An electron is constrained to the central perpendicular axis of a ring of charge of radius 2.2 m and charge 0.021 mC. Suppose the electron is released from rest a distance 0.050 m from the ring center. It then oscillates through the ring center. Calculate its period. (The electron is always much closer to the ring center than a radius.)

T = 1.12 10⁻⁷ s

Explanation:

This exercise must be solved in parts. Let's start looking for the electric field in the axis of the ring.

All the charge dq is at a distance r

           dE = k dq / r²

Due to the symmetry of the ring, the field perpendicular to the axis is canceled, leaving only the field in the direction of the axis, if we use trigonometry

            cos θ =

             dEₓ = dE cos θ

              cos θ = x / r

substituting

                dEₓ =

                DEₓ = k dq x / r³

let's use the Pythagorean theorem to find the distance r

             r² = x² + a²

where a is the radius of the ring

we substitute

              dEₓ =

we integrate

               ∫ dEₓ =k \frac{x}{(x^2 + a^2 ) ^{3/2} }  ∫ dq

               Eₓ =

In the exercise indicate that the electron is very central to the center of the ring

                x << a

                Eₓ =

if we expand in a series

we keep the first term if x<<a

                 Eₓ =

the force is

                 F = q E

                 F =

this is a restoring force proportional to the displacement so the movement is simple harmonic,

                 F = m a

the solution is of type

                  x = A cos (wt + Ф)

with angular velocity

                w² =

angular velocity and period are related

                 w = 2π/ T

we substitute

               4π² / T² = \frac{keQ}{m a^3}

                T = 2π  

let's calculate

                T = 2π

                 T = 2π pi

                 T = 2π  17.9 10⁻⁹ s

                 T = 1.12 10⁻⁷ s

what experimental evidence led scientists to change from the previous model to this one?

a few of the positive particles aimed at a gold foil seemed to bounce back.

the colors of light emitted from heated atoms had very specific energies.

experiments with water vapor showed that elements combine in specific proportions.

cathode rays were bent in the same way whenever a magnet was brought near them.

this is an example of "committed" time.

committed time, as contracted time, takes need over essential and available time since it is seen as profitable work. it alludes to the time allotted to keep up a home and family. at the point when a man is commuting using committed time this individual may feel that the drive is an obligation to family, for example, strolling youngsters to class or driving a mate to work. contracted and committed time clients may feel that their commute could really compare to the commute of important or leisure time clients on the grounds that their commute is profitable work. in this manner, they might be more disposed to pick a mechanized method of movement.