An electron is accelerated horizontally from rest in a television picture tube by a potential difference
of 1.00 × 104 V.
(a) Find the final velocity after the original acceleration period.
(b) It then passes between two horizontal plates 6.0 cm long and 1.0 cm apart which have a
potential difference of 200 V. At what angle will the electron be traveling after it passes
between the plates?
any help is really appreciated..im extremely lost with questions like this,. thanks.
Sure! I can help you with that. Let's break it down step by step.
(a) To find the final velocity of the electron after the original acceleration period, we can use the kinematic equation for velocity:
v = u + at
Where:
v is the final velocity
u is the initial velocity (which is 0 in this case because the electron starts from rest)
a is the acceleration
t is the time taken
In this case, the potential difference of 1.00 × 104 V is equivalent to the electric potential energy (EPE) gained by the electron. This EPE is converted into kinetic energy, given by:
EKE = (1/2)mv^2
Where:
m is the mass of the electron
v is the final velocity
Since the electron is accelerated by a potential difference, the kinetic energy gained will be equal to the initial electric potential energy:
EKE = EPE
Substituting the equations for EKE and EPE:
(1/2)mv^2 = qV
Where:
q is the charge of the electron
V is the potential difference
The charge of the electron (q) is a fundamental constant with a value of approximately 1.6 × 10^-19 C.
Solving for v:
v^2 = (2qV) / m
v = sqrt((2qV) / m)
Now we have the formula to find the final velocity of the electron after the original acceleration period.
(b) To find the angle at which the electron will be traveling after passing between the plates, we can use the conservation of energy.
The electric potential energy lost by the electron between the plates is converted into kinetic energy. The change in electric potential energy is given by:
ΔEPE = qΔV
Where:
q is the charge of the electron
ΔV is the potential difference across the plates
The kinetic energy gained by the electron will be equal to the change in electric potential energy:
EKE = ΔEPE
Substituting the equations for EKE and ΔEPE:
(1/2)mv_f^2 - (1/2)mv_i^2 = qΔV
Where:
m is the mass of the electron
v_f is the final velocity after passing between the plates
v_i is the initial velocity before passing between the plates
ΔV is the potential difference across the plates
Since the electron is initially accelerated horizontally, the initial velocity (v_i) is equal to the final velocity (v) we found in part (a).
Substituting v_i = v:
(1/2)mv_f^2 - (1/2)mv^2 = qΔV
Now we can solve this equation for the final velocity (v_f). However, keep in mind that the electron will be traveling at an angle after passing between the plates, so we need to consider the horizontal and vertical components of velocity.
v_f = sqrt((2qΔV) / m) * sinθ
Where:
θ is the angle at which the electron will be traveling after passing between the plates.
To find the angle (θ), we can use simple trigonometry. The horizontal component of velocity (v_h) will be equal to the horizontal distance between the plates divided by the time taken to pass between them.
v_h = d / t
Where:
d is the horizontal distance between the plates (6.0 cm = 0.06 m)
t is the time taken to pass between the plates
The vertical component of velocity (v_v) can be found using the formula:
v_v = v_f * cosθ
Now we can use the components of velocity (v_h and v_v) to find the angle (θ) using trigonometry:
sinθ = v_v / v_f
θ = arcsin(v_v / v_f)
I hope this helps! Let me know if you have any further questions.
To solve this problem, we'll need to use some basic principles of physics, such as the equations of motion and the Coulomb's law. Here's how you can approach each part of the problem:
(a) Finding the final velocity after the original acceleration period:
1. Determine the initial velocity (u) of the electron. The problem states that it starts from rest, so the initial velocity is zero.
2. Note down the potential difference (V) applied, which is 1.00 × 10^4 V.
3. Utilize the equation of motion that relates final velocity (v), initial velocity (u), acceleration (a), and distance (s):
v^2 = u^2 + 2as
4. Since the electron is accelerated only horizontally, the vertical distance traveled would be zero. Therefore, the distance (s) mentioned in the equation is not relevant here.
5. Identify the acceleration (a) of the electron, which is caused by the potential difference applied. We can calculate it using the equation:
V = a * d
where V is the potential difference and d is the horizontal distance. In this case, d is not mentioned, so we'll assume it doesn't influence the acceleration.
6. Equate the equation from step 3 to zero, since the initial velocity is zero and the distance is zero along the vertical direction:
v^2 = 0 + 2 * a * 0
v^2 = 0
v = 0
Therefore, the final velocity (v) after the original acceleration period is 0.
(b) Finding the angle at which the electron will be traveling after it passes between the plates:
1. Calculate the electric field (E) between the two plates using the Coulomb's law:
E = V / d
where V is the potential difference and d is the distance between the plates. In this case, V is 200 V and d is 1.0 cm = 0.01 m.
2. Determine the force (F) experienced by the electron in the electric field, using the equation:
F = q * E
where q is the charge of the electron. The charge of an electron is approximately -1.6 × 10^-19 C.
3. Apply the force (F) to the electron for the distance it travels between the plates, which is 6.0 cm = 0.06 m. The work done (W) is given by:
W = F * s * cosθ
where s is the distance and θ is the angle between the force and displacement vectors. In this case, θ is the angle at which the electron will be traveling after it passes between the plates.
4. Since the electron is not initially moving vertically, there is no vertical displacement. Therefore, we can set s = 0.06 m.
5. Rearrange the equation from step 3 to solve for cosθ:
cosθ = W / (F * s)
cosθ = (W / F) / s
6. Substitute the values obtained from previous steps into the equation in step 5 and calculate cosθ.
7. Finally, take the inverse cosine (arccos) of cosθ to find the angle θ at which the electron will be traveling after it passes between the plates.
I hope this explanation helps you understand the steps involved in solving this problem. If you have any further questions, feel free to ask.