To find the ratio of the magnitude of E2 to the magnitude of E1, we can use the equation for displacement under constant acceleration:
$\Delta x = v_i t + \frac{1}{2} a t^2$
where $\Delta x$ is the displacement, $v_i$ is the initial velocity, $t$ is the time, and $a$ is the acceleration.
Let's break down the problem into two parts, before and after the field changes:
Before the field changes:
The proton is accelerated in the direction of the electric field E1, pointing due North. As the proton is released from rest, its initial velocity is 0.
Using the above equation, we can find the displacement of the proton before the field changes:
$\Delta x_1 = \frac{1}{2} a_1 t_1^2$
where $\Delta x_1$ is the displacement, $a_1$ is the acceleration due to E1, and $t_1$ is the time before the field changes.
After the field changes:
The proton is accelerated in the direction of the electric field E2, pointing due South. Since the proton returns to its starting point, its final displacement is 0.
Using the above equation, we can find the displacement of the proton after the field changes:
$\Delta x_2 = 0 = v_{i2} t_2 + \frac{1}{2} a_2 t_2^2$
where $\Delta x_2$ is the displacement, $v_{i2}$ is the initial velocity after the field changes, $a_2$ is the acceleration due to E2, and $t_2$ is the time after the field changes.
Now, let's solve the equations and find the values of $a_1$ and $a_2$:
From the equation $\Delta x_1 = \frac{1}{2} a_1 t_1^2$, we know that the displacement before the field changes is equal to 0 since the proton returns to its starting point. Therefore, we can write:
$0 = \frac{1}{2} a_1 t_1^2$
This leads to $a_1 = 0$ since $t_1 \neq 0$.
For the second part, solving the equation $\Delta x_2 = v_{i2} t_2 + \frac{1}{2} a_2 t_2^2$, we know that $v_{i2}$ is equal to the final velocity before the field changes. Hence, $v_{i2} = a_1 t_1$. Substituting this into the equation, we get:
$0 = a_1 t_1 t_2 + \frac{1}{2} a_2 t_2^2$
Since $a_1 = 0$, the equation simplifies to:
$0 = \frac{1}{2} a_2 t_2^2$
Solving for $a_2$, we find:
$a_2 = 0$
From the above results, we can see that both $a_1$ and $a_2$ are 0, implying that there is no acceleration and the electric fields do not affect the proton's motion.
Therefore, the ratio of the magnitude of E2 to the magnitude of E1 is also 0.