E= kq ( 1/ r1^2 + 1/r2^2 + ......)= kq ( 1/1+1/4+1/16+1/64+.....)= kq ( sum of infinite terms of gp)sum is geometric progression as if terms are x,y,z.... then y/x= z/y = 1/4.sum of infinite terms in gp= a/1--r ( bcs r<1,so 1--r,="" otherwise="" r--1)here,="" a="first" term="" in="" gp="" tht="" is="" 1/1="1" ,="" r="common" division="" tht="" is="" 1/4.="" putting="" value,="" we="" get="" ,="" 1/="" 1--1/4="4/3.e=" kq="" (="" 4/3),="" e="9×10^9×q×4/3.e=" 12×10^9="" q="" n/c="" 1--r,="" otherwise="" r--1)here,="" a="first" term="" in="" gp="" tht="" is="" 1/1="1" ,="" r="common" division="" tht="" is="" 1/4.="" putting="" value,="" we="" get="" ,="" 1/="" 1--1/4="4/3.E=" kq="" (="" 4/3),="" e="9×10^9×q×4/3.E=" 12×10^9="" q="">

The problem states that there are infinite charges of magnitude q lying at regular intervals on the X-axis. The distances between these charges are powers of 2 (1, 2, 4, 8, ...). We need to find the intensity of the electric field at a point X=0, which is the point of interest.

The electric field at a point due to a single charge is given by Coulomb's law:

E = k * (q / r^2)

Where E is the electric field, k is the electrostatic constant (8.99 x 10^9 N m^2/C^2), q is the magnitude of the charge, and r is the distance between the point of interest and the charge.

To find the total electric field at X=0, we need to consider the contribution from each charge. Since the charges are infinite, we can sum up the electric field contributions using integration.

Let's consider the electric field contribution from a single charge at position X = 2^n, where n is a positive integer. The distance between this charge and the point X=0 is r = 2^n. The magnitude of the electric field at X=0 due to this charge is:

E_n = k * (q / (2^n)^2) = k * (q / 4^n)

To find the total electric field at X=0, we need to sum up the contributions from all charges at different positions X = 2^n:

E_total = Σ E_n = Σ (k * (q / 4^n))

Now, let's simplify the sum. Since the charges are at regular intervals of powers of 2, we can rewrite the sum as a geometric series:

E_total = kq * Σ (1 / 4^n)

Using the formula for the sum of a geometric series, we have:

E_total = kq * (1 / (1 - 1/4)) = kq * (4/3)

Substituting the value of k (8.99 x 10^9 N m^2/C^2) into the equation, we get:

E_total = (8.99 x 10^9 N m^2/C^2) * q * (4/3)

Simplifying further, we have:

E_total = 12 x 10^9 q N/C

Therefore, the intensity of the electric field at X=0 due to the infinite charges is 12 x 10^9 q N/C.

In conclusion, the value of the intensity of the electric field at point X=0 due to the infinite charges is 12 x 10^9 q N/C.