PHYS 251 Lecture - RC Circuits / Magnetic Fields

Capacitor Concepts

  • Basic Formulas for Capacitors:
    • Energy stored in a capacitor:
      U = rac{1}{2}C( riangle V)^2
    • Charge on capacitor:
      Q = C riangle V
    • Capacitance formula:
      C = rac{oldsymbol{ ext{ε}}_0 oldsymbol{ ext{κ}}A}{d}

Capacitor Behavior

  • Charging a Capacitor:

    • Connected to a battery of voltage V0 , the capacitor collects charge Q until: Q = C V0
    • Voltage across capacitor:
      riangle VC = V0

  • Discharging a Capacitor:

    • Upon closing a switch, the charged capacitor discharges through a resistor:
    • Current during discharge:
      i = rac{V_0}{R}
    • Capacitor behaves like a wire when fully charged and blocks current when uncharged.

Time Constant ( \tau )

  • Definition:

    • Time constant \tau indicates the rate of charging or discharging:
      \tau = RC

  • Charging Process:

    • After one time constant, capture 63.2% of the maximum charge:
    • Voltage across goes to:
      VC(t) = V0(1 - e^{-t/\tau})

  • Discharging Process:

    • After one time constant, drop to 36.8% of maximum voltage:
    • Voltage across goes to:
      VC(t) = V0 e^{-t/\tau}

  • Significance of Multiple ( \tau ):

    • C is considered essentially (dis)charged after 5 time constants (less than 1% charge remains).

Charge and Current Dynamics

  • Charge Dynamics:
    • Current only flows during the charging or discharging phase, reflecting the change in charge over time.
    • The relationship:
      i(t) = \left| \frac{\Delta q}{\Delta t} \right|

Magnetism Basics

  • Introduction to Magnetism:

    • Earth's surface magnetic field: \sim 100 \mu T = 1 \text{ Gauss}
    • Common magnets: Refrigerator magnet \sim 10 mT = 100 \text{ Gauss} , Electromagnet \sim 1 T , MRI magnets \sim 2 T .

  • Compass Behavior:

    • Compass behaves as a magnetic dipole aligning with magnetic fields (B fields).

Relationship of Electric Current and Magnetism

  • Key Observation:
    • Electric current creates magnetic fields, shown by compass needle deflection.
  • Right-Hand Rule (RHR):
    • Thumb indicates current direction; fingers show magnetic field direction around the wire.

Current and Magnetic Fields

  • Vector Addition in Magnetic Fields:
    • Magnetic fields generated by currents add as vectors; direction determines effect on a compass.

Visualization of Magnetic Fields

  • Field Representation:
    • Various notations represent magnetic field directions:
    • Out of Page: O, Into Page: X
  • Examples of representing fields and understanding interactions between them based on current direction.