Use the following values for mass and charge: an electron has mass me = 9.11×10-31 kg and charge -e, a proton has mass mp = 1.67×10-27 kg and charge +e, an alpha particle has mass malpha = 6.65×10-27 kg and charge +2e, where e = 1.60×10-19 C. An electron is released from rest in a vacuum between two flat, parallel metal plates that are 13.0 cm apart and are maintained at a constant electric potential difference of 780 Volts. If the electron is released at the negative plate, what is its speed just before it strikes the positive plate?
To determine the speed of the electron just before it strikes the positive plate, we can use the principles of electrostatics and conservation of energy.
First, let's find the electric field between the metal plates. The electric field (E) can be calculated using the electric potential difference (V) and the distance between the plates (d) using the formula:
E = V / d
Substituting the given values, we have:
E = 780 V / 0.13 m = 6000 V/m
The electric field between the plates will accelerate the electron. Using the equation:
F = q * E
where F is the force, q is the charge, and E is the electric field, we can determine the force acting on the electron. Since the electron has a charge of -e, the force is:
F = (-e) * E
Next, we'll use the equation for force:
F = m * a
where m is the mass of the electron and a is its acceleration. Since the electron starts from rest, its initial velocity (v₀) is 0. Solving for the acceleration:
a = F / m
Plugging in the values:
a = [(-e) * E] / me
Now, we can calculate the acceleration (a) using the values given:
a = [(-1.60×10^(-19) C) * (6000 V/m)] / (9.11×10^(-31) kg) ≈ -1.05 × 10^20 m/s²
The negative sign indicates that the acceleration is opposite to the direction of the electric field (which is from positive to negative).
Since the force applied is constant, we can use the constant acceleration equation:
v = v₀ + a * t
where v is the final velocity, v₀ is the initial velocity, a is the acceleration, and t is the time. In this case, we're interested in the final velocity just before the electron strikes the positive plate, so v₀ = 0.
Using the equation, we can solve for v:
v = 0 + a * t
To find the time it takes for the electron to travel between the plates (t), we'll use the equation for displacement:
d = v₀ * t + 0.5 * a * t²
The initial velocity is 0, so the equation simplifies to:
d = 0.5 * a * t²
Rearranging the equation, we can solve for t:
t = √(2 * d / a)
Plugging in the given values:
t = √(2 * 0.13 m / -1.05 × 10^20 m/s²) ≈ 3.09 × 10^(-11) s
Now, we can find the final velocity (v):
v = a * t = (-1.05 × 10^20 m/s²) * (3.09 × 10^(-11) s) ≈ -3.24 × 10^9 m/s
The negative sign indicates that the final velocity is in the opposite direction to the initial velocity.
Therefore, the speed of the electron just before it strikes the positive plate is approximately 3.24 × 10^9 m/s.