A beam of electrons traveling with a speed of 3.00 x 107 m/s enters a uniform, downward electric field of magnitude 1.17 x 104 N/C between the deflection plates of an oscilloscope (the figure below ). The initial velocity of the electrons is perpendicular to the field. The plates are L = 6.50 cm long. What is the magnitude of the change in velocity of the electrons while they are between the plates?
How far are the electrons deflected in the �y-direction while between the plates?
To find the magnitude of the change in velocity of the electrons between the plates, we need to use the equation for the acceleration of a charged particle in an electric field:
a = qE/m
Where:
- a is the acceleration of the electrons
- q is the charge of the electron (-1.6 x 10^-19 C)
- E is the electric field strength (1.17 x 10^4 N/C)
- m is the mass of the electron (9.1 x 10^-31 kg)
Now, we can plug in the values:
a = (qE) / m
= (-1.6 x 10^-19 C) * (1.17 x 10^4 N/C) / (9.1 x 10^-31 kg)
≈ -1.60 x 10^15 m/s^2
Since the acceleration is negative, we know that the electrons are decreasing in velocity. The magnitude of the acceleration is the same as the magnitude of the change in velocity.
Change in velocity = |a| = 1.60 x 10^15 m/s^2
Now, to find how far the electrons are deflected in the y-direction, we can use the equation for displacement:
s = 1/2 * a * t^2
Where:
- s is the displacement (deflection in the y-direction)
- a is the acceleration (from the previous calculation)
- t is the time it takes for the electrons to travel between the plates
To find the time, we can use the equation for the distance traveled by an object with constant velocity:
d = v * t
Where:
- d is the distance between the plates (6.50 cm) = 0.065 m
- v is the initial velocity of the electrons (3.00 x 10^7 m/s)
Rearranging the equation, we can solve for t:
t = d/v
= (0.065 m) / (3.00 x 10^7 m/s)
≈ 2.17 x 10^-9 s
Now we can substitute the values into the displacement equation:
s = 1/2 * a * t^2
= 1/2 * (-1.60 x 10^15 m/s^2) * (2.17 x 10^-9 s)^2
≈ -3.23 x 10^-4 m
Since the displacement is negative, we know that the electrons are deflected in the negative y-direction. The magnitude of the displacement is |s| = 3.23 x 10^-4 m.
Therefore, the magnitude of the change in velocity of the electrons between the plates is 1.60 x 10^15 m/s^2, and the electrons are deflected approximately 3.23 x 10^-4 m in the negative y-direction.
To solve these questions, we can use the equations of motion for a charged particle under the influence of an electric field.
Let's start by finding the magnitude of the change in velocity of the electrons while between the plates:
1. The electric field force on a charged particle is given by F = q * E, where F is the force, q is the charge of the particle, and E is the electric field strength.
2. The force experienced by an electron is F = -e * E, where e is the elementary charge (-1.6 x 10^-19 C) and E is the electric field strength (1.17 x 10^4 N/C).
3. The electric force on an object can be related to its acceleration using Newton's second law: F = m * a, where m is the mass of the object and a is the acceleration.
4. The acceleration (a) of an electron is given by a = F / m, where m is the mass of the electron (9.11 x 10^-31 kg).
5. The change in velocity (Δv) of the electron can be found using the equation Δv = a * t, where t is the time.
Now let's calculate the magnitude of the change in velocity:
Given:
Initial velocity (v) = 3.00 x 10^7 m/s
Electric field strength (E) = 1.17 x 10^4 N/C
Length between plates (L) = 6.50 cm = 0.065 m
Step 1: Calculate the time (t) taken to cross the plates.
t = L / v
t = 0.065 m / 3.00 x 10^7 m/s
t ≈ 2.17 x 10^-9 s
Step 2: Calculate the acceleration (a) of the electron.
a = (-e * E) / m
a = (-1.6 x 10^-19 C * 1.17 x 10^4 N/C) / 9.11 x 10^-31 kg
a ≈ -1.96 x 10^14 m/s^2 (accelerating downwards)
Step 3: Calculate the magnitude of the change in velocity (Δv).
Δv = a * t
Δv = -1.96 x 10^14 m/s^2 * 2.17 x 10^-9 s
Δv ≈ -4.25 x 10^5 m/s
Therefore, the magnitude of the change in velocity of the electrons between the plates is approximately 4.25 x 10^5 m/s.
Now let's find out how far the electrons are deflected in the y-direction while between the plates:
Step 1: Calculate the displacement (s) in the y-direction.
s = v * t
s = 3.00 x 10^7 m/s * 2.17 x 10^-9 s
s ≈ 6.51 x 10^-2 m
Therefore, the electrons are deflected approximately 6.51 x 10^-2 m in the y-direction while between the plates.