The motion of a charged spring oscillator in a uniform electric field is governed by the interplay between the external electric force and the elastic restoring force. The electric field, denoted as $ \mathbf{E} = E \sin A \, \mathbf{i} - E \cos A \, \mathbf{k} $, exerts a force on the charge $ q $, resulting in $ \mathbf{F} = q\mathbf{E} $. The spring provides an opposing force $ \mathbf{f} = -k l \, \mathbf{i} $, where $ l(t) $ represents the displacement along the x-axis.
Using Newton’s second law, the equation of motion becomes:
$$
\frac{d^2 l}{dt^2} + X^2 l = A,
$$
where $ X^2 = \frac{k}{m} $ and $ A = qE \sin A $. This is a second-order linear non-homogeneous differential equation. Its general solution is:
$$
l(t) = C \cos(Xt + U) + \frac{A}{X^2},
$$
with constants $ C $ and $ U $ determined by initial conditions. Assuming $ l(0) = l_0 $ and $ \frac{dl}{dt}(0) = 0 $, we find $ C = l_0 - \frac{A}{X^2} $ and $ U = 0 $. Thus, the specific solution becomes:
$$
l(t) = \left(l_0 - \frac{A}{X^2}\right)\cos(Xt) + \frac{A}{X^2}.
$$
The velocity of the oscillator is then:
$$
v(t) = -\left(l_0 - \frac{A}{X^2}\right)X \sin(Xt),
$$
indicating that the direction of motion is opposite to the positive x-axis.
To analyze the electromagnetic field generated by the moving charge, we consider the electric field at a point $ p(x, y, z) $ due to the oscillator. From literature, the electric field is given by:
$$
\mathbf{E}_{cp} = \frac{q}{4\pi\varepsilon_0 r^2} \cdot \frac{\mathbf{r}}{r^3} \quad \text{(approximated for far distances)}.
$$
When the distance $ r $ is much larger than the amplitude $ |l| $, higher-order terms like $ l^2 $ can be neglected. Using a Taylor expansion, the field simplifies to:
$$
\mathbf{E}_{cp} \approx \frac{q}{4\pi\varepsilon_0 r^3} (1 - \frac{2xl}{r^2}) \cdot \mathbf{r}.
$$
Given that the oscillator moves slowly compared to the speed of light, the electromagnetic field can be treated as quasi-static. The magnetic field at point $ p $ is derived using the Lorentz transformation:
$$
\mathbf{B}_p = \frac{v}{c^2} \mathbf{E}_p,
$$
where $ v $ is the velocity of the oscillator and $ c $ is the speed of light.
From the equations, it is clear that the motion of the charged spring oscillator remains simple harmonic under the influence of both the electric and elastic forces. The equilibrium position shifts to $ \frac{A}{X^2} $, while the amplitude of oscillation is $ l_0 - \frac{A}{X^2} $.
At large distances from the oscillator, the total electric field consists of three components: one static due to the spatial coordinates, one dynamic due to the time-varying displacement, and one external electric field. The magnetic field also varies periodically with time, confirming that the system generates a dynamic electromagnetic field.
In summary, the interaction between the external electric field and the elastic force results in a periodic motion, which in turn produces a time-dependent electromagnetic field. This demonstrates how even a simple mechanical system can generate complex electromagnetic effects when charged and subjected to external influences.
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