CBSE Class 12 Physics Revision Notes: Moving Charges And Magnetism

Key Concepts to Remember

• Lorentz Force: The total force experienced by a charged particle moving in both electric and magnetic fields (F = q(E + v x B)).
• Right-Hand Thumb Rule: Used to determine the direction of the magnetic field produced by a current-carrying wire.
• Fleming's Left-Hand Rule: Used to determine the direction of force on a current-carrying conductor placed in an external magnetic field.
• Biot-Savart Law: A fundamental law describing the magnetic field generated by a steady current.
• Ampere's Circuital Law: Provides an alternative, often simpler, way to calculate magnetic fields for symmetric current distributions.
• Magnetic Force does no Work: The magnetic force is always perpendicular to the velocity of the charged particle, hence W = F · dl = Fv cos90° = 0.
• Magnetic Dipole Moment: Represents the strength and orientation of a magnetic source; for a current loop, M = IAN.
• Cyclotron Principle: Charged particles can be accelerated to high energies using perpendicular electric and magnetic fields.
• Moving Coil Galvanometer: Detects and measures small electric currents based on the torque experienced by a current loop in a magnetic field.

Essential Definitions

Magnetic Field (B)

A region around a moving electric charge or a magnet where another moving charge or magnet experiences a magnetic force. Its SI unit is Tesla (T).

Lorentz Force

The total electromagnetic force on a point charge due to an electromagnetic field. It is the sum of the electric force (qE) and the magnetic force (q(v x B)).

Biot-Savart Law

A law that gives the magnetic field dB at a point due to a small current element Idl at a distance r from it: dB = (μ₀/4π) * (Idl sinθ / r²).

Ampere's Circuital Law

States that the line integral of the magnetic field B around any closed loop is equal to μ₀ times the net current I_enc passing through the area enclosed by the loop: ∮ B · dl = μ₀I_enc.

Magnetic Dipole Moment (M)

A vector quantity that measures the strength and orientation of a magnetic source. For a current loop, M = NIA (where N is number of turns, I is current, A is area), with direction given by the right-hand rule.

Cyclotron

A particle accelerator that uses a rapidly alternating electric field and a constant magnetic field to accelerate charged particles to very high kinetic energies.

Velocity Selector

A device that uses perpendicular electric and magnetic fields to select charged particles moving at a specific velocity, allowing them to pass undeflected.

The Lorentz Force: Unifying Electric and Magnetic Effects

The Lorentz force is a fundamental concept in electromagnetism, describing the total force experienced by a charged particle moving within an electromagnetic field. It elegantly combines both electric and magnetic forces into a single expression: F = q(E + v x B). Here, q is the charge of the particle, E is the electric field, v is the velocity of the particle, and B is the magnetic field.

The electric part of the force, F_E = qE, acts in the direction of the electric field (for positive charges) and is independent of the particle's velocity. This force can do work on the particle, changing its kinetic energy.

In contrast, the magnetic force, F_M = q(v x B), behaves quite differently. It is always perpendicular to both the velocity vector (v) and the magnetic field vector (B). This perpendicularity has crucial implications: the magnetic force never does any work on the charged particle (W = F_M · v dt = 0) because the force is always perpendicular to the displacement. Consequently, the magnetic force cannot change the kinetic energy or speed of the particle; it can only change its direction of motion. The direction of the magnetic force is determined by the right-hand rule for cross products (or Fleming's Left-Hand Rule for current-carrying wires). For example, if a positive charge moves perpendicular to a uniform magnetic field, the magnetic force provides the necessary centripetal force, causing the particle to move in a circular path. If the velocity has a component parallel to the magnetic field, the path becomes helical. Understanding the Lorentz force is key to analyzing the motion of charged particles in various fields, forming the basis for devices like cyclotrons and mass spectrometers.

