Capacitance

"Capacitor": stores electric charge

                         electric potential energy

Parallel plate capacitor

Two plates of area A, spacing d, with equal and opposite charge ±Q

Each plate is a conductor, at constant potential. Potential difference between the plates is V. Net charge is zero, field outside is small.

The charge Q and the potential difference V are proportional:

Q = CV

The constant C is "capacitance"

Unit: 1 Coulomb/Volt = 1 Farad  (F)

After M. Faraday, 1830

Circuit symbol:  

Typical circuit values mF or pF, large capacitors needed in power supplies.

The electric field between plates is uniform :

     (magnitude)

where s is the surface charge density Q/A

(This equation is derived by integration or Gauss's Law - not discussed here)

The potential difference between the plates:

Capacitance C = Q/V

Parallel plate capacitor capacitance depends on area and plate separation.

C measures capacity: the amount of charge (and energy)  per Volt (potential difference across the plates) the device can hold.

For large C, we need area A large and separation d small.

Any arrangement with two plates can form a capacitor: cylindrical or spherical.

In each case, the capacitance depends on geometry of the arrangement.

Example: The plates of parallel -plate capacitor in vacuum are 5 mm apart and 2 m² in area A. Potential difference of 10 kV is applied across the capacitor. Find:

a)      the capacitance

b)      the charge on each plate

c)      the magnitude of the electric field between the plates

Capacitors in series and in parallel

Equivalent capacitance of a combination of capacitors.

Two capacitors in parallel

Potential difference V appears across both capacitors:

Q₁ = C₁V

Q₂ = C₂V

The total charge Q = Q₁  +  Q₂

Q = (C₁ +C₂) V

Q = CV

where total capacitance

C = C₁ + C₂

Capacitances add for capacitors in parallel.

Two capacitors in series

Conservation of charge requires that the net charge be zero around the circuit and on each isolated segment.

Q₁ = Q₂ = Q  (charge drawn from the battery)

Total voltage V = V₁ + V₂

Also Q = C₁V₁ = C₂V₂

 

Capacitances  in series add as reciprocals.

Energy in a capacitor

To charge up a capacitor requires work to be done and energy supplied. Charge must be transported against the potential difference. Consider small amount of charge dq transferred through potential V:

Work done :   dW = Vdq

Total work done

Final potential energy  (U = W)

since Q = CV

The potential energy may be used when the capacitor is discharged across an external resistance. The energy is stored in the electric field between the plates.

Energy density

We can work out the energy per unit volume stored in the field:

but

,      V  = Ed

depends only on the square of the electric field strength. This formula holds generally, for the energy density in an electric field.