Lecture 23 - Hall Effect & Force on Current Carrying Wire

Lecture Goals

  • Hall Effect
    • Understand the Hall Effect and its significance.
    • Review concepts of magnetic field, force, and Lorentz force law.
    • Recap cross product and right-hand rule for vectors.
    • Know how magnetic force affects charged particles.
    • Conceptualize path behavior of particles in uniform magnetic fields.
    • Calculate properties of paths quantitatively.
    • Explain the origin and generation of Hall voltage.
    • Calculate the magnitude and sign of Hall voltage for specific setups.
    • Determine unknown magnetic field, charge carrier density, or drift velocity using Hall voltage measurements.

Hall Effect

  • Definition: A voltage ($V_H$) generated in a conductor carrying current ($I$) in a magnetic field ($B$) perpendicular to the current.
  • Charge Movement:
    • For positive charge carriers ($+q$) moving through a magnetic field ($B$) – using the right-hand rule (RHR) to find the direction of force.
    • For electrons ($-e$), the direction of movement is opposite.

Charge Deflection

  • Charge carriers are initially deflected due to magnetic force, leading to a voltage difference.
  • Buildup of charge on conductor edges generates an induced electric field (Hall electric field).

Measuring Hall Voltage

  • Voltage readings help identify charge carrier types based on potential differences between conductor edges.
  • Potential difference is calculated as $| riangle V_H| = |I| |B| |d|$ for given setups.
  • Examples specific for different geometries, such as rectangular strips and semicircular conductors.

Force on Current Carrying Wire

  • A current-carrying wire in a magnetic field experiences a force ($FB$) given by: FB = I L imes B
    • Direction determined by the right-hand rule.
  • If wires carry currents in the same direction, they attract; if in opposite directions, they repel.

Important Formulas

  • Current Density: $I = n |q| A v_d$
  • Drift Velocity Measurement:
    |V_H| = rac{IB}{nt|q|}
    |A| = t imes d
  • Force among Wires: uses the magnetic field equations detailed above.

Example Calculations

  • Various examples including calculating Hall voltage for copper strips and p-type semiconductor probes to find magnetic field strengths.
  • Applies for experiments with drift velocity in Hall effect probes and using magnetic force calculations for circular loops (semicircular conductors).

Summary Points

  • The Hall effect provides insights into charge carrier behavior and quantity in materials under magnetic influences. Measurements using voltage can elucidate underlying material properties.
  • Force experienced by wires depends on current direction and its relation to the magnetic field, highlighting interactions that can stabilize or destabilize systems depending on configurations.