Ten positively charged particles are held fixed along the x -axis at positions x = 10 cm , 20 cm , 30 cm , , 100 cm . The first particle possesses a charge of 1.0 10⁻⁸ C , the second 8 10⁻⁸ C , the third 27 10⁻⁸ C , and so forth, up to the tenth particle which has a charge of 1000 10⁻⁸ C . Determine the magnitude of the electric force acting on a 1 C charge located at the origin.

Step-by-step solution

The position of the n^ th particle is x_n = n 10 cm = n 0.1 m . The charge of the n^ th particle follows the pattern q_n = n^3 10⁻⁸ C . The electric force exerted by the n^ th particle on a 1 C charge at the origin is given by Coulomb's law: F_n = 1 4 ₀ Q q_n x_n^2 Substituting the given values: F_n = (9 10^9) 1 (n^3 10⁻⁸) (n 0.1)^2 F_n = (9 10^9) n^3 10⁻⁸ n^2 10⁻² F_n = 9 10^9 n 10⁻⁶ = 9 10^3 n N The total electric force F is the sum of the forces from all 10 particles: F = _ n=1 ¹⁰ F_n = _ n=1 ¹⁰ 9 10^3 n F = 9 1