Lorentz Force and Hall Effect (E10) - Experiment Notes

Ziel des Versuchs

• Untersuchung der Lorentzkraft: The primary goal is to conduct a quantitative investigation of the fundamental principles behind the Lorentz force, a cornerstone concept in electromagnetism. This investigation uses two distinct experimental setups to observe its effects directly.

• Part 1: Induced Voltage in a Moving Conductor: This part focuses on the generation of an electromotive force (EMF) when a conductor moves through a magnetic field, leading to a measurable voltage. This experiment demonstrates the direct conversion of mechanical energy into electrical energy via the Lorentz force.

• Part 2: Hall Effect: This section is dedicated to examining the Hall effect in a semiconductor material, which allows for the determination of the sign and density of charge carriers within the material, providing insights into its conducting properties.

Theoretischer Hintergrund

Kraft auf geladene Teilchen

• Electric Field: In the presence of an electric field $\vec{E}$, any electric charge $q$, whether stationary or in motion, experiences an electric force given by the equation $\vec{F}_E = q \vec{E}$. This force acts along the direction of the electric field for positive charges and opposite for negative charges.

• Magnetic Field (Lorentz Force): Unlike the electric field, a magnetic field $\vec{B}$ (or sometimes described by magnetic field strength $\vec{H}$) exerts a force only if the charged particles are in motion relative to the field. This force, known as the Lorentz force, is always perpendicular to both the velocity of the charge and the magnetic field direction. Crucially, the magnetic force does no work on a moving charge as it is always perpendicular to the displacement.

• Lorentz Force Equation: The total Lorentz force acting on a charged particle moving through a combined electric and magnetic field is described by the vector sum: $\vec{F} = q (\vec{E} + \vec{v} \times \vec{B})$ (1a) When only a magnetic field is considered, the magnetic component of the Lorentz force is: $\vec{F}_B = q \vec{v} \times \vec{B}$ (1b)

  • $\vec{F}_B$: The magnetic component of the Lorentz force (in Newtons).

  • $q$: The electric charge of the particle (in Coulombs).

  • $\vec{v}$: The velocity vector of the charge (in meters per second).

  • $\vec{B}$: The magnetic flux density (or magnetic field strength, in Teslas).

Part 1: Induced Voltage in a Moving Conductor

• Setup: Consider a conductor, which can be envisioned as a closed volume $V$ containing mobile free charge carriers, moving with a constant velocity $\vec{v}$ through a uniform magnetic field $\vec{B}$. The interaction is maximized when $\vec{v}$ and $\vec{B}$ are perpendicular.

• Charge Separation: As the conductor moves, the free charge carriers within it (e.g., electrons in a metal) experience the Lorentz force $\vec{F}_B = q \vec{v} \times \vec{B}$. This force causes a macroscopic displacement of these charge carriers, pushing them to one side of the conductor and leaving opposite charges on the other side. This process is known as charge separation. Since these charge carriers are confined within the conductor's volume, they accumulate at its boundaries.

• Electric Field Generation: The accumulation of separated charges at the boundaries creates an internal electrostatic field $\vec{E}_ind$ within the conductor. This induced electric field opposes the further separation of charges by exerting an electric force $\vec{F}_E = q \vec{E}_ind$ on the charge carriers.

• Equilibrium: Charge separation continues until the electrostatic force due to the induced electric field precisely balances the magnetic Lorentz force. At this point, the net force on the charge carriers becomes zero, and charge flow (current) ceases if the circuit is open. This equilibrium condition is expressed as:
  $q \vec{E}_ind = q \vec{v} \times \vec{B}$ (2)
  Thus, the induced electric field equals $\vec{E}_ind = \vec{v} \times \vec{B}$.

• Measurement: The equilibrium electric field $\vec{E}_ind$ established across the conductor can be directly detected as a potential difference or voltage $U$ between two points within the conductor. This induced voltage represents the electromotive force (EMF) generated by the Lorentz force.

• Experimental Arrangement (Rotating Disk):

  • In this specific setup, a highly conductive metal disk is rotated at a constant angular frequency $f$ (or angular velocity $\omega = 2\pi f$) within a uniform magnetic field.

  • The magnetic field $\vec{B}$ is rigorously oriented perpendicular to the disk's plane of rotation, ensuring that the velocity vector of any point on the disk is always perpendicular to $\vec{B}$, maximizing the Lorentz force.

  • The induced voltage $U$ is measured via two stationary sliding contacts (typically carbon brushes) which make electrical contact with the disk at specified radial distances $r_1$ and $r_2$ from the center of rotation.

• Voltage Derivation: The voltage between the two contacts at $r_1$ and $r_2$ is derived by integrating the induced electric field along the radial path. Knowing that the tangential velocity $v$ at a radius $r$ is $v = \omega r = 2\pi f r$, and the induced electric field magnitude is $|\vec{E}_ind| = |\vec{v} \times \vec{B}| = vB$ (since $\vec{v}$ and $\vec{B}$ are perpendicular): $\frac{dU}{dr} = E_{ind} = vB = 2\pi f Br$ Integrating this differential voltage $dU$ across the radial path from $r_1$ to $r_2$ yields: $U = \int_{r_1}^{r_2} (2\pi f Br) dr = 2\pi f B \left[ \frac{r^2}{2} \right]_{r_1}^{r_2} = 2\pi f B \left( \frac{r_2^2}{2} - \frac{r_1^2}{2} \right) = \pi f B (r_2^2 - r_1^2)$ (3a) Alternatively, using the provided form $U = 2\pi f B r_{avg} \Delta r$: Let $r_{avg} = (r_1 + r_2)/2$ and $\Delta r = r_2 - r_1$. Substituting these into the formula (3a): $U = \pi f B (r_2 - r_1)(r_2 + r_1) = \pi f B (\Delta r)(2r_{avg}) = 2\pi f B r_{avg} \Delta r$ (3b)

  • $r_{avg} = (r_1 + r_2)/2$: The average radius.

  • $\Delta r = r_2 - r_1$: The difference in radii.

  • $f$: The rotational frequency of the disk (in Hertz).

  • $B$: The magnetic flux density (in Teslas).

• Induced Voltage vs. Terminal Voltage: The voltage $U$ calculated above represents the electromotive force (EMF), which is the ideal voltage generated by the Lorentz force before any current is drawn from the system. It is the open-circuit voltage. The terminal voltage, on the other hand, is the actual voltage measured when a current flows through an external circuit connected to the setup. This terminal voltage will be less than the EMF due to any internal resistance of the conductor and the contacts, causing a voltage drop ($V_{terminal} = EMF - IR_{internal}$), similar to a real battery.