Electrostatics: Capacitors, and connecting them in parallel and serial

A micro-course in electrostatics covering the concept of capacitance, capacitors and connecting them in parallel and serial for high school students.

 

Capacitors

Capacitor is a two-terminal electrical component, made of two electrical conductors separated by a insulator (or dielectric); capacitor has capacity of storing electric energy in a electric field between its plates.

                                            A capacitor, attached to a battery.
 


When a capacitor is attached to a battery, an electric field develops in the space between two conductors, causing positive charge to collect on one plate, and negative charge to collect on the other plate. While the capacitor is charging, electric potential difference between its conductors, increases from zero to a maximum value v (voltage of battery), and the magnitude of electric charge on conductors, increases from zero to a maximum value 'q'.

Capacitance

At any given time through charging and after charging, the ratio of the magnitude of electric charge 'q' on each conductor to the electric potential difference 'v' between them, is given by,

where 'c' is a constant value called capacitance. Each capacitor is characterized by its capacitance.
The SI unit of capacitance is farad (F), where one farad is equal to one coulomb per volt (1 C/V).

Electric potential energy stored in a capacitor

While charging, when electric potential difference between two conductors of a capacitor is " v' " , by adding extra charge of "dq' " , electric potential energy of the capacitor varies as, 

So the total electric potential energy stored in capacitor after charging, is given by this equation,

but c=q'/v', so the equation above can be written as,

 

 

The simplest form of capacitor

The simplest form of capacitor is made of two conductive parallel plate with the same surface area. If the distance between the plates is 'd', and their surface area is 'A', the capacitance of this capacitor is given by,

where 'ε' is permittivity of insulator between two plate, and it is equal to ε₀ if there is nothing between these two plates (ε₀ is permittivity of vacuum, or air approximately).
Due to the uniform distribution of charge on these parallel plates, electric field between them is also uniform; so,

where 'E' is perpendicular to the plates, and its direction is from positive plate to the negative plate. 

Connecting Capacitors in Parallel

 

                    Capacitors with capacitance of  c₁ , c₂ , c₃ , ..., c_n , connected in parallel.


When capacitors with capacitance of c₁, c₂, c₃, ..., c_n, are connected in parallel, the electric potential difference between two terminals of them, is 'v'; so the electric charges stored on them are as follows:

The equivalent capacitance, c_eq , between points A and B, can be represented by,

where 'q' is the sum of electric charges stored in capacitors, i.e.,

So we have,

Connecting Capacitors in Serial

                            Capacitors with capacitance of  c₁ , c₂ , c₃ , ..., c_n , connected in serial.



When capacitors with capacitance of c₁, c₂, c₃, ..., c_n, are connected in serial, the electric potential difference between two terminals of them, are v₁, v₂, v₃, ..., v_n, but electric charges stored on them, are the same. So we have,

The equivalent capacitance, c_eq, between points A and B, can be represented by,

where 'v' is the sum of electric potential drops on capacitors, given by,

So we have,