Magnetism and Moving Charges

Overview

This lecture provides a comprehensive overview of moving charges and magnetism, covering key experiments, laws, formulas, and applications essential for exams, including numerical problem-solving strategies.

Introduction to Moving Charges & Magnetism

• Moving charges generate magnetic fields, transitioning from electrostatics to electrodynamics.
• Three effects of current through a conductor: heating, chemical (covered in Chemistry), and magnetic (focus of this chapter).

Oersted’s Experiment

• Hans Christian Oersted discovered that a current-carrying conductor produces a magnetic field detectable with a compass.
• When current flows, the compass needle deflects, proving the presence of a magnetic field.

Biot–Savart Law

• Biot–Savart Law quantifies the magnetic field produced by a small current element.
• Magnetic field (dB) ∝ current (I), length element (dl), and sinθ, and inversely ∝ square of distance (r²).
• Vector form: dB = (μ₀/4π) * (I dl × r̂)/r², direction by right-hand rule.
• The magnetic field is always perpendicular to both the direction of current and position vector.

Magnetic Field Due to Current-Carrying Conductor

• For a straight wire: B = (μ₀I)/(2πr).
• For a circular loop at center: B = (μ₀I)/(2r); with n turns, B = (μ₀nI)/(2r).
• On the loop’s axis: B = (μ₀nIr²)/(2(r² + a²)^(3/2)), a = distance from center.

Ampere’s Circuital Law

• Line integral of magnetic field around a closed loop equals μ₀ times net current enclosed.
• Used to derive magnetic field for straight wires and inside solenoids.

Magnetic Field Due to Solenoid

• Inside a long solenoid: B = μ₀nI, where n = number of turns per unit length.
• Magnetic field inside is uniform; outside is nearly zero.

Motion of Charged Particle in Magnetic Field

• Force on a moving charge: F = q(v × B).
• If velocity is parallel to B, force = 0; if perpendicular, moves in a circle (radius r = mv/qB).
• Time period T = 2πm/qB; frequency f = qB/2πm.
• For arbitrary angle, path is helical.

Force on a Current-Carrying Conductor

• F = I (L × B), where L = length vector of conductor.
• Maximum force when conductor is perpendicular to magnetic field.

Force Between Parallel Current-Carrying Wires

• Force per unit length: F/l = (μ₀I₁I₂)/(2πd).
• Parallel currents attract; opposite currents repel.
• Definition of 1 ampere: current that produces 2 × 10⁻⁷ N/m force between wires 1 m apart.

Torque on a Current Loop (Magnetic Dipole)

• A current loop in a magnetic field experiences torque: τ = nIAB sinθ.
• Net force on loop is zero; only torque acts to rotate it.

Moving Coil Galvanometer

• Measures small currents via deflection of a coil in a magnetic field.
• Principle: current-carrying loop in magnetic field experiences torque.
• Sensitivity increased by higher magnetic field, more turns, larger area, and lower spring constant.
• Current sensitivity: deflection per unit current; voltage sensitivity: deflection per unit voltage.

Conversion of Galvanometer

• To ammeter: connect low-resistance shunt parallel to galvanometer.
• To voltmeter: connect high resistance in series with galvanometer.

Key Terms & Definitions

• Electrodynamics — Study of moving charges and associated magnetic fields.
• Oersted’s Experiment — Showed current creates a magnetic field.
• Biot–Savart Law — Formula for magnetic field due to a small current element.
• Ampere’s Circuital Law — Relates magnetic field in a closed loop to enclosed current.
• Solenoid — Coil producing uniform magnetic field inside.
• Galvanometer — Device to detect and measure small currents.
• Shunt — Low-resistance used for ammeter conversion.
• Torque — Rotational effect produced by a force.

Action Items / Next Steps

• Review Shivdas’s book for problem practice on this chapter.
• Attempt numerical examples on Biot–Savart Law, solenoids, and force between wires.
• Revise derivation steps for key formulas (especially Biot–Savart, Ampere’s Law, force/torque expressions).
• Prepare definitions and differences between galvanometer, ammeter, and voltmeter.