Question

an infinite number of point charges each equal to 4 micro Coulomb  are placed along the x-axis at x=1 m,x=2 m,x=4 m,x=8 m.the electric potential at the origin due to all charges is 72000V.....if the consecutive charges of opposite sign are plaed then the electric potential at the origin is
1)72000 V
2)1490 V
3)24000 V
4)7200 V.......plz explain with clear solution

Answer 1

Let q = 4 μC
distances of the charges from origin are :  1m, 2m, 2² m, 2³ m, 2⁴ m .....

Potential at Origin = V (x=0)
  = $\frac{q}{4 \pi \epsilon_0} [\frac{1}{2^0} + \frac{1}{2^1}+\frac{1}{2^2}+.....] \\\\=\frac{q}{4 \pi \epsilon_0} * 1 * [ \frac{1}{1-\frac{1}{2}}] \\\\=4 * 10^{-6} *9 * 10^9*2\\\\=7.2*10^4\ V$
 
Now alternate charges are of opposite sign.
The charge at x = 1 is  q, at x = 2 m is -q, at x = 3m is q, at x = 4m is -q etc....

$V(x=0) = \frac{q}{4 \pi \epsilon_0} [\frac{1}{2^0} - \frac{1}{2^1}+\frac{1}{2^2}-\frac{1}{2^3}+\frac{1}{2^4}-\frac{1}{2^5}+.....] \\\\=\frac{q}{4 \pi \epsilon_0} * [\frac{1}{2^0}+\frac{1}{2^2}+\frac{1}{2^4}+....] - [\frac{1}{2^1}+\frac{1}{2^3}+\frac{1}{2^5}+....]\\\\= 9 * 10^9 * 4 * 10^{-6} * [ \frac{1}{1-\frac{1}{4}} - \frac{1}{2}*\frac{1}{1-\frac{1}{2}}] \\\\=3.6*10^{4} * [ \frac{4}{3}-1]=12,000\ Volts$

When the charges of opposite sign are placed side by side, then the net potential will be less than if all charges are of same sign..  

the answer is not matching.. perhaps i have not understood what "consecutive charges of opposite sign are paced then...."  means...