If an electron travels 0.150 m from an electron gun to a TV screen in 19.0 ns, what voltage was used to accelerate it? (Note that the voltage you obtain here is lower than actually used in TVs to avoid the necessity of relativistic corrections.)
To find the voltage used to accelerate the electron, we can use the equation for the electric potential energy of a charged particle:
ΔPE = q ΔV
where:
ΔPE is the change in potential energy
q is the charge of the electron
ΔV is the change in voltage
We can calculate the charge of an electron using the elementary charge:
e = 1.602 × 10^-19 C
Now, let's solve this step-by-step.
Step 1: Calculate the time (Δt)
The time (Δt) is given as 19.0 ns. However, we need to convert it to seconds:
Δt = 19.0 × 10^-9 s
Step 2: Calculate the velocity (v)
The distance traveled (0.150 m) is given, and we can use it to calculate the velocity using the equation:
v = Δx / Δt
v = 0.150 m / (19.0 × 10^-9 s)
Step 3: Calculate the change in potential energy (ΔPE)
The change in potential energy (ΔPE) corresponds to the kinetic energy gained by the electron. Using the equation for kinetic energy:
KE = (1/2)mv^2
where:
m is the mass of the electron
v is the velocity
The mass of an electron (m) is approximately:
m = 9.109 × 10^-31 kg
Now we can calculate the change in potential energy:
ΔPE = (1/2)mv^2
Step 4: Calculate the change in voltage (ΔV)
We can rearrange the equation ΔPE = q ΔV to solve for ΔV:
ΔV = ΔPE / q
Plugging in the values for ΔPE and q, we get:
ΔV = (ΔPE) / (1.602 × 10^-19 C)
Finally, we can calculate the voltage used to accelerate the electron by substituting all the values into the equation for ΔV.
To determine the voltage used to accelerate the electron, we can apply the formula for the acceleration of a charged particle in an electric field. The formula is:
acceleration = (voltage * charge) / mass
Since we already have the distance traveled and the time taken by the electron, we can calculate the initial velocity of the electron using the formula:
initial velocity = distance / time
Next, we need to calculate the acceleration of the electron using the formula:
acceleration = (final velocity - initial velocity) / time
Now, we can rearrange the equation for acceleration to solve for voltage:
voltage = (acceleration * mass) / charge
Let's calculate each step to find the voltage value:
Step 1: Calculate initial velocity
Given: distance = 0.150 m, time = 19.0 ns (convert to seconds)
initial velocity = 0.150 m / (19.0 ns * 10^-9 s/ns) = 7.89 x 10^6 m/s
Step 2: Calculate acceleration
Since we know the electron's initial velocity, we assume the final velocity is 0 m/s (as it hits the TV screen).
acceleration = (0 m/s - 7.89 x 10^6 m/s) / (19.0 ns * 10^-9 s/ns) = -4.15 x 10^14 m/s^2
Step 3: Calculate voltage
Given: charge of an electron = 1.6 x 10^-19 C, mass of an electron = 9.11 x 10^-31 kg
voltage = (-4.15 x 10^14 m/s^2 * 9.11 x 10^-31 kg) / (1.6 x 10^-19 C) = -2.34 x 10^6 V
Therefore, the voltage used to accelerate the electron is approximately -2.34 x 10^6 volts. The negative sign indicates that the electron is negatively charged and is attracted towards the positive electrode in the electron gun.