Lecture Notes: Capacitors, Circuits, and Introduction to Magnetism

Quiz Solution: Capacitors and Circuits

• Capacitors in Circuit:

  • Two capacitors with equal charge but opposite polarity.
  • Charge: +Q and -Q.
  • Initial net charge inside a surface is zero.
  • Final state: capacitors uncharged.

• Current in the Circuit:

  • Current (I) must be the same throughout the circuit.
  • Denote charges on capacitors as Q1(t) and Q2(t).
  • Current I(t) = -dQ1(t)/dt = dQ2(t)/dt.

• Kirchhoff’s Voltage Law (KVL):

  • Potential differences calculated across capacitors and resistor:
    • +Q2(t)/C for first capacitor.
    • -Q1(t)/C for second capacitor.
    • -I(t) * R across the resistor.
  • Both capacitors are initially related and have equal charges (Q1 = Q2).

• Differential Equation:

  • Derived equation: 2Q1(t)/C = I(t) * R.
  • Solve for Q1(t): Q1(t) = Qe^(-2t/RC).

• Current as a Function of Time:

  • I(t) = -dQ1/dt = 2Q/RC * e^(-2t/RC).
  • Current flows counterclockwise.

• Time Constant:

  • Time constant τ = RC/2, important for determining charge decay.
  • Two capacitors in series have equivalent capacitance of C/2.*

Introduction to Magnetism

History and Basics

• Origin:

  • Magnetism discovered in ancient Magnesia (modern-day Manisa).
  • Stones attracted each other (magnets).

• Magnetic Poles:

  • Magnets have two poles: North and South.
  • Attempt to isolate poles results in smaller magnets, not monopoles.

• Magnetic Field Creation:

  • Current creates magnetic fields.
  • Magnetic fields apply force on currents.

• Electromagnetic Waves:

  • Changing magnetic fields induce electric fields and vice versa.
  • Electromagnetic waves consist of both electric and magnetic fields.

• Relativity and Unification:

  • Electric and magnetic fields depend on observer's velocity.
  • Einstein's contribution to understanding fields.

Magnetic Field and Forces

• Definition of Magnetic Field (B):

  • Acts on moving charges.
  • Force on a charge: F = q(v x B).
  • Force is perpendicular to velocity and magnetic field.

• Units of Magnetic Field:

  • Tesla (T) as the unit of magnetic field.
  • Earth's magnetic field is about 1 Gauss = 10^-4 T.

Motion of Charged Particles in Magnetic Field

• Circular Motion:

  • Charged particles in magnetic fields undergo circular motion.
  • Radius (r) of motion: r = mv/qB.
  • Period (T) of motion: T = 2πm/qB, independent of velocity.

• Cyclotron Frequency:

  • ω_c = qB/m, frequency independent of velocity.
  • Cyclotron, the first particle accelerator, used this principle.

Interesting Applications and Facts

• Magnetism in Materials:

  • Permanent magnets and trial-and-error in producing strong magnets.
  • Quantum mechanics necessary for understanding magnetism at atomic levels.

• Practical Considerations:

  • Large electromagnets needed for high Tesla magnetic fields.
  • Cyclotron as a historical and practical application of magnetic field principles.

The lecture transitioned from solving a circuit problem involving capacitors and resistors to introducing magnetism and its principles, including historical context, physical laws, and applications.