Answer
Verified
440.1k+ views
Hint: Draw a free body diagram indicating the forces acting on the charged ball. Equate the forces and use the relation between electric field and permittivity of the medium.
Formula used:
\[{F_e} = qE\]
Here, q is the charge of on the ball and E is the electric field.
Complete step by step answer:
The charged ball is placed in a uniform electric field experiences an electric force of magnitude,
\[{F_e} = qE\]
Here, q is the charge of on the ball and E is the electric field.Draw a free body diagram of the forces acting on the charged ball as shown in the figure below,
Balance the forces acting on the charged ball in the vertical direction as follows,
\[T\cos \theta = mg\] ……. (1)
Balance the forces acting on the charged ball in the horizontal direction as follows,
\[T\sin \theta = {F_e}\]
\[ \Rightarrow T\sin \theta = qE\] ……. (2)
Divide equation (2) by equation (1). \[\dfrac{{T\sin \theta }}{{T\cos \theta }} = \dfrac{{qE}}{{mg}}\]
\[ \Rightarrow \tan \theta = \left( {\dfrac{q}{{mg}}} \right)E\] ……. (3)
We know that the electric field is the ratio of surface charge density \[\sigma \]and permittivity of the medium \[\varepsilon \].
\[E = \dfrac{\sigma }{\varepsilon }\]
Substitute \[E = \dfrac{\sigma }{\varepsilon }\] in equation (3).
\[\left( {\dfrac{q}{{mg}}} \right)\dfrac{\sigma }{\varepsilon } = \tan \theta \]
\[ \Rightarrow \sigma = \left( {\dfrac{{\varepsilon mg}}{q}} \right)\tan \theta \]
\[\therefore \sigma \propto \tan \theta \]
So, the correct answer is “Option D”.
Note:
Specify the correct directions of forces acting on the charged ball whether they are along the positive direction of the y-axis or along the negative direction of the y-axis and the same for the x-axis. Here, the electric force should be along the positive direction of the x-axis.
Formula used:
\[{F_e} = qE\]
Here, q is the charge of on the ball and E is the electric field.
Complete step by step answer:
The charged ball is placed in a uniform electric field experiences an electric force of magnitude,
\[{F_e} = qE\]
Here, q is the charge of on the ball and E is the electric field.Draw a free body diagram of the forces acting on the charged ball as shown in the figure below,
Balance the forces acting on the charged ball in the vertical direction as follows,
\[T\cos \theta = mg\] ……. (1)
Balance the forces acting on the charged ball in the horizontal direction as follows,
\[T\sin \theta = {F_e}\]
\[ \Rightarrow T\sin \theta = qE\] ……. (2)
Divide equation (2) by equation (1). \[\dfrac{{T\sin \theta }}{{T\cos \theta }} = \dfrac{{qE}}{{mg}}\]
\[ \Rightarrow \tan \theta = \left( {\dfrac{q}{{mg}}} \right)E\] ……. (3)
We know that the electric field is the ratio of surface charge density \[\sigma \]and permittivity of the medium \[\varepsilon \].
\[E = \dfrac{\sigma }{\varepsilon }\]
Substitute \[E = \dfrac{\sigma }{\varepsilon }\] in equation (3).
\[\left( {\dfrac{q}{{mg}}} \right)\dfrac{\sigma }{\varepsilon } = \tan \theta \]
\[ \Rightarrow \sigma = \left( {\dfrac{{\varepsilon mg}}{q}} \right)\tan \theta \]
\[\therefore \sigma \propto \tan \theta \]
So, the correct answer is “Option D”.
Note:
Specify the correct directions of forces acting on the charged ball whether they are along the positive direction of the y-axis or along the negative direction of the y-axis and the same for the x-axis. Here, the electric force should be along the positive direction of the x-axis.
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
Which are the Top 10 Largest Countries of the World?
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE