The potential difference between the plates of a capacitor is 335 V. Midway between the plates, a proton and an electron are released. The electron is released from rest. The proton is projected perpendicularly toward the negative plate with an initial speed. The proton strikes the negative plate at the same instant that the electron strikes the positive plate. Ignore the attraction between the two particles, and find the initial speed of the proton.
To solve this problem, we need to consider the motion of both the proton and the electron as they move between the plates of the capacitor.
1. First, let's analyze the motion of the electron:
- The electron is released from rest, so its initial velocity is zero.
- We can use the equations of motion to determine the time taken by the electron to reach the positive plate.
- The distance between the plates is not given, so we need to find it using the potential difference and the equation for the electric field between the capacitor plates.
- The potential difference (V) between the plates is given as 335 V.
- The electric field (E) between the plates is given by the formula E = V/d, where d is the distance between the plates.
- Once we find the value of d, we can use the equation of motion, s = ut + (1/2)at^2, to calculate the time taken by the electron to reach the positive plate (where s is the distance traveled, u is the initial velocity, a is the acceleration, and t is the time).
- In this case, the acceleration of the electron is due to the electric field between the plates, and it is given by a = eE/m, where e is the charge of the electron and m is its mass.
- We know the values of e and m, so we can substitute them in the equation to find the acceleration.
- Finally, we can solve the equation of motion to find the time taken by the electron to reach the positive plate.
2. Next, let's analyze the motion of the proton:
- We are given that the proton strikes the negative plate at the same instant that the electron strikes the positive plate.
- Since the plates are parallel, the distance traveled by both particles is the same.
- Therefore, the time taken by the proton to reach the negative plate is the same as the time taken by the electron to reach the positive plate.
- We can use this time value to find the initial speed of the proton.
- The initial velocity of the proton is perpendicular to the electric field, so it experiences no electric force.
- Therefore, the force on the proton is solely due to its initial velocity and its perpendicular motion.
- Using the equation F = ma, where F is the force, m is the mass of the proton, and a is its acceleration, we can find the acceleration of the proton.
- We know the value of m, so we can substitute it in the equation to find the acceleration.
- Finally, we can use the equation v = u + at, where v is the final velocity (zero in this case since the proton strikes the plate), u is the initial velocity, a is the acceleration, and t is the time.
By following these steps, we can determine the initial speed of the proton in the given scenario.
To solve this problem, we first need to understand the behavior of charged particles in an electric field.
The potential difference between the plates of the capacitor represents the electric field strength between the plates. The electric field is directed from the positive plate to the negative plate.
For the electron:
1. The initial velocity of the electron is zero since it is released from rest.
2. The electron experiences a force due to the electric field, causing it to accelerate towards the positive plate.
3. The electric force on the electron is given by Fe = qe * E, where qe is the charge of the electron and E is the electric field strength.
4. The work done on the electron by the electric field is equal to the change in its kinetic energy, given by W = ΔKE = Fe * d, where d is the distance covered by the electron.
5. The electric potential difference between the plates, V, is equal to the work done per unit charge, which is V = W / qe.
For the proton:
1. The proton is projected perpendicularly towards the negative plate with an initial speed v.
2. The proton experiences a force due to the electric field, causing it to decelerate towards the negative plate.
3. The electric force on the proton is given by Fp = qp * E, where qp is the charge of the proton and E is the electric field strength.
4. The work done on the proton by the electric field is equal to the change in its kinetic energy, given by W = ΔKE = -Fp * d, since the force is in the opposite direction of motion.
5. The electric potential difference between the plates, V, is equal to the work done per unit charge, which is V = W / qp.
Since the electron and proton strike the plates at the same instant, the distances they travel are the same. Therefore, we can equate their electric potential differences:
V = W / qe = -W / qp
Now, we can solve for the initial speed of the proton, v:
qe * E = -qp * E
qe * V / d = -qp * V / d
qe = -qp
v = sqrt(2 * |qe| * V / mp),
where mp is the mass of the proton.
In this case, the charge of the proton is equal in magnitude but opposite in sign to the charge of the electron. Therefore, qe = -qp.
Let's substitute the given values to calculate the initial speed of the proton.