Two parallel, metal plates with separation distance
d = 1.00 cm
carry charges of equal magnitude but opposite sign. The plates are oriented horizontally. Assume the electric field between the plates is uniform, and it has a magnitude of 1,860 N/C. A charged particle with mass 2.00 ✕ 10−16 kg and charge 1.03 ✕ 10−6 C is projected from the center of the bottom negative plate with an initial speed of 1.04 ✕ 105 m/s at an angle of 37.0° above the horizontal. Assume the plates are square with side
length = 20.0 cm.
(a)
Describe the trajectory of the particle.
a helix
a straight line
a parabola
a circle
Correct: Your answer is correct.
(b)
Which plate does it strike?
negative plate
positive plate
It does not strike either plate.
Correct: Your answer is correct.
(c)
Where does it strike, relative to its starting point? (Enter the horizontal distance from the initial position, in m. If it does not strike either plate, enter 0.)
.00109

Correct: Your answer is correct.
m
(d)
What If? What is the minimum initial speed (in m/s) the particle must have in order to change the answer to part (b)?

To determine the minimum initial speed required for the particle to reach the positive plate, we'll need to calculate the conditions under which the particle's vertical motion is sufficient for it to cover the distance between the plates.

### Key Information:
- **Distance between plates (\(d\))**: \(0.01\) m
- **Electric field (\(E\))**: \(1860\) N/C
- **Charge of the particle (\(q\))**: \(1.03 \times 10^{-6}\) C
- **Mass of the particle (\(m\))**: \(2.00 \times 10^{-16}\) kg
- **Angle of projection (\(\theta\))**: \(37.0^\circ\)
- **Side length of the plates**: \(0.20\) m (only relevant for ensuring particle stays within boundaries)

### Forces Acting on the Particle:
- The electric force acting on the particle is \(F = qE\).
- The acceleration due to the electric force is \(a = \frac{F}{m} = \frac{qE}{m}\).

### Vertical Motion:
The vertical component of the initial velocity is \(v_{y0} = v_0 \sin \theta\).

The vertical motion is influenced by the electric force, which provides a constant downward acceleration \(a = \frac{qE}{m}\). 

### Equation of Motion:
The vertical displacement of the particle as a function of time \(t\) is given by:
\[
y(t) = v_{y0}t - \frac{1}{2} a t^2
\]
To reach the positive plate, the vertical displacement must equal the distance \(d\):
\[
d = v_0 \sin \theta \cdot t - \frac{1}{2} \cdot \frac{qE}{m} \cdot t^2
\]
Substituting known values and solving for \(v_0\), we need the particle to at least reach \(y = 0.01\) m at the time when \(t = \frac{d}{v_{y0}} \) (this is the time to reach the plate without considering acceleration):

### Simplifying:
Substituting the known values:
\[
0.01 = v_0 \sin(37^\circ) \cdot t - \frac{1}{2} \cdot \frac{1.03 \times 10^{-6} \cdot 1860}{2.00 \times 10^{-16}} \cdot t^2
\]
Here, \(a = \frac{qE}{m} = \frac{1.03 \times 10^{-6} \cdot 1860}{2.00 \times 10^{-16}} \).

To find \(v_0\), we set the condition that the particle just reaches \(y = 0.01\) m at the time \(t\) it takes for the horizontal displacement to be negligible.

However, the minimum speed occurs when \(v_{y0} = 0\) under the effect of gravity alone.

Thus we will solve for \(v_0\) considering vertical motion via the quadratic equation above.

After substituting and simplifying, the condition for \(v_0\) yields:
\[
v_0^2 \geq \frac{2qd}{m \sin^2(\theta) } 
\]

### Calculation:
Let's calculate this:

- \(a = \frac{1.03 \times 10^{-6} \times 1860}{2.00 \times 10^{-16}} = 9.57 \times 10^{9}\) m/s²

The value \(v_0 \) becomes:
\[
v_0 \geq \sqrt{\frac{2 \times 9.57 \times 10^{9} \times 0.01}{\sin^2(37^\circ)} }
\]

### Solving for \(v_0\):

Let's compute this value using Python to get the minimum initial speed.The minimum initial speed required for the particle to reach the positive plate is approximately **727,297 m/s**.

Would you like more details or have any questions? Here are some related questions you might find useful:

1. How does changing the angle of projection affect the required initial speed?
2. What would happen if the electric field strength was doubled?
3. How would the trajectory change if the mass of the particle was increased?
4. What is the significance of the horizontal component of the initial speed in this scenario?
5. How would the outcome differ if the plates were not parallel?

**Tip:** When solving problems involving electric fields and motion, carefully consider both the horizontal and vertical components of velocity and acceleration.

Explore the trajectory of a charged particle projected in a uniform electric field between two parallel metal plates. Learn how to determine whether the particle strikes the plates, its trajectory type, and the minimum initial speed required to reach the positive plate. Key formulas include electric force and equations of motion. Suitable for Grades 11-12.

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