Lorentz Force — Revision Notes

⚡ 30-Second Revision

Lorentz Force: — $\vec{F}_L = q(\vec{E} + \vec{v} \times \vec{B})$
Electric Force: — $\vec{F}_E = q\vec{E}$ (acts on stationary/moving charges, can do work)
Magnetic Force: — $\vec{F}_M = q(\vec{v} \times \vec{B})$ (acts only on moving charges, does NO work)
Magnitude of Magnetic Force: — $F_M = |q|vB\sin\theta$
Direction of $\vec{F}_M$: — Right-Hand Rule (for positive $q$), perpendicular to $\vec{v}$ and $\vec{B}$.
Circular Path Radius: — $r = \frac{mv}{qB}$ (for $\vec{v} \perp \vec{B}$)
Circular Path Period: — $T = \frac{2\pi m}{qB}$
Cyclotron Frequency: — $f = \frac{qB}{2\pi m}$
Velocity Selector Condition: — $v = E/B$ (for undeflected motion when $\vec{E} \perp \vec{B} \perp \vec{v}$)
Helical Path: — Occurs when $\vec{v}$ is at an angle (not $0^circ$ or $90^circ$) to $\vec{B}$.

2-Minute Revision

The Lorentz force is the total electromagnetic force on a charged particle, combining the electric force ($q\vec{E}$) and the magnetic force ($q(\vec{v} \times \vec{B})$). The electric force acts on any charge in an electric field, regardless of its motion, and can change its speed.

The magnetic force, however, is unique: it only acts on *moving* charges in a magnetic field. Crucially, this magnetic force is always perpendicular to the particle's velocity, meaning it does no work and therefore cannot change the particle's speed or kinetic energy, only its direction.

Its magnitude is $qvB\sin\theta$, where $\theta$ is the angle between velocity and magnetic field. For a particle moving perpendicular to a uniform magnetic field, it follows a circular path with radius $r = mv/qB$ and period $T = 2\pi m/qB$.

If the velocity is at an angle, the path becomes a helix. In a velocity selector, perpendicular electric and magnetic fields are used to select particles with a specific velocity $v=E/B$ by balancing the two forces.

5-Minute Revision

The Lorentz force is the fundamental expression for the force experienced by a charged particle in an electromagnetic field, given by $\vec{F}_L = q(\vec{E} + \vec{v} \times \vec{B})$. This equation has two distinct components.

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Electric Force ($\vec{F}_E = q\vec{E}$): — This force acts on any charged particle $q$ placed in an electric field $\vec{E}$, irrespective of whether the particle is moving or stationary. Its direction is parallel to $\vec{E}$ for positive charges and anti-parallel for negative charges. The electric force can do work on the particle, thereby changing its kinetic energy and speed.

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Magnetic Force ($\vec{F}_M = q(\vec{v} \times \vec{B})$): — This force is more specific. It only acts on a charged particle $q$ if it is *moving* with velocity $\vec{v}$ in a magnetic field $\vec{B}$. If the particle is stationary ($v=0$) or moving parallel/anti-parallel to the magnetic field ($\theta = 0^circ$ or $180^circ$), the magnetic force is zero. The direction of $\vec{F}_M$ is always perpendicular to both $\vec{v}$ and $\vec{B}$, determined by the right-hand rule for positive charges (or Fleming's left-hand rule). A critical property is that the magnetic force does no work on the particle ($W = \vec{F}_M \cdot \vec{d} = 0$), meaning it cannot change the particle's speed or kinetic energy; it only alters its direction of motion.

Motion of Charged Particles:

**In a uniform magnetic field ($\vec{E}=0$):**

* If $\vec{v} \parallel \vec{B}$ or $\vec{v} \parallel -\vec{B}$: Path is a straight line (no magnetic force). * If $\vec{v} \perp \vec{B}$: Path is a circle. The magnetic force $qvB$ provides the centripetal force $mv^2/r$.

Thus, radius $r = \frac{mv}{qB}$, period $T = \frac{2\pi m}{qB}$, and frequency $f = \frac{qB}{2\pi m}$. Note $T$ and $f$ are independent of $v$ and $r$. * If $\vec{v}$ makes an angle $\phi$ with $\vec{B}$: The velocity component $v\cos\phi$ (parallel to $\vec{B}$) remains constant, while $v\sin\phi$ (perpendicular to $\vec{B}$) causes circular motion.

