Isn't the answer infinity?

Electricity and Magnetism Level 4

A charge $+16$ C is fixed at each of the points $x=3, 9, 15, \ldots, \infty$ on the $x$ -axis, and a charge $-16$ C is fixed at each of the points $x=6, 12, 18, \ldots, \infty$ on the $x$ -axis. Then find the potential at the origin due to these charges.

If the potential is of the form $\frac { A\ln B }{ C\pi { \varepsilon }_{ 0 } }$ , find $A+B+C$ , where $A$ and $C$ are co-prime and $B$ is prime.

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The answer is 9.

1 solution

Aditya Kumar
Aug 12, 2015

Potential energy at the origin is given by:

$V=\frac { 16 }{ 4\pi { \varepsilon }_{ 0 } } \left( \frac { 1 }{ 3 } -\frac { 1 }{ 6 } +\frac { 1 }{ 9 } -\frac { 1 }{ 12 } +... \right) \\ \quad =\frac { 16 }{ 4\pi { \varepsilon }_{ 0 } } .\frac { 1 }{ 3 } \left( \frac { 1 }{ 1 } -\frac { 1 }{ 2 } +\frac { 1 }{ 3 } -\frac { 1 }{ 4 } +... \right) \\ \quad =\frac { 4.ln(2) }{ 3\pi { \varepsilon }_{ 0 } } \\ \therefore \quad A+B+C=4+2+3=9$

Shouldn't you be telling why that series is ln(2)?

Agnishom Chattopadhyay - 5 years, 10 months ago

It is an expansion of ln(2). I learnt it from Wikipedia

Aditya Kumar - 5 years, 10 months ago

Hint: Use the fact that harmonic series can be expressed as

$H(n)=ln(n)+\gamma+\alpha_0$

Where $\gamma$ is some constant and $\alpha_0$ is some zero-sequence.

Вук Радовић - 5 years, 9 months ago

This series the expansion of ln(1+x).Here the value of x is 1.

Akshay Sharma - 5 years, 5 months ago

How the value of x is one !

Rohit Yadv - 6 months, 4 weeks ago

Good Question!! Solved it the same way. But check your solution. A kiddish error. 4+2+3 = 9. :D

Abhinav Dixit - 5 years, 10 months ago

Thanks for the report it was really kiddish :P

Aditya Kumar - 5 years, 10 months ago

However,this series is conditionally convergent,and it's terms can't be reordered.So can you explain why you have used this particular arrangement:see riemann's rearrangement theorem.

Aditya Anand - 5 years, 8 months ago

It is proportional to inverse of distance squared. So, where has the square gone. Please let me know

Rishabh Joshi - 2 years, 1 month ago

I am in second grade P

Bibi Fajroo - 1 year, 6 months ago

Isn't the answer infinity?

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The answer is 9.

1 solution

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