Answer
398.7k+ views
Hint The capacitance of a capacitor is directly proportional to the dielectric constant and inversely proportional to the distance between the plates. Compare the two capacitances to each other by division.
Formula used: In this solution we will be using the following formulae;
\[C = \dfrac{{K\varepsilon A}}{d}\] where \[C\] is the capacitance of a capacitor, \[K\] is the dielectric constant of the material between the capacitor plates, \[\varepsilon \] is the permittivity of free space, \[A\] is the area of the capacitor plates, and \[d\] is the distance between the plates.
Complete Step-by-Step solution:
According to the question, the first air is between the plates, hence, the dielectric constant is 1. Generally, the capacitance of a capacitor is given by
\[C = \dfrac{{K\varepsilon A}}{d}\] where\[K\] is the dielectric constant of the material between the capacitor plates, \[\varepsilon \] is the permittivity of free space, \[A\] is the area of the capacitor plates, and \[d\] is the distance between the plates.
Hence, initially,
\[C = \dfrac{{\varepsilon A}}{d} = 8 \times {10^{ - 12}}\]
Then, capacitance after a dielectric material is added, and the distance between the plate is halved will be given as
\[{C_2} = \dfrac{{2K\varepsilon A}}{d}\]
Then, we compare the two capacitances by dividing, we have
\[\dfrac{{{C_2}}}{C} = \dfrac{{2K\varepsilon A}}{d} \div \dfrac{{\varepsilon A}}{d}\]
\[ \Rightarrow \dfrac{{{C_2}}}{C} = \dfrac{{2K\varepsilon A}}{d} \times \dfrac{d}{{\varepsilon A}}\]
Which by cancellation will give the expression
\[\dfrac{{{C_2}}}{C} = 2K\]
\[ \Rightarrow {C_2} = 2KC\]
By inserting all known values, we have
\[{C_2} = 2 \times 6 \times 8 \times {10^{ - 12}} = 9.6 \times {10^{ - 11}}F\]
Note: Noting that in the final expression the area, and permittivity of free space where absent, this implies that we can find the expression without their knowledge. Hence, alternatively, from the knowledge that the capacitance is proportional to dielectric constant but inversely to distance, we can just write that generally,
\[C = k\dfrac{K}{d}\] where \[k\] is an arbitrary constant. Hence,
\[\dfrac{{{C_2}}}{C} = k\dfrac{K}{{\dfrac{d}{2}}} \div k\dfrac{1}{d} = k\dfrac{{2K}}{d} \times \dfrac{d}{k}\]
\[ \Rightarrow {C_2} = 2KC\]
Formula used: In this solution we will be using the following formulae;
\[C = \dfrac{{K\varepsilon A}}{d}\] where \[C\] is the capacitance of a capacitor, \[K\] is the dielectric constant of the material between the capacitor plates, \[\varepsilon \] is the permittivity of free space, \[A\] is the area of the capacitor plates, and \[d\] is the distance between the plates.
Complete Step-by-Step solution:
According to the question, the first air is between the plates, hence, the dielectric constant is 1. Generally, the capacitance of a capacitor is given by
\[C = \dfrac{{K\varepsilon A}}{d}\] where\[K\] is the dielectric constant of the material between the capacitor plates, \[\varepsilon \] is the permittivity of free space, \[A\] is the area of the capacitor plates, and \[d\] is the distance between the plates.
Hence, initially,
\[C = \dfrac{{\varepsilon A}}{d} = 8 \times {10^{ - 12}}\]
Then, capacitance after a dielectric material is added, and the distance between the plate is halved will be given as
\[{C_2} = \dfrac{{2K\varepsilon A}}{d}\]
Then, we compare the two capacitances by dividing, we have
\[\dfrac{{{C_2}}}{C} = \dfrac{{2K\varepsilon A}}{d} \div \dfrac{{\varepsilon A}}{d}\]
\[ \Rightarrow \dfrac{{{C_2}}}{C} = \dfrac{{2K\varepsilon A}}{d} \times \dfrac{d}{{\varepsilon A}}\]
Which by cancellation will give the expression
\[\dfrac{{{C_2}}}{C} = 2K\]
\[ \Rightarrow {C_2} = 2KC\]
By inserting all known values, we have
\[{C_2} = 2 \times 6 \times 8 \times {10^{ - 12}} = 9.6 \times {10^{ - 11}}F\]
Note: Noting that in the final expression the area, and permittivity of free space where absent, this implies that we can find the expression without their knowledge. Hence, alternatively, from the knowledge that the capacitance is proportional to dielectric constant but inversely to distance, we can just write that generally,
\[C = k\dfrac{K}{d}\] where \[k\] is an arbitrary constant. Hence,
\[\dfrac{{{C_2}}}{C} = k\dfrac{K}{{\dfrac{d}{2}}} \div k\dfrac{1}{d} = k\dfrac{{2K}}{d} \times \dfrac{d}{k}\]
\[ \Rightarrow {C_2} = 2KC\]
Watch videos on
A parallel plate capacitor with air between the plates has a capacitance of \[8\mu F\] (\[1\mu F \times {10^{ - 12}}\]). What will be the capacitance if the distance between the plates is reduced by half, and the space between them is filled with a substance of dielectric constant 6?
![icon](https://i.ytimg.com/vi_webp/jgCMPJXYuXk/hqdefault.webp)
Electrostatic Potential and Capacitance Class 12 Physics - NCERT EXERCISE 2.5 | Vishal Kumar Sir
Subscribe
likes
94 Views
11 months ago
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)