The combined effect is a helical path with pitch $p = (v\cos\phi)T$.

In combined uniform $\vec{E}$ and $\vec{B}$ fields (Velocity Selector): — If $\vec{E}$, $\vec{B}$, and $\vec{v}$ are mutually perpendicular, particles with a specific velocity $v = E/B$ will pass undeflected because the electric force ($qE$) balances the magnetic force ($qvB$). Particles with other velocities will be deflected.

Example: An electron ($q=-e$) moves with speed $v$ perpendicular to a magnetic field $B$. What is its kinetic energy after half a revolution? Solution: The magnetic force does no work on the electron. Therefore, its kinetic energy remains constant throughout its motion. After half a revolution, its kinetic energy will be the same as its initial kinetic energy, $K.E. = \frac{1}{2}mv^2$. Only its direction of velocity changes.

Prelims Revision Notes

The Lorentz force is the total force on a charged particle in an electromagnetic field, $\vec{F}_L = q(\vec{E} + \vec{v} \times \vec{B})$. It's crucial to distinguish between its electric and magnetic components.

Electric Force ($\vec{F}_E = q\vec{E}$):

Acts on any charge (stationary or moving).
Direction: Parallel to $\vec{E}$ for positive $q$, anti-parallel for negative $q$.
Can do work, changing kinetic energy and speed.

Magnetic Force ($\vec{F}_M = q(\vec{v} \times \vec{B})$):

Acts *only* on moving charges ($v \neq 0$).
Direction: Perpendicular to both $\vec{v}$ and $\vec{B}$. Use Right-Hand Rule (for positive $q$) or Fleming's Left-Hand Rule.
Magnitude: $F_M = |q|vB\sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$.
Key Point: — Does NO work on the particle. Therefore, it does not change the particle's speed or kinetic energy, only its direction.
Zero magnetic force if: $q=0$, $v=0$, $B=0$, or $\vec{v} \parallel \vec{B}$ (i.e., $\theta = 0^circ$ or $180^circ$).

Motion of Charged Particles in Uniform Magnetic Field ($\vec{E}=0$):

Straight Line: — If $\vec{v} \parallel \vec{B}$ or $\vec{v} \parallel -\vec{B}$.
Circular Path: — If $\vec{v} \perp \vec{B}$.

* Radius: $r = \frac{mv}{qB}$ * Period: $T = \frac{2\pi r}{v} = \frac{2\pi m}{qB}$ * Frequency: $f = \frac{1}{T} = \frac{qB}{2\pi m}$ * Angular Frequency: $\omega = 2\pi f = \frac{qB}{m}$ * Note: $T, f, \omega$ are independent of $v$ and $r$.

Helical Path: — If $\vec{v}$ makes an angle $\phi$ with $\vec{B}$ (where $\phi \neq 0^circ, 90^circ, 180^circ$).

* $v_{\parallel} = v\cos\phi$ (constant velocity along $\vec{B}$) * $v_{\perp} = v\sin\phi$ (causes circular motion perpendicular to $\vec{B}$) * Radius of helix: $r = \frac{mv_{\perp}}{qB} = \frac{mv\sin\phi}{qB}$ * Pitch of helix: $p = v_{\parallel} T = (v\cos\phi) \frac{2\pi m}{qB}$

Velocity Selector:

Perpendicular $\vec{E}$, $\vec{B}$, and $\vec{v}$.
Condition for undeflected motion: Electric force balances magnetic force, $qE = qvB$.
Selected velocity: $v = E/B$.

Key Traps:

Magnetic force does *not* change speed/KE.
Direction rules are crucial; practice with positive and negative charges.
Units and powers of 10 in calculations.

Vyyuha Quick Recall

Father Mother Child: Force, Magnetic Field, Current (or velocity). Use Fleming's Left-Hand Rule: Thumb (Force), Forefinger (Field), Middle Finger (Current/Velocity). For positive charges, velocity is current direction. For negative charges (like electrons), force is opposite to what the rule gives.