Magnetic Fields due to Currents

Motion of a Charged Particle in a Magnetic Field

1. — When a charged particle (charge q, mass m) enters a uniform magnetic field B perpendicularly (v ⊥ B), the magnetic force F_M = qvB acts as the centripetal force. This results in circular motion. Radius of circular path: r = mv / qB Angular frequency (cyclotron frequency): ω = qB / m Time period: T = 2πm / qB Frequency: f = qB / 2πm
2. — If the velocity v makes an angle θ with B, resolve v into two components: v_parallel = v cosθ (parallel to B) and v_perpendicular = v sinθ (perpendicular to B). v_parallel component causes linear motion along the field direction (no force). v_perpendicular component causes circular motion in a plane perpendicular to B. The combined motion is a helical path. Radius of helix: r = mv_perpendicular / qB = (mv sinθ) / qB * Pitch of helix: The distance moved along the magnetic field direction in one rotation: p = v_parallel × T = (v cosθ) × (2πm / qB)
3. — In a region with both uniform electric field E and magnetic field B perpendicular to each other and to the particle's velocity v. Electric force: F_E = qE Magnetic force: F_M = qvB * For undeflected motion, F_E = F_M, so qE = qvB, which gives v = E/B. Only particles with this specific velocity pass through undeflected.
4. — Uses perpendicular electric and magnetic fields to accelerate charged particles. Magnetic field forces particles into circular paths within 'dees'. Electric field between the dees accelerates particles twice per cycle. Resonance condition: The frequency of the applied alternating voltage must match the cyclotron frequency: f_osc = f_cyclotron = qB / 2πm. Maximum kinetic energy: K_max = (q²B²R²) / 2m (where R is the radius of the dees).

Worked Examples

• {"description":"An electron moving with a velocity of 2 x 10⁶ m/s enters a uniform magnetic field of 0.2 T perpendicular to its path. Calculate the radius of the circular path it follows. (Given: mass of electron m_e = 9.1 x 10⁻³¹ kg, charge e = 1.6 x 10⁻¹⁹ C).","solution":"Given: v = 2 x 10⁶ m/s, B = 0.2 T, m_e = 9.1 x 10⁻³¹ kg, q = 1.6 x 10⁻¹⁹ C.\nSince v ⊥ B, the electron moves in a circle. The magnetic force provides the centripetal force:\nqvB = mv²/r\nr = mv / qB\nr = (9.1 x 10⁻³¹ kg 2 x 10⁶ m/s) / (1.6 x 10⁻¹⁹ C 0.2 T)\nr = (18.2 x 10⁻²⁵) / (0.32 x 10⁻¹⁹)\nr = 56.875 x 10⁻⁶ m = 5.69 x 10⁻⁵ m (approx)."}

Exam Traps & Key Learnings

Always pay close attention to the vector nature of magnetic field (B), velocity (v), and force (F). A common mistake is misinterpreting the direction of the force. Use the Right-Hand Rule (for current and B-field) or Fleming's Left-Hand Rule (for force on current) consistently. Remember that the magnetic force itself does no work, so it cannot change the speed of a charged particle, only its direction. When applying Ampere's Circuital Law, ensure your chosen Amperian loop is symmetric and encloses the current correctly. Practice drawing diagrams for force directions. For galvanometers, understand the concept of radial magnetic field and why it ensures uniform torque, making the scale linear.

Practice Questions with Solutions

• Q: What is the main difference in the work done by electric force and magnetic force on a moving charged particle? A: Electric force can do work and change the kinetic energy of a particle, while magnetic force does no work and only changes the direction of motion.
• Q: State the conditions under which a charged particle moving in a uniform magnetic field will follow a helical path. A: A helical path occurs when the charged particle's velocity vector has components both parallel and perpendicular to the magnetic field.
• Q: A current is flowing east along a long straight wire. What is the direction of the magnetic field directly above the wire? A: Using the Right-Hand Thumb Rule, if the current flows east, the magnetic field directly above the wire points North.
• Q: What is the principle behind a velocity selector? A: A velocity selector works on the principle that if electric and magnetic forces on a charged particle are equal and opposite, only particles with a specific velocity v = E/B will pass undeflected.

Frequently Asked Questions