Respuesta :
Answer:
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 =[tex]\frac{dE_x}{dE}[/tex]
dEₓ = dE cos θ
cos θ = x / r
substituting
dEₓ = [tex]k \frac{dq}{r^2 } \ \frac{x}{r}[/tex]
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 = [tex]k \frac{x}{(x^2 + a^2 ) ^{3/2} } \ dq[/tex]
we integrate
∫ dEₓ =k \frac{x}{(x^2 + a^2 ) ^{3/2} } ∫ dq
Eₓ = [tex]k \ Q \ \frac{x}{(x^2+a^2)^{3/2}}[/tex]
In the exercise indicate that the electron is very central to the center of the ring
x << a
Eₓ = [tex]k \ Q \frac{x}{a^3 \ ( 1 +(x/a)^2)^{3/2})}[/tex]
if we expand in a series
[tex](\ 1+ (x/a)^2 \ )^{-3/2} = 1 - \frac{3}{2} (\frac{x}{a} )^2[/tex]
we keep the first term if x<<a
Eₓ = [tex]\frac{ k Q}{a^3} \ x[/tex]
the force is
F = q E
F = [tex]- \frac{kQ }{a^3} \ x[/tex]
this is a restoring force proportional to the displacement so the movement is simple harmonic,
F = m a
[tex]- \frac{keQ}{a^3} \x = m \frac{d^2 x}{dt^2 }[/tex]
[tex]\frac{d^2 x}{dt^2} = \frac{keQ}{ma^3} \ x[/tex]
the solution is of type
x = A cos (wt + Ф)
with angular velocity
w² = [tex]\frac{keQ}{m a^3}[/tex]
angular velocity and period are related
w = 2π/ T
we substitute
4π² / T² = \frac{keQ}{m a^3}
T = 2π [tex]\sqrt{\frac{m a^3 }{keQ} }[/tex]
let's calculate
T = 2π [tex]\sqrt{ \frac{ 9.1 \ 10^{-31} \ 2.2^3 }{9 \ 10^9 \ 1.6 \ 10^{-19} \ 0.021 \ 10^{-3} } }[/tex]
T = 2π pi [tex]\sqrt{320.426 \ 10^{-18} }[/tex]
T = 2π 17.9 10⁻⁹ s
T = 1.12 10⁻⁷ s
