Lorentz Force: Motion of Charged Particles in E and B Fields

1. The Lorentz Force

When a particle with charge $q$ moves with velocity $\mathbf{v}$ through a region with an electric field $\mathbf{E}$ and a magnetic field $\mathbf{B}$, the total force it experiences is the vector sum of the electric and magnetic forces. This is the Lorentz force:

The two components are:

1. Electric Force ${\mathbf{F}}_E=q\mathbf{E}$: This force is parallel to the electric field (for a positive charge) and is independent of the particle's velocity.
2. Magnetic Force ${\mathbf{F}}_B=q(\mathbf{v}\times \mathbf{B})$: This force depends on the particle's velocity and is directed perpendicular to both $\mathbf{v}$ and $\mathbf{B}$, as determined by the right-hand rule.

A Crucial Property: Work and Energy

A key distinction between the two forces lies in the work they do. The rate at which work is done on the particle (i.e., the power delivered) is $\mathbf{F}\cdot \mathbf{v}$. Let's look at the magnetic force component:

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

Because the vector $(\mathbf{v}\times \mathbf{B})$ is, by definition of the cross product, perpendicular to $\mathbf{v}$, their dot product is always zero.

This means a static magnetic field does no work on a charged particle. Consequently, a magnetic field alone cannot change a particle's kinetic energy or its speed; it can only change the direction of its velocity. The electric field, in contrast, can and does do work, changing the particle's kinetic energy.

To solidify these foundational concepts, please review the following brief sections from this provided text.

2. Motion in a Uniform Magnetic Field ($\mathbf{E}=0$)

Let's start by analyzing the case where only a uniform magnetic field $\mathbf{B}$ is present. The equation of motion is $m\mathbf{a}=q(\mathbf{v}\times \mathbf{B})$.

Case A: Velocity Perpendicular to $\mathbf{B}$

If a particle enters the field with its velocity $\mathbf{v}$ perpendicular to $\mathbf{B}$, the magnetic force will be constant in magnitude ($F_B=|q|vB$) and always perpendicular to $\mathbf{v}$. A force of constant magnitude that is always perpendicular to the velocity is the definition of a centripetal force. The particle will undergo uniform circular motion.

Equating the magnetic force to the centripetal force required for circular motion of radius $r$:
$|q|vB=\frac{m{v}^2}{r}$

From this, we can derive two important quantities:

• Cyclotron Radius: $r=\frac{mv}{|q|B}$. The radius is proportional to the momentum of the particle.
• Cyclotron Angular Frequency: ${\omega }_c=\frac{v}{r}=\frac{|q|B}{m}$.

Notice that ${\omega }_c$ is independent of the particle's velocity and the radius of its orbit. This remarkable fact is the operating principle of the cyclotron particle accelerator.

Case B: Velocity at an Arbitrary Angle to $\mathbf{B}$

If the velocity vector $\mathbf{v}$ is not perpendicular to $\mathbf{B}$, we can decompose it into components parallel ($v_{\parallel }$) and perpendicular ($v_{\perp }$) to the field.

• The parallel component $v_{\parallel }$ experiences no magnetic force ($\mathbf{v}\times \mathbf{B}=0$), so the particle continues to move along the field line with constant velocity $v_{\parallel }$.
• The perpendicular component $v_{\perp }$ results in circular motion in the plane perpendicular to $\mathbf{B}$, as described above.

The superposition of these two motions—uniform motion along an axis and circular motion around it—results in a helical path. The distance the particle travels along the field line during one full circle is called the pitch of the helix, given by $p=v_{\parallel }T=v_{\parallel }\frac{2\pi }{{\omega }_c}$.

A helical path is traced by a charged particle when its initial velocity has components both parallel and perpendicular to the magnetic field.

3. Motion in Combined Electric and Magnetic Fields

Now for the more complex and interesting scenarios involving both $\mathbf{E}$ and $\mathbf{B}$ fields. The trajectory depends critically on the orientation of the fields and the particle's initial velocity.

The following video is an excellent, mathematically rigorous exploration of several key cases. You'll find it aligns well with your preference for detailed derivations.

Let's dissect the two most significant scenarios involving perpendicular fields from that video.

Scenario 1: The Velocity Selector

Consider a region with crossed (perpendicular) fields, $\mathbf{E}$ and $\mathbf{B}$. A particle with velocity $\mathbf{v}$ enters, also perpendicular to both fields. The net force is $\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$.

