Problem: An infinite number of charges, each of charge 1 microcoulomb, are placed on the x-axis with coordinates x=1,2,4,8, infinity. If a charge of 1C is kept at the origin, what is the net force acting on 1C?

Solution:

To solve this problem, we need to use Coulomb's Law, which states that the force between two charges is proportional to the product of the charges and inversely proportional to the square of the distance between them. We can use this law to calculate the force between the charge at the origin and each of the other charges on the x-axis.

Step 1: Calculate the distance between the charge at the origin and each of the other charges on the x-axis.

The distance between the charge at the origin and the charge at x=1 is 1 meter. The distance between the charge at the origin and the charge at x=2 is 2 meters. Similarly, the distances between the charge at the origin and the charges at x=4, x=8, and infinity are 4 meters, 8 meters, and infinity, respectively.

Step 2: Calculate the force between the charge at the origin and each of the other charges on the x-axis.

Using Coulomb's Law, we can calculate the force between the charge at the origin and each of the other charges on the x-axis. The force between the charge at the origin and the charge at x=1 is given by:

F = k * (q1 * q2) / r^2

where k is the Coulomb's constant (9 x 10^9 N m^2 C^-2), q1 is the charge of the charge at the origin (1 C), q2 is the charge of the charge at x=1 (1 microcoulomb or 1 x 10^-6 C), and r is the distance between the two charges (1 m).

Plugging in the values, we get:

F1 = (9 x 10^9) * (1 C * 1 x 10^-6 C) / (1 m)^2

F1 = 9 N

Similarly, the forces between the charge at the origin and the charges at x=2, x=4, x=8, and infinity are given by:

F2 = (9 x 10^9) * (1 C * 1 x 10^-6 C) / (2 m)^2

F2 = 2.25 N

F4 = (9 x 10^9) * (1 C * 1 x 10^-6 C) / (4 m)^2

F4 = 0.5625 N

F8 = (9 x 10^9) * (1 C * 1 x 10^-6 C) / (8 m)^2

F8 = 0.140625 N

Finf = (9 x 10^9) * (1 C * 1 x 10^-6 C) / (infinity)^2

Finf = 0 N

Note that the force between the charge at the origin and the charge at infinity is zero because the distance between them is infinite.

Step 3: Calculate the net force acting on the