Force on Moving Charge — Revision Notes

⚡ 30-Second Revision

Magnetic Force: — $\vec{F}_B = q(\vec{v} \times \vec{B})$
Magnitude: — $F_B = |q|vB\sin\theta$
Direction: — Right-Hand Rule (for positive $q$), reverse for negative $q$.
Zero Force: — If $\vec{v} = 0$ or $\vec{v} \parallel \vec{B}$ (i.e., $\theta = 0^\circ$ or $180^\circ$).
Maximum Force: — If $\vec{v} \perp \vec{B}$ (i.e., $\theta = 90^\circ$), $F_{max} = |q|vB$.
Work Done: — Magnetic force does NO work ($W=0$). Kinetic energy and speed are constant.
**Circular Path ($\vec{v} \perp \vec{B}$):**

* Radius: $r = \frac{mv}{|q|B}$ * Time Period: $T = \frac{2\pi m}{|q|B}$ * Frequency: $f = \frac{|q|B}{2\pi m}$

**Helical Path ($\vec{v}$ at angle $\theta$ to $\vec{B}$):**

* Pitch: $p = (v\cos\theta) T = \frac{2\pi m v \cos\theta}{|q|B}$

Velocity Selector: — For undeflected motion, $v = E/B$ (when $\vec{E} \perp \vec{B} \perp \vec{v}$).

2-Minute Revision

The magnetic force on a moving charge is a fundamental concept described by the Lorentz force law's magnetic component: $\vec{F}_B = q(\vec{v} \times \vec{B})$. This force acts only on moving charges and is always perpendicular to both the velocity $\vec{v}$ and the magnetic field $\vec{B}$.

Its magnitude is $F_B = |q|vB\sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$. The direction is found using the right-hand rule for positive charges, reversing it for negative charges.

Crucially, because the force is perpendicular to velocity, it does no work on the particle, meaning the particle's kinetic energy and speed remain constant; only its direction of motion changes.

Key scenarios to remember for NEET include:

1
Zero Force: — Occurs if the charge is stationary ($v=0$) or if its velocity is parallel or anti-parallel to the magnetic field ($\theta = 0^\circ$ or $180^\circ$).
2
Circular Motion: — If a charged particle enters a uniform magnetic field perpendicularly ($\theta = 90^\circ$), it follows a circular path. The radius is $r = mv/(|q|B)$, and the cyclotron frequency is $f = |q|B/(2\pi m)$. These formulas are vital.
3
Helical Motion: — If the velocity has components both parallel and perpendicular to the field, the particle follows a helical path. The pitch of the helix is $p = (v\cos\theta)T$.
4
Velocity Selector: — In regions with crossed electric and magnetic fields, particles with a specific velocity $v = E/B$ pass undeflected, as the electric and magnetic forces balance each other.

5-Minute Revision

The magnetic force on a moving charge is a cornerstone of electromagnetism, described by the magnetic component of the Lorentz force: $\vec{F}_B = q(\vec{v} \times \vec{B})$. Here, $q$ is the charge, $\vec{v}$ is its velocity, and $\vec{B}$ is the magnetic field.

The magnitude of this force is $F_B = |q|vB\sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$. The direction is critical and is determined by the right-hand rule for positive charges (fingers along $\vec{v}$, curl towards $\vec{B}$, thumb gives $\vec{F}$), with the direction reversed for negative charges.

Key Properties & Implications:

No Work Done: — Since $\vec{F}_B$ is always perpendicular to $\vec{v}$, the magnetic force does no work on the particle ($W = \vec{F}_B \cdot \vec{d} = 0$). This means the kinetic energy and speed of the particle remain constant; only its direction of motion changes.
Conditions for Force: — A magnetic force only acts if the charge is moving ($v \neq 0$) and its velocity has a component perpendicular to the magnetic field ($\sin\theta \neq 0$). If $\vec{v} \parallel \vec{B}$ (or anti-parallel), the force is zero.

Particle Trajectories in Uniform Magnetic Fields:

1
Circular Motion: — If a charged particle's velocity is perpendicular to a uniform magnetic field ($\theta = 90^\circ$), the magnetic force provides the necessary centripetal force, causing the particle to move in a uniform circular path. Equating $F_B = |q|vB$ with $F_c = mv^2/r$ gives the radius: $r = \frac{mv}{|q|B}$. The time period $T = \frac{2\pi r}{v} = \frac{2\pi m}{|q|B}$ and frequency $f = \frac{1}{T} = \frac{|q|B}{2\pi m}$ are independent of speed and radius, depending only on the charge-to-mass ratio and field strength.