The electric force and magnetic force will be in opposite directions. If we adjust the magnitudes of the fields or the velocity of the particle just right, the net force can be zero.
$qE=qvB⟹v=\frac{E}{B}$

Only particles with exactly this speed will pass through the region undeflected. Particles that are too slow will be deflected by the electric field, and particles that are too fast will be deflected by the magnetic field. This device is known as a velocity selector and is a critical component in many experiments, including the mass spectrometer.

Scenario 2: Cycloidal Motion and $\mathbf{E}\times \mathbf{B}$ Drift

This is the most mathematically rich case (Case 4 in the video). A particle starts from rest in crossed $\mathbf{E}$ and $\mathbf{B}$ fields.
The equations of motion, as derived in the video and in Problem 18 of the handout Electromagnetism IV: Lorentz Force, are:
$m\ddot{x}=q(E-B\dot{z})$
$m\ddot{z}=qB\dot{x}$
(Assuming $\mathbf{E}=E\hat{x}$ and $\mathbf{B}=B\hat{y}$).

Solving this system of coupled differential equations yields parametric equations for a cycloid. A cycloid is the curve traced by a point on the rim of a rolling wheel.

The trajectory of a particle starting from rest in crossed E and B fields is a cycloid.

An essential insight here is the particle's average motion. While it undergoes complex cycloidal loops, its net displacement is in the direction of $\mathbf{E}\times \mathbf{B}$. This is a general phenomenon known as the $\mathbf{E}\times \mathbf{B}$ drift, which is fundamental to plasma physics.

4. Applications

The Lorentz force is not just a theoretical concept; it's the principle behind several crucial technologies.

Application 1: The Mass Spectrometer

This is a perfect engineering application of our analysis. A mass spectrometer separates ions based on their mass-to-charge ratio. It typically has two stages:

1. Velocity Selector: Ensures a beam of ions, regardless of mass, enters the next stage with the exact same velocity.
2. Deflection Chamber: The ions enter a region with only a uniform magnetic field. Here, they undergo circular motion with a radius $r=mv/qB$.

Since $v$, $q$, and $B$ are the same for different isotopes of an element, the radius of the path is directly proportional to the mass ($r\propto m$). Heavier isotopes travel in wider circles than lighter ones, allowing them to be separated and detected.

The following video provides an excellent conceptual overview and walks through a typical problem.

Application 2: The Hall Effect

The Lorentz force also acts on charge carriers moving within a conductor. This leads to the Hall effect. Consider a flat conducting strip carrying a current $I$ placed in a perpendicular magnetic field $\mathbf{B}$.

• The magnetic field exerts a Lorentz force on the moving charge carriers (e.g., electrons), deflecting them to one side of the strip.
• This accumulation of charge creates a transverse electric field, ${\mathbf{E}}_H$, called the Hall field.
• The charge buildup continues until the electric force from the Hall field exactly cancels the magnetic force: $q{E}_H=q{v}_{d}B$, where $v_d$ is the drift velocity of the charges.
• This creates a measurable potential difference across the width of the strip, the Hall voltage $V_H=E_{H}w=v_{d}Bw$.

Crucially, the direction of the Hall voltage depends on the sign of the charge carriers. This was the first experimental proof that charge carriers in some metals are effectively positive (due to "holes" in the electronic band structure), a non-intuitive result of quantum mechanics.

Conclusion

Today we've explored the rich dynamics of charged particles under the influence of the Lorentz force. This single equation describes a wide array of physical phenomena, from the beautiful helical aurora displays to the precise functioning of mass spectrometers.

Key Takeaways:

• The total force on a charge is given by the Lorentz Force Law: $\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$.
• A static magnetic field does no work and cannot change a particle's kinetic energy; it only changes the direction of motion.
• Motion in a uniform $\mathbf{B}$-field is circular or helical, characterized by the cyclotron frequency ${\omega }_c=|q|B/m$.
• Crossed $\mathbf{E}$ and $\mathbf{B}$ fields can be used to create a velocity selector ($v=E/B$) or can lead to complex cycloidal motion with an average $\mathbf{E}\times \mathbf{B}$ drift.
• The Lorentz force is the principle behind key applications like mass spectrometers and the Hall effect.

Preview of the Next Lesson:
We have just analyzed the motion of individual charged particles. Next, we will scale up to examine the collective behavior of charge in macroscopic circuits. We will study RL, LC, and RLC circuits, analyzing their transient and steady-state responses. You will see that the behavior of these circuits is described by second-order ordinary differential equations, very similar in mathematical structure to the equations of motion we solved today. This will beautifully connect the microscopic dynamics of charges to the macroscopic behavior of electronic components.