Question:

A parallel plate air capacitor of capacitance C is connected to a cell of emf V and then disconnected from it. A dielectric slab of dielectric constant K, which can just fill the air gap of the capacitor, is now inserted in it. Which of the following is incorrect?

Updated On: May 5, 2024

The change in energy stored is $\frac{1}{2}CV^{2}\left(\frac{1}{K}-1\right)$
The charge on the capacitor is not conserved
The potential difference between the plates decreases K times
The energy stored in the capacitor decreases K times

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The Correct Option is B

Solution and Explanation

$ q = CV \Rightarrow V = q/c $
Due to dielectric insertion, new capacitance $ C_2 = CK $
Initial energy stored in capacitor, $ U_1 = \frac{q^2}{2C} $
Final energy stored in capacitor, $ U_2 = \frac{q^2}{2KC} $
Change in energy stored, $ \Delta U = U_2 - U_1 $
$ \Delta U = \frac{q^2}{2C} \bigg(\frac{1}{K} - 1 \bigg) = \frac{1}{2} CV^2 \bigg(\frac{1}{K} - 1 \bigg) $
New potential difference between plates
$ V' = \frac{q}{CK} = \frac{V}{K} $

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Concepts Used:

Electrostatic Potential and Capacitance

Electrostatic Potential

The potential of a point is defined as the work done per unit charge that results in bringing a charge from infinity to a certain point.

Some major things that we should know about electric potential:

They are denoted by V and are a scalar quantity.
It is measured in volts.

Capacitance

The ability of a capacitor of holding the energy in form of an electric charge is defined as capacitance. Similarly, we can also say that capacitance is the storing ability of capacitors, and the unit in which they are measured is “farads”.

The capacitor is in Series and in Parallel as defined below;

In Series

Both the Capacitors C₁ and C₂ can easily get connected in series. When the capacitors are connected in series then the total capacitance that is C_totalis less than any one of the capacitor’s capacitance.

In Parallel

Both Capacitor C₁ and C₂ are connected in parallel. When the capacitors are connected parallelly then the total capacitance that is C_total is any one of the capacitor’s capacitance.