MOVING CHARGES AND MAGNETISM

Notes on Moving Charges and Magnetism

4.1 Introduction to Moving Charges and Magnetism

• Basic Understanding: Electricity and magnetism, though recognized for millennia, were unified in the 19th century with Oersted’s discovery that electric currents create magnetic fields. This realization bridged the gap between electricity and magnetism.
• Key Historical Figures:
  • Hans Christian Oersted observed the deflection of a compass needle near a current-carrying wire.
  • James Clerk Maxwell united electricity and magnetism in 1864, leading to the understanding of electromagnetic waves and innovations in technology.
  • Hertz, Bose, and Marconi made significant developments in radio waves.

4.2 Magnetic Force and Fields

4.2.1 Sources and Fields

• Electric Field Recap: An electric field (E) generated by a charge (Q) can exert a force (F) on another charge (q). The equation is given by:
  [ F = qE = \frac{qQ}{4\pi \epsilon_0 r^2} ]
• Magnetic Field Definition: The magnetic field (B) is similar to electric fields but emerges from moving charges (currents). It is a vector field defined at each point in space, capable of changing over time and obeying the principle of superposition.

4.2.2 Lorentz Force

• The force (F) on a point charge (q) in the electric (E) and magnetic (B) fields is expressed as: [ F = q[E + v \times B] ]
  • The magnetic force is perpendicular to both the velocity (v) of the charge and the magnetic field (B).
  • The vector product impacts the magnitude and direction of the force.

4.2.3 Force on Current-Carrying Conductors

• The force experienced by a current-carrying wire in a magnetic field can be expressed as: [ F = I l \times B ]
  where I is the current, l is the length vector of the wire, and B is the magnetic field vector.

4.3 Motion in a Magnetic Field

• Circular Motion: A charged particle moving perpendicular to a uniform magnetic field describes a circular path due to the centripetal force provided by the magnetic force.
  • The radius (r) of this circular path can be calculated using: [ r = \frac{mv}{qB} ]
  • The angular frequency of rotation is given by: [ \omega = \frac{qB}{m} ]
• If there’s a component of motion along the magnetic field, the particle travels in a helical motion.

4.4 Biot-Savart Law and Magnetic Field due to a Current

• Biot-Savart Law: Describes the magnetic field generated by an electric current. The law states: [ dB = \frac{\mu_0 I}{4\pi r^2} , dl \times \hat{r} ]
  where dB is the magnetic field produced by an element of current (I), at a distance (r).
• This leads to the integration over a current distribution to find the total magnetic field.

4.5 Magnetic Field of Circular Currents

• The magnetic field at the center of a circular loop of radius R carrying current I is given by: [ B = \frac{\mu_0 I}{2R} ]

4.6 Ampère's Circuital Law

• Ampère's Circuital Law: Relates the integrated magnetic field around a closed loop to the electric current passing through the loop. [ \int B \cdot dl = \mu_0 I ]
• Simplified forms apply under conditions of symmetry, similar to Gauss's Law for electric fields.

4.7 Solenoids

• A solenoid is a long coil of wire. The magnetic field within a long solenoid is uniform and proportional to the current and turns per unit length: [ B = \mu_0 n I ]
  where n is the number of turns per unit length.

4.8 Force Between Parallel Currents

• Parallel currents in wires attract each other, while anti-parallel currents repel each other. This principle is crucial in defining the ampere, the unit of current.

4.9 Torque on Current Loops

• A current loop placed in a magnetic field experiences a torque trying to align it with the field. The torque (τ) is given by: [ \tau = m \times B ]
  where m is the magnetic moment of the loop.

4.10 Moving Coil Galvanometer

• A galvanometer can measure current and voltages due to the torque acting on its coil within a magnetic field. It can be modified into an ammeter or voltmeter using shunt or series resistors, respectively.

Summary of Key Concepts

• The chapter concludes with definitions, equations, and physical quantities relevant to the study of electromagnetism, specifically focusing on the interaction between moving charges and magnetic fields.

1. Oersted's Discovery: Electric current creates a magnetic field.
2. Lorentz Force: Expresses the force on a charge in electric and magnetic fields.
3. Biot-Savart Law: Relates the magnetic field to current elements.
4. Ampère's Law: Integrates the magnetic field around a loop to the current enclosed.
5. Magnetic Field in Solenoid: Given by B = μ₀nI; uniform inside a long solenoid.
6. Force Between Currents: Like currents attract, antiparallel repel; defines the ampere.
7. Moving Coil Galvanometer: Device to measure current/voltage using torque in a magnetic field.
8. Magnetic Moment: For a loop, m = NIA; determines torque in a magnetic field.
9. Centripetal Motion: Charged particles move in circular paths in magnetic fields.