Lorentz Force

The Lorentz force describes the total force acting on a charged particle as it moves through electric and magnetic fields. It unifies the effects of both fields into a single formula, showing how a charge can bend, speed up, or change its path under electromagnetic influence. This concept plays a crucial role in physics and is widely applied in devices such as electric motors, generators, particle accelerators, and plasma systems.

1.0Basics of Lorentz Force

The Lorentz Force is the total force experienced by a charged particle when it moves through electric and magnetic fields.

A particle with charge q may be affected by:

• an electric field ($\vec{E}$), which exerts a force based on charge and field strength.
• and a magnetic field ($\vec{B}$) which exerts a force based on the particle’s velocity and the magnetic field.

Note: The Lorentz Force accounts for both.

2.0Components of the Lorentz Force

Electric Force: The force on a charge in an electric field is: ${\vec{F}}_E=q\vec{E}$

Magnetic Force: The force on a moving charge in a magnetic field is: ${\vec{F}}_B=q(\vec{v}\times \vec{B})$

The total Lorentz Force is the vector sum of these two forces.

The complete Lorentz Force equation is: $\vec{F}=q(\vec{E}+\vec{v}\times \vec{B})$

3.0Derivation of the Lorentz Force

Force on a Moving Charge in a Magnetic Field

• A magnetic field alone does not exert force on a stationary charge. It only acts on moving charges.

Experiments (e.g., deflection of electron beams) show

• The magnetic force is perpendicular to both velocity and the magnetic field.
• The magnitude depends on the charge and the component of velocity perpendicular to the magnetic field.

Magnitude of magnetic force is: ${\vec{F}}_B=q(\vec{v}\times \vec{B})\sin \theta$

Force on a Charge in an Electric Field

• The electric force on a charge is simpler and derived from Coulomb’s Law.
• A charge q in an electric field $\vec{E}$ experiences a force: ${\vec{F}}_E=q\vec{E}$

Electric fields can accelerate both stationary and moving charges.

Now consider a region containing both electric and magnetic fields. Since both forces act simultaneously and independently, the total force is:

$\vec{F}={\vec{F}}_E+{\vec{F}}_B$

$\vec{F}=q\vec{E}+q(\vec{v}\times \vec{B})$

$\vec{F}=q(\vec{E}+\vec{v}\times \vec{B})$

4.0Lorentz Force in Vector Form

$\vec{v}\times \vec{B}\left[\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ v_x& v_y& v_z\\ B_x& B_y& B_z\end{array}\right]$

$\vec{F}=q\left[\begin{array}{c}{E}_x+(v_y{B}_z-v_z{B}_y)\\ E_y+(v_z{B}_x-v_x{B}_z)\\ E_z+(v_x{B}_y-v_y{B}_x)\end{array}\right]$

Direction of the Lorentz Force (Right-Hand Rule)

• Point your fingers in the direction of velocity $\vec{v}$
• Curl them toward the magnetic field $\vec{B}$
• Your thumb points in the direction of force for a positive charge.
• For a negative charge, the force direction is opposite.

5.0Velocity Selector

In the presence of both electric field $\vec{E}$ and magnetic field $\vec{B}$, the total force on a charged particle is $\vec{F}=q(\vec{E}+\vec{v}\times \vec{B})$.

• This is called the Lorentz force. By using both electric and magnetic fields, particles moving at specific velocities can be selected. J.J. Thomson applied this principle to measure the charge-to-mass ratio of electrons. The schematic of Thomson’s apparatus is shown in the figure.
• The electrons with charge q=-e and mass m are emitted from the cathode C and then accelerated toward slit A.Let the potential difference between A and C be $V_A-V_C=\Delta V$. The change in potential energy is equal to the external work done in accelerating the electrons. $\Delta U=W_{ext}=q\Delta V=-e\Delta V$. By energy conservation, the kinetic energy gained is$\Delta K=-\Delta U=\frac{m{v}^2}{2}$. Thus the speed of the electron is given by $v=\sqrt{\frac{2e\Delta V}{m}}$
• If the electrons further pass through a region where there exists a downward uniform electric field, the electrons, being negatively charged, will be deflected upward. However, if in addition to the electric field, a magnetic field directed into the page is also applied, then the electrons will move in a straight path.
• From the equation we see that when the condition for the cancellation of the two forces is given by eE = evB. Which implies $v=\frac{E}{B}$

In other words, only those particles with speed $v=\frac{E}{B}$ will be able to move in a straight line. Combining the two equations, we obtain$\frac{e}{m}=\frac{E^2}{2(\Delta V)B^2}$

By measuring E, ΔV and B, the change-to-mass ratio can be readily determined. The most precise measurement to date is $\frac{e}{m}=1.758820174(71)\times 10^{11}C/kg$

Illustration-1.A particle with charge q and mass m starts from the origin with an initial velocity $\vec{v}=v_0\hat{j}$. It moves under the influence of a uniform electric field $\vec{E}=E_0\hat{i}$ and a uniform magnetic field $\vec{B}=B_0\hat{i}$. After what time will the particle’s speed increase to $2v_0$?

Solution: Charged particle will moves in a Helical path

$\vec{F}=q\left(\vec{E}+\vec{v}\times \vec{B}\right)$

$\vec{a}=\frac{qE\hat{i}}{m}+\frac{q{v}_0\times B}{m}$

$v_x=\frac{qE}{b}t$

$v=\sqrt{v_x^2+v_0^2}$

$2v_0=\sqrt{{\left(\frac{qE}{m}t\right)}^2+v_0^2}$

$\sqrt{3}{v}_0=\frac{qE}{m}t\Rightarrow t=\frac{\sqrt{3}m{v}_0}{qE}$

6.0Applications of Lorentz Force

• Electric Motors and Generators: Magnetic Lorentz forces on current-carrying conductors create rotation in motors, while generators use the reverse effect to produce electricity.
• Particle Accelerators and Cyclotrons: Magnetic fields bend the paths of charged particles, and electric fields speed them up, enabling high-energy acceleration.
• Mass Spectrometry: Charged particles travel in curved paths in a magnetic field; the radius depends on mass, enabling precise mass separation.
• Hall Effect Devices: The Lorentz force generates a Hall voltage in conductors, allowing accurate measurement of magnetic field strength.
• Plasma and Fusion Systems: Magnetic fields confine and guide charged particles in plasmas—critical in tokamaks, MHD systems, and various industrial processes.