Lorentz Force Examples: Problems and Solutions

📚 Quick Study Guide: Lorentz Force

• ✍️ Definition: The Lorentz force is the total force experienced by a charged particle moving through both an electric field ($\mathbf{E}$) and a magnetic field ($\mathbf{B}$). It's a fundamental principle in electromagnetism.
• 🧪 The Formula: The combined Lorentz force ($\mathbf{F}$) on a charge ($q$) moving with velocity ($\mathbf{v}$) in an electric field ($\mathbf{E}$) and a magnetic field ($\mathbf{B}$) is given by:

  $$\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})$$
• 💡 Components: The total Lorentz force is the vector sum of two parts:
  • ⚡ Electric Force ($\mathbf{F}_E$): The force due to the electric field only. It's always parallel to the electric field for a positive charge and opposite for a negative charge.

    $$\mathbf{F}_E = q\mathbf{E}$$

  • 🧲 Magnetic Force ($\mathbf{F}_B$): The force due to the magnetic field. It acts perpendicular to both the velocity vector ($\mathbf{v}$) and the magnetic field vector ($\mathbf{B}$). This means it does no work on the particle.

    $$\mathbf{F}_B = q(\mathbf{v} \times \mathbf{B})$$

• 🧭 Direction of Magnetic Force (Right-Hand Rule):
  • For a positive charge: Point your fingers in the direction of \(\mathbf{v}\), curl them towards \(\mathbf{B}\). Your thumb points in the direction of \(\mathbf{F}_B\).
  • For a negative charge: Use the right-hand rule, then reverse the direction of the force.
• ✅ Key Conditions for Magnetic Force: For a magnetic force to exist, the particle must:
  • ➕ Be charged ($\textbf{q} \neq 0$).
  • 🏃 Be moving ($\textbf{v} \neq 0$).
  • 🌌 Be in a magnetic field ($\textbf{B} \neq 0$).
  • 📐 Have a velocity component perpendicular to the magnetic field ($\textbf{v}$ not parallel to $\textbf{B}$). If $\mathbf{v}$ is parallel or anti-parallel to $\mathbf{B}$, then $\mathbf{v} \times \mathbf{B} = 0$, and the magnetic force is zero.
• 📏 Units:
  • Force ($\mathbf{F}$): Newtons (N)
  • Charge ($q$): Coulombs (C)
  • Electric Field ($\mathbf{E}$): Newtons per Coulomb (N/C) or Volts per meter (V/m)
  • Velocity ($\mathbf{v}$): meters per second (m/s)
  • Magnetic Field ($\mathbf{B}$): Tesla (T)
• 🛰️ Applications: Lorentz force is crucial in technologies like mass spectrometers, cyclotrons, particle accelerators, electric motors, and even explains phenomena like the aurora borealis.

🧠 Practice Quiz

1. Which of the following conditions is NOT necessary for a charged particle to experience a magnetic force?

   A. The particle must be moving.

   B. The particle must be in a magnetic field.

   C. The particle must have a net charge.

   D. The particle's velocity must be parallel to the magnetic field.

2. A proton (positive charge) moves with a velocity \(\mathbf{v}\) in the +x direction. It enters a uniform magnetic field \(\mathbf{B}\) pointing in the +y direction. In what direction will the magnetic force act on the proton?

   A. +x direction

   B. +y direction

   C. +z direction

   D. -z direction

3. An electron (charge $-1.6 \times 10^{-19}$ C) moves at a speed of $2.0 \times 10^6$ m/s perpendicular to a uniform magnetic field of 0.5 T. What is the magnitude of the magnetic force on the electron?

   A. $1.6 \times 10^{-13}$ N

   B. $8.0 \times 10^{-14}$ N

   C. $3.2 \times 10^{-13}$ N

   D. $1.6 \times 10^{-25}$ N

4. A positively charged particle is placed in a uniform electric field \(\mathbf{E}\) pointing towards the right. If the particle is initially at rest, in what direction will it accelerate?

   A. Towards the left

   B. Towards the right

   C. Upwards

   D. It will remain at rest.

5. Which statement correctly describes the work done by the magnetic component of the Lorentz force on a charged particle?

   A. It always does positive work, increasing the particle's kinetic energy.

   B. It always does negative work, decreasing the particle's kinetic energy.

   C. It does no work on the particle, only changing its direction of motion.

   D. It does work only if the particle moves parallel to the magnetic field.

6. An alpha particle (charge $+2e$, where $e = 1.6 \times 10^{-19}$ C) moves through a region with an electric field of $1.0 \times 10^3$ N/C (upwards) and a magnetic field. If the alpha particle experiences a net upward force of $6.4 \times 10^{-16}$ N, what is the magnitude of the electric force acting on it?

   A. $1.6 \times 10^{-16}$ N

   B. $3.2 \times 10^{-16}$ N

   C. $4.8 \times 10^{-16}$ N

   D. $6.4 \times 10^{-16}$ N

7. In a velocity selector, electric and magnetic fields are arranged such that a charged particle moving at a specific velocity passes through undeflected. If the electric field (E) is upwards and the magnetic field (B) is into the page, what is the direction of the velocity (v) for a positive particle to pass undeflected?

   A. To the right

   B. To the left

   C. Upwards

   D. Downwards