Moving Charges and Magnetism

• Introduction

  • Magnetic Force (Origin and nature)
  • Oersted's Experiment (Electricity and magnetism link)

• Magnetic Force

  • Lorentz Force ($\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$)
  • Magnetic Force on Moving Charge ($\vec{F} = q(\vec{v} \times \vec{B})$)
    • Magnitude: $F = qvB \sin\theta$
    • Right-Hand Rule for direction
  • Motion in a Magnetic Field
    • Uniform Magnetic Field
    • Circular Path ($r = \frac{mv}{qB}$)
    • Helical Path (Uniform + parallel components)

• Motion in Combined Electric and Magnetic Fields

  • Velocity Selector ($v = \frac{E}{B}$)
  • Cyclotron
    • Principle and working
    • Frequency of oscillator ($f = \frac{qB}{2\pi m}$)
    • Limitations (Relativistic effects)

• Magnetic Field Due to Current Element: Biot-Savart Law

  • Biot-Savart Law Vector Form ($d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$)
  • Permeability of Free Space ($\mu_0$)
  • Application: Circular Current Loop
    • Field at the center ($B = \frac{\mu_0 I}{2R}$)
    • Field along the axis ($B = \frac{\mu_0 I R^2}{2(x^2 + R^2)^{3/2}}$)

• Ampere’s Circuital Law

  • Statement ($\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enclosed}}$)
  • Applications
    • Long Straight Wire ($B = \frac{\mu_0 I}{2\pi r}$)
    • Solenoid ($B = \mu_0 n I$)
    • Toroid ($B = \mu_0 n I$)

• Force Between Two Parallel Currents

  • Parallel Currents
    • Same direction (Attractive force)
    • Opposite direction (Repulsive force)
  • Definition of Ampere (SI unit of current)

• Torque on Current Loop & Magnetic Dipole

  • Torque on Rectangular Loop ($\tau = NIAB \sin\theta$ or $\vec{\tau} = \vec{M} \times \vec{B}$)
  • Magnetic Dipole Moment ($M = NIA$)
  • Current Loop as Magnetic Dipole

• Moving Coil Galvanometer

  • Principle (Torque on current loop in magnetic field)
  • Construction & Working
    • Radial magnetic field
    • Restoring torque ($k\theta = NIAB$)
  • Current Sensitivity ($IS = \frac{\theta}{I} = \frac{NAB}{k}$)
  • Voltage Sensitivity ($VS = \frac{\theta}{V} = \frac{NAB}{kR}$)
  • Conversion of Galvanometer
    • Ammeter (Using a small shunt resistance $S$)
    • Voltmeter (Using a high series resistance $R$)