* *Example:* An electron ($m_e = 9.1 \times 10^{-31}\,\text{kg}$, $q = -1.6 \times 10^{-19}\,\text{C}$) moves at $10^7\,\text{m/s}$ perpendicular to a $0.05\,\text{T}$ field. $r = \frac{(9.1 \times 10^{-31})(10^7)}{(1.6 \times 10^{-19})(0.05)} \approx 1.14\,\text{mm}$.

1
Helical Motion: — If $\vec{v}$ makes an angle $\theta$ with $\vec{B}$ (where $0 < \theta < 90^\circ$), the velocity component $v_\parallel = v\cos\theta$ moves along the field (no force), while $v_\perp = v\sin\theta$ causes circular motion. The combination is a helix. The pitch (distance traveled along $\vec{B}$ per revolution) is $p = v_\parallel T = (v\cos\theta) \frac{2\pi m}{|q|B}$.

Velocity Selector: In a region with perpendicular $\vec{E}$ and $\vec{B}$ fields, if a charged particle passes undeflected, the electric force $q\vec{E}$ balances the magnetic force $q(\vec{v} \times \vec{B})$. This implies $qE = qvB$, so $v = E/B$. Only particles with this specific velocity pass straight through.

Prelims Revision Notes

1
Magnetic Force Formula: — The force on a charge $q$ moving with velocity $\vec{v}$ in a magnetic field $\vec{B}$ is $\vec{F}_B = q(\vec{v} \times \vec{B})$.
2
Magnitude of Force: — $F_B = |q|vB\sin\theta$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$.
3
Direction of Force: — Use the Right-Hand Rule for $\vec{v} \times \vec{B}$. For positive charge, force is in thumb direction. For negative charge, force is opposite to thumb direction.
4
Zero Force Conditions:

* Charge is stationary ($v=0$). * Velocity is parallel or anti-parallel to the magnetic field ($\theta = 0^\circ$ or $180^\circ$).

1
Maximum Force Condition: — Velocity is perpendicular to the magnetic field ($\theta = 90^\circ$), $F_{max} = |q|vB$.
2
Work and Energy: — Magnetic force does NO work on the charged particle. Consequently, the kinetic energy and speed of the particle remain constant. Only the direction of velocity changes.
3
**Circular Motion (when $\vec{v} \perp \vec{B}$):**

* Magnetic force provides centripetal force: $|q|vB = \frac{mv^2}{r}$. * Radius of path: $r = \frac{mv}{|q|B}$. (Proportional to momentum, inversely proportional to charge and B-field). * Angular frequency (cyclotron frequency): $\omega = \frac{v}{r} = \frac{|q|B}{m}$. (Independent of $v$ and $r$). * Time period: $T = \frac{2\pi}{\omega} = \frac{2\pi m}{|q|B}$. * Frequency: $f = \frac{1}{T} = \frac{|q|B}{2\pi m}$.

1
**Helical Motion (when $\vec{v}$ makes angle $\theta$ with $\vec{B}$):**

* Resolve velocity: $v_\parallel = v\cos\theta$ (along B, no force), $v_\perp = v\sin\theta$ (perpendicular to B, causes circular motion). * Radius of helix: $r = \frac{m v_\perp}{|q|B} = \frac{m v \sin\theta}{|q|B}$. * Pitch of helix: $p = v_\parallel T = (v\cos\theta) \left(\frac{2\pi m}{|q|B}\right)$.

1
Velocity Selector: — For a charged particle to pass undeflected through perpendicular $\vec{E}$ and $\vec{B}$ fields, the electric force must balance the magnetic force: $qE = qvB \implies v = E/B$. The direction of $\vec{v}$ must be perpendicular to both $\vec{E}$ and $\vec{B}$ such that $\vec{E} = -(\vec{v} \times \vec{B})$.
2
Units: — Ensure all quantities are in SI units (C, m/s, T, kg, N) for calculations.

Vyyuha Quick Recall

Father Mother I (Force, Magnetic field, Current/Velocity) for Fleming's Left-Hand Rule. For the Right-Hand Rule for $\vec{v} \times \vec{B}$ direction: Very Big Force (Thumb for Force, Fingers for Velocity, Curl for B-field).