Question

What is the dimensional formula of capacitance?

Answer 1

Hint: Capacity is a system 's capability to store an electrical charge. It is given by the ratio of the increase in a system's electrical charge to the corresponding change in its electrical potential.

Complete step-by-step answer:

Capacitors consist of two parallel conductive plates, usually metal, which are prevented from touching each other and separated by a dielectric insulating layer. When an electrical current flow is applied to these pipes, charging one surface with a positive charge and the other plate with an equal and opposite negative charge proportional to the supply voltage. A condenser has the capacity to store an electrical charge Q of electrons.

The capacitor capacity to store this electric charge Q between its plates is proportional to the voltage V applied to a capacitor.

$Q = C \times V$
It can also be written as
Capacitance, C= Charge$\times$ $Voltag{e^{ - 1}}$ ---(i)
We know, Charge = Current $\times$ Time
Dimensional formula of charge is $\left[ {{I^1}{T^1}} \right]$---(ii)
Voltage = Electric Field $\times$ Distance ---(iii)
Also, Electric field = Force $\times$ $Charg {e^{ - 1}}$

The dimensional formula of force and charge is given by $\left[ {{M^1}{L^1}{T^{ - 2}}} \right]$
dimensional formula of charge is given by $\left[ {{I^1}{T^1}} \right]$

Hence the dimensional formula for Electric field is

$\left[ {{M^1}{L^1}{T^{ - 2}}} \right]$ $\times$ ${\left[ {{I^1}{T^1}} \right]^{ - 1}}$
=$\left[ {{M^1}{L^1}{T^{ - 3}}{I^{ - 1}}} \right]$--(iv)

In substituting (iv) and (iii) we get,

The dimensional formula of Voltage=$\left[ {{M^1}{L^1}{T^{ - 3}}{I^{ - 1}}} \right]$ $\times$ $\left[ {{L^1}} \right]$=$\left[ {{M^1}{L^2}{T^{ - 3}}{I^{ - 1}}} \right]$---(v)

On substituting equation (v) and (ii) in equation (i) we get,

Capacitance = Charge$\times$ $Voltag{e^{ - 1}}$
Hence, C=${\left[ {{I^1}{T^1}} \right]^{}}$ $\times$ ${\left[ {{M^1}{L^2}{T^{ - 3}}{I^{ - 1}}} \right]^{ - 1}}$=$\left[ {{M^{ - 1}}{L^{ - 2}}{T^4}{I^2}} \right]$

Therefore, the Capacitance is dimensionally represented as

$\left[ {{M^{ - 1}}{L^{ - 2}}{T^4}{I^2}} \right]$

Note: Once a condenser is fully charged, there is a possible difference between the plates and the greater the area of the plates or the greater the distance between them, the greater the capacitance of the condenser.