetic field which is perpendicular to its direction of motion travels in a circular path

​To select particles travelling at exactly the desired the speed v, the electric and magnetic force must therefore be equal, but in opposite directions