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  • Can someone help me with this, Infinite charges are lying at x = 1, 2, 4, 8…meter on X-axis and the value of each charge is Q. The value of intensity of electric field and potential at point x = 0 due to these charges will be respectively

Infinite charges are lying at x = 1, 2, 4, 8…meter on X-axis and the value of each charge is Q. The value of intensity of electric field and potential at point x = 0 due to these charges will be respectively

  • Option 1)

    12\times 10^{9}QN/C, 1.8\times 10^{4}V

     

     

     

     

     

  • Option 2)

    Zero, 1.2\times 10^{4}V

  • Option 3)

    6\times 10^{9}QN/C, 9\times 10^{3}V\:

  • Option 4)

    4\times 10^{9}QN/C, 6\times 10^{3}V

 

Answers (1)

best_answer

As we learned

 

Potential of a System of Charge -

V=\sum_{i=1}^{i=n}\frac{kQ_{i}}{r_{i}}

-

 

 By the superposition, Net electric field at origin

E=kQ\left [ \frac{1}{1^{2}} +\frac{1}{2^{2}}+\frac{1}{4^{2}}+\frac{1}{8^{2}}+\cdots \infty \right ]

E=kQ\left [ 1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\cdots \infty \right ]

1+ \frac{1}{4} +\frac{1}{16}+\frac{1}{64}+\cdots \infty  is an infinite geometrical progression it’s sum can be obtained by using the formula

S_{\infty }=\frac{q}{1-r}; Where a = First term, r = Common ratio.

Here a=1 and  r=\frac{1}{4} so, 1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\cdots \infty =\frac{1}{1-1/4}=\frac{4}{3}.

Hence  E=9\times 10^{9\times }Q\times \frac{4}{3}=12\times 10^{9}Q N/C

Electric potential at origin   V=\frac{1}{4\pi \varepsilon _{0}}\left [ \frac{1\times 10^{-6}}{1} +\frac{1\times 10^{-6}}{2}+\frac{1\times 10^{-6}}{4}+\frac{1\times 10^{-6}}{8}+\cdots \infty \right ]

9\times 10^{9}\times 10^{-6}\left [ 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8} +\cdots \infty \right ] = 9\times 10^{3}\left [ \frac{1}{1-\frac{1}{2}} \right ]=1.8\times 10^{4} volt

 


Option 1)

12\times 10^{9}QN/C, 1.8\times 10^{4}V

 

 

 

 

 

Option 2)

Zero, 1.2\times 10^{4}V

Option 3)

6\times 10^{9}QN/C, 9\times 10^{3}V\:

Option 4)

4\times 10^{9}QN/C, 6\times 10^{3}V

Posted by

Avinash

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