Capacitance Equations to Know for Electromagnetism I

Capacitance is key in understanding how capacitors store electric charge. These notes cover essential equations, including how geometry and materials affect capacitance, energy storage, and the role of dielectrics, all crucial concepts in Electromagnetism I.

1. Capacitance definition: C = Q/V

   • Capacitance (C) measures a capacitor's ability to store charge (Q) per unit voltage (V).
   • It is expressed in farads (F), where 1 F = 1 C/V.
   • A higher capacitance indicates a greater ability to store charge.

2. Parallel plate capacitor: C = εA/d

   • The capacitance of a parallel plate capacitor depends on the area (A) of the plates and the distance (d) between them.
   • ε (epsilon) represents the permittivity of the material between the plates.
   • Increasing plate area or decreasing distance increases capacitance.

3. Capacitors in parallel: C_total = C1 + C2 + C3 + ...

   • In a parallel configuration, the total capacitance is the sum of individual capacitances.
   • This arrangement allows for increased total charge storage.
   • The voltage across each capacitor remains the same.

4. Capacitors in series: 1/C_total = 1/C1 + 1/C2 + 1/C3 + ...

   • In a series configuration, the total capacitance is less than the smallest individual capacitor.
   • The charge (Q) is the same across all capacitors, but the voltage divides among them.
   • This arrangement reduces the overall capacitance.

5. Energy stored in a capacitor: U = 1/2 CV^2

   • The energy (U) stored in a capacitor is proportional to the capacitance and the square of the voltage.
   • This equation shows that energy increases significantly with higher voltage.
   • Energy is measured in joules (J).

6. Cylindrical capacitor: C = 2πεL / ln(b/a)

   • The capacitance of a cylindrical capacitor depends on the length (L) and the logarithm of the ratio of the outer (b) and inner (a) radii.
   • This formula is useful for capacitors with cylindrical geometry.
   • The permittivity (ε) of the material between the cylinders also affects capacitance.

7. Spherical capacitor: C = 4πε / (1/a - 1/b)

   • The capacitance of a spherical capacitor is determined by the radii of the inner (a) and outer (b) spheres.
   • This formula highlights the relationship between geometry and capacitance.
   • The permittivity (ε) of the medium between the spheres is also a factor.

8. Dielectric effect on capacitance: C = κC0

   • The presence of a dielectric material increases capacitance by a factor of κ (kappa), the dielectric constant.
   • Dielectrics reduce the electric field within the capacitor, allowing more charge storage.
   • Different materials have different dielectric constants, affecting capacitance.

9. Capacitance of parallel plates with dielectric: C = κε0A/d

   • This equation modifies the parallel plate capacitance formula to include the dielectric constant (κ).
   • ε0 is the permittivity of free space, a constant value.
   • The dielectric increases capacitance compared to a vacuum or air-filled capacitor.

10. Gauss's law for capacitors: ΦE = Q/ε0

• Gauss's law relates the electric flux (ΦE) through a closed surface to the charge (Q) enclosed.
• ε0 is the permittivity of free space, fundamental in electrostatics.
• This law is essential for understanding electric fields around capacitors.