Answer
Verified
434.1k+ views
Hint: To solve this question, first find the force acting in any one of the charges in the corner of the square soap film. Then find the force on the soap film due to its surface film. Since the system of charges and the planar film is in equilibrium, we will equate the two forces and then we can find our answer.
Complete answer:
There are four charges at the corners of a square.
The side of the square is a.
Let us draw a diagram of the system given in the question.
Now, the force in any of the charges due to one of the neighbouring charges,
${{F}_{1}}={{F}_{2}}=\dfrac{k{{q}^{2}}}{{{a}^{2}}}$
Now, the force on any charge due to the charge in the diagonal position is,
${{F}_{3}}=\dfrac{k{{q}^{2}}}{{{\left( \sqrt{2}a \right)}^{2}}}$
So, the net force on any of the charges in the corner of the square is,
$\begin{align}
& F={{F}_{3}}+\sqrt{\left( {{F}_{1}}^{2}+{{F}_{2}}^{2} \right)} \\
& F=\dfrac{k{{q}^{2}}}{2{{a}^{2}}}+\sqrt{2}\dfrac{k{{q}^{2}}}{{{a}^{2}}} \\
& F=\dfrac{{{q}^{2}}}{{{a}^{2}}}\times \text{constant}\text{.} \\
\end{align}$
The force on the side due to the surface tension on the soap film can be mathematically expressed as,
$\begin{align}
& F=\text{ surface tension }\times \text{ side length} \\
& \text{F=}\gamma \text{a} \\
\end{align}$
Since the system as a whole is in equilibrium, the outward force due to forces of the charges and the inward force due to the surface tension of the soap film will be equal.
Equating the two forces,
$\begin{align}
& \dfrac{{{q}^{2}}}{{{a}^{2}}}\times \text{constant=}\gamma \text{a} \\
& {{a}^{3}}=\dfrac{{{q}^{2}}}{\gamma }\times \text{constant} \\
& a\text{=}{{\left[ \dfrac{{{q}^{2}}}{\gamma }\times \text{constant} \right]}^{\dfrac{1}{3}}} \\
& a=K{{\left[ \dfrac{{{q}^{2}}}{\gamma } \right]}^{\dfrac{1}{3}}} \\
\end{align}$
Now, given in the question that the expression for the side length of the square is,
$a=k{{\left[ \dfrac{{{q}^{2}}}{\gamma } \right]}^{\dfrac{1}{N}}}$
Comparing the above two expression we get that,
$\begin{align}
& \dfrac{1}{3}=\dfrac{1}{N} \\
& N=3 \\
\end{align}$
So, the value of N will be 3.
The correct option is (A).
Note:
Surface tension is the tendency of liquid surfaces to shrink into the minimum surface area. So, the surface tension exerts an inward force in the liquid surface. Force due to surface tension depends on the value of surface tension and the side length or the perimeter of the liquid surface.
Complete answer:
There are four charges at the corners of a square.
The side of the square is a.
Let us draw a diagram of the system given in the question.
Now, the force in any of the charges due to one of the neighbouring charges,
${{F}_{1}}={{F}_{2}}=\dfrac{k{{q}^{2}}}{{{a}^{2}}}$
Now, the force on any charge due to the charge in the diagonal position is,
${{F}_{3}}=\dfrac{k{{q}^{2}}}{{{\left( \sqrt{2}a \right)}^{2}}}$
So, the net force on any of the charges in the corner of the square is,
$\begin{align}
& F={{F}_{3}}+\sqrt{\left( {{F}_{1}}^{2}+{{F}_{2}}^{2} \right)} \\
& F=\dfrac{k{{q}^{2}}}{2{{a}^{2}}}+\sqrt{2}\dfrac{k{{q}^{2}}}{{{a}^{2}}} \\
& F=\dfrac{{{q}^{2}}}{{{a}^{2}}}\times \text{constant}\text{.} \\
\end{align}$
The force on the side due to the surface tension on the soap film can be mathematically expressed as,
$\begin{align}
& F=\text{ surface tension }\times \text{ side length} \\
& \text{F=}\gamma \text{a} \\
\end{align}$
Since the system as a whole is in equilibrium, the outward force due to forces of the charges and the inward force due to the surface tension of the soap film will be equal.
Equating the two forces,
$\begin{align}
& \dfrac{{{q}^{2}}}{{{a}^{2}}}\times \text{constant=}\gamma \text{a} \\
& {{a}^{3}}=\dfrac{{{q}^{2}}}{\gamma }\times \text{constant} \\
& a\text{=}{{\left[ \dfrac{{{q}^{2}}}{\gamma }\times \text{constant} \right]}^{\dfrac{1}{3}}} \\
& a=K{{\left[ \dfrac{{{q}^{2}}}{\gamma } \right]}^{\dfrac{1}{3}}} \\
\end{align}$
Now, given in the question that the expression for the side length of the square is,
$a=k{{\left[ \dfrac{{{q}^{2}}}{\gamma } \right]}^{\dfrac{1}{N}}}$
Comparing the above two expression we get that,
$\begin{align}
& \dfrac{1}{3}=\dfrac{1}{N} \\
& N=3 \\
\end{align}$
So, the value of N will be 3.
The correct option is (A).
Note:
Surface tension is the tendency of liquid surfaces to shrink into the minimum surface area. So, the surface tension exerts an inward force in the liquid surface. Force due to surface tension depends on the value of surface tension and the side length or the perimeter of the liquid surface.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Write the differences between monocot plants and dicot class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE