Moving Charges and Magnetism

4.1 INTRODUCTION

Electricity and magnetism have been known as separate phenomena for more than 2000 years. However, it was only around 1820 that their intimate connection was discovered. Danish physicist Hans Christian Oersted, during a lecture demonstration, noticed that a current in a straight wire caused a deflection in a nearby magnetic compass needle. This deflection was tangential to an imaginary circle centered on the wire and lying in a plane perpendicular to the wire. Reversing the current reversed the needle's orientation. The deflection increased with current magnitude and proximity of the needle to the wire. Iron filings sprinkled around the wire arranged themselves in concentric circles, revealing the magnetic field pattern. Oersted concluded that moving charges or currents produce a magnetic field in the surrounding space. Following Oersted's discovery, intense experimentation led to the unification of electricity and magnetism by James Maxwell in 1864, who also realized that light is an electromagnetic wave. Radio waves were discovered by Hertz and produced by J.C. Bose and G. Marconi by the end of the 19th century. The 20th century saw remarkable scientific and technological progress due to the understanding of electromagnetism and the invention of devices for electromagnetic wave production, amplification, transmission, and detection. This chapter explores how magnetic fields exert forces on moving charged particles and current-carrying wires, how currents produce magnetic fields, the acceleration of particles in cyclotrons, and the detection of currents and voltages using galvanometers. The convention adopted is that a current or field emerging out of the plane of the paper is depicted by a dot (⊙), and going into the plane by a cross (⊗). Figures 4.1(a) and 4.1(b) illustrate these two situations respectively.

• Electricity and magnetism were initially considered separate phenomena for over 2000 years.
• Oersted discovered in 1820 that electric currents produce magnetic fields affecting compass needles.
• Magnetic field lines around a current-carrying wire form concentric circles perpendicular to the wire.
• Maxwell unified electricity and magnetism and identified light as an electromagnetic wave.
• Radio waves were discovered and produced by Hertz, Bose, and Marconi.
• The chapter introduces forces on moving charges, magnetic fields due to currents, cyclotron acceleration, and galvanometers.
• 📌 Magnetic field: A vector field produced by moving charges or currents, influencing magnetic compass needles.
• 📌 Lorentz force: The force on a charged particle moving in electric and magnetic fields (to be detailed later).
• 📌 Electromagnetic waves: Waves consisting of oscillating electric and magnetic fields, including light and radio waves.

4.2 MAGNETIC FORCE

This section begins by recalling the electric field E produced by a static charge Q, given by E = Q r̂ / (4πε₀ r²), where r̂ is the unit vector along the displacement vector r. The force on a test charge q in this field is F = qE. The electric field is a vector field defined at each point in space and can vary with time. It obeys the principle of superposition, meaning fields due to multiple charges add vectorially. Analogously, moving charges or currents produce a magnetic field B(r), also a vector field obeying superposition. The force on a charge q moving with velocity v in the presence of electric field E and magnetic field B is given by the Lorentz force: F = q [E(r) + v × B(r)] = F_electric + F_magnetic The magnetic force depends on the charge q, velocity v, and magnetic field B. It is perpendicular to both v and B, given by the right-hand rule for the vector cross product. The force is zero if the velocity is parallel or antiparallel to B or if the charge is stationary. The SI unit of magnetic field B is the tesla (T), defined as the magnetic field that exerts a force of one newton on a charge of one coulomb moving at one meter per second perpendicular to the field. The smaller unit gauss (G) is also used, where 1 G = 10⁻⁴ T. The Earth's magnetic field is about 3.6 × 10⁻⁵ T. Extending this to a current-carrying conductor, the force on a straight rod of length l carrying current I in a magnetic field B is F = l × B, where l is a vector along the direction of current with magnitude l. For wires of arbitrary shape, the total force is the vector sum (integral) of forces on infinitesimal elements.

• Electric field E due to a static charge Q is given by E = Q r̂ / (4πε₀ r²).
• Magnetic field B is produced by moving charges or currents and is a vector field.
• Lorentz force on a charge q moving with velocity v in fields E and B is F = q [E + v × B].
• Magnetic force is perpendicular to both velocity and magnetic field; zero if velocity is parallel to B or charge is stationary.
• Unit of magnetic field is tesla (T); 1 T causes 1 N force on 1 C charge moving at 1 m/s perpendicular to B.
• Force on a current-carrying conductor in magnetic field is F = l × B, where l is vector along current.
• 📌 Lorentz force: Total force on a charge moving in electric and magnetic fields.
• 📌 Magnetic field (B): Vector field produced by moving charges or currents.
• 📌 Right-hand rule: Method to determine direction of magnetic force or field.