Answer:
(a) The total charge on the rod can be found by multiplying the charge density by the length of the rod:
Q = λL = (100 nC/m)(0.5 m) = 50 nC.
(b) The electric field at the origin due to the charged rod can be found using the formula for the electric field due to a charged rod:
E = (λ/(4πε₀)) ln((L+a)/a)
where ε₀ is the permittivity of free space, ln is the natural logarithm, and a is the distance from the origin to the rod.
Plugging in the given values, we have:
E = (100 nC/m)/(4πε₀) ln((0.5 m + 0.1 m)/0.1 m) ≈ 5.03 × 10^7 N/C, in the negative x-direction (since the rod is negatively charged and the electric field points toward the positive charges).
(c) The electric force on an electron placed at the origin can be found using the formula for the electric force on a point charge:
F = qE
where q is the charge of the particle and E is the electric field.
Plugging in the given values, we have:
F = (-1.6 × 10^-19 C)(5.03 × 10^7 N/C) ≈ -8.05 × 10^-12 N, in the positive x-direction (since the electron is negatively charged and the force is in the direction of the electric field).
(d) The magnitude of the electron's initial acceleration at the origin can be found using Newton's second law:
F = ma
where F is the force on the electron and m is its mass.
Solving for a, we have:
a = F/m = (-8.05 × 10^-12 N)/(9.11 × 10^-31 kg) ≈ -8.84 × 10^18 m/s^2.
The negative sign indicates that the electron is accelerating in the positive x-direction (opposite to the direction of the electric field). However, this acceleration is so large that relativistic effects must be taken into account, and classical mechanics is not sufficient to describe the motion of the electron.