Answer
437.1k+ views
Hint: To solve this question we need to consider both the charges which are present in the question that is q and Q. Then we have to find the total force at any of the three vertices of the triangle. And after equating them, we will get the required result.
Complete answer:
Let us assume an equilateral triangle ABC having AB=BC=CA=a units. Charge Q is placed at three of its vertices and another charge q has been placed at the centre.
We have been given that the system of charges is in equilibrium. So, at any position where the charges have been placed should be balanced.
Let us find out the net force acting on the charge Q positioned at the point C. Since the charges placed at the vertices are of same sign and magnitude, so, the forces exerted by them will be equal in magnitude.
Now, the forces exerted by the two charges placed at vertices A and B will exert equal forces at the charge placed at C, suppose it as F.
We know that the force exerted by a charge particle $q_1$ on another charge particle $q_2$ placed at a distance r is given by $F=\dfrac{kq_1 q_2}{r^2}$
So, the force exerted by the charges Q at vertices A and B, $F=\dfrac{kQ^2}{a^2}$ ………. (i)
And the force exerted by the charge q will be, $F_q =\dfrac{kqQ}{ \left(\dfrac{a}{\sqrt{3}}\right)^2}$.......... (ii) (since $OC=\dfrac{a}{\sqrt{3}}$)
Now, we can see that the vertical component of the forces exerted by charges Q at point C are equal and opposite in nature and for balancing their horizontal component the charge q should be negative in nature.
Thus, we can write that $F_q = 2Fcos30^o$
Substituting the values from equation (i) and (ii), we get
$\dfrac{kqQ}{ \left(\dfrac{a}{\sqrt{3}}\right)^2}=2\times\dfrac{kq_1 q_2}{r^2}cos30^o$
$\dfrac{3kqQ}{ a^2}=2\times\dfrac{kq_1 q_2}{r^2}\times \dfrac{\sqrt{3}}{2}$
$\implies \dfrac{q}{Q}=\dfrac{1}{\sqrt{3}}$
So, the correct answer is “Option B”.
Note:
If we will consider the fourth charge as positive, then the net force at any of the three vertices will not be zero and hence the system will not be in equilibrium. The net force at the centre will be zero as because of the symmetry, since the triangle is an equilateral triangle.
Complete answer:
![seo images](https://www.vedantu.com/question-sets/78d28504-aa64-4802-8683-b09f8822980b352953868575663981.png)
Let us assume an equilateral triangle ABC having AB=BC=CA=a units. Charge Q is placed at three of its vertices and another charge q has been placed at the centre.
We have been given that the system of charges is in equilibrium. So, at any position where the charges have been placed should be balanced.
Let us find out the net force acting on the charge Q positioned at the point C. Since the charges placed at the vertices are of same sign and magnitude, so, the forces exerted by them will be equal in magnitude.
Now, the forces exerted by the two charges placed at vertices A and B will exert equal forces at the charge placed at C, suppose it as F.
We know that the force exerted by a charge particle $q_1$ on another charge particle $q_2$ placed at a distance r is given by $F=\dfrac{kq_1 q_2}{r^2}$
So, the force exerted by the charges Q at vertices A and B, $F=\dfrac{kQ^2}{a^2}$ ………. (i)
And the force exerted by the charge q will be, $F_q =\dfrac{kqQ}{ \left(\dfrac{a}{\sqrt{3}}\right)^2}$.......... (ii) (since $OC=\dfrac{a}{\sqrt{3}}$)
Now, we can see that the vertical component of the forces exerted by charges Q at point C are equal and opposite in nature and for balancing their horizontal component the charge q should be negative in nature.
Thus, we can write that $F_q = 2Fcos30^o$
Substituting the values from equation (i) and (ii), we get
$\dfrac{kqQ}{ \left(\dfrac{a}{\sqrt{3}}\right)^2}=2\times\dfrac{kq_1 q_2}{r^2}cos30^o$
$\dfrac{3kqQ}{ a^2}=2\times\dfrac{kq_1 q_2}{r^2}\times \dfrac{\sqrt{3}}{2}$
$\implies \dfrac{q}{Q}=\dfrac{1}{\sqrt{3}}$
So, the correct answer is “Option B”.
Note:
If we will consider the fourth charge as positive, then the net force at any of the three vertices will not be zero and hence the system will not be in equilibrium. The net force at the centre will be zero as because of the symmetry, since the triangle is an equilateral triangle.
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)