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.)
s=at²/2
a=2s/t²
F=ma
eU =Fs =mas =m2s²/t²
U= 2ms²/et² =
=2•9.1•10⁻³¹•(0.15)²/1.6•10⁻¹⁹•(19•10⁻⁹)²=
=709 V
To calculate the voltage used to accelerate the electron, we can use the equation for the acceleration of an electron in an electric field:
a = (2 * d) / (t^2)
Where:
a is the acceleration of the electron
d is the distance traveled by the electron
t is the time taken to travel the distance
Given:
d = 0.150 m
t = 19.0 ns = 19.0 × 10^(-9) s
Substituting these values into the equation:
a = (2 * 0.150 m) / (19.0 × 10^(-9) s)^2
Next, we can use the equation for acceleration in terms of voltage:
a = (e * V) / m
Where:
a is the acceleration of the electron
e is the charge of the electron (1.6 × 10^(-19) C)
V is the applied voltage
m is the mass of the electron (9.11 × 10^(-31) kg)
Substituting these values into the equation:
(e * V) / m = (2 * 0.150 m) / (19.0 × 10^(-9) s)^2
Simplifying the equation:
V = (2 * 0.150 m * m) / (e * (19.0 × 10^(-9) s)^2)
Now, we can substitute the known values:
V = (2 * 0.150 m * 9.11 × 10^(-31) kg) / (1.6 × 10^(-19) C * (19.0 × 10^(-9) s)^2)
Calculating the result:
V ≈ 2584.273 V
Therefore, the voltage used to accelerate the electron is approximately 2584.273 V.
To find the voltage used to accelerate the electron, we can use the equation for the energy gained by a charged particle accelerated through a potential difference. The equation is:
E = qV
Where:
E is the energy gained by the electron
q is the charge of the electron (1.6 x 10^-19 coulombs)
V is the voltage used to accelerate the electron
We know that the electron traveled a distance of 0.150 m in a time of 19.0 ns. We can use these values to find the velocity of the electron.
Velocity (v) = Distance (d) / Time (t)
v = 0.150 m / 19.0 x 10^-9 s
Now, we need to find the kinetic energy (K) gained by the electron using the equation:
K = (1/2)mv^2
Where:
m is the mass of the electron (9.11 x 10^-31 kg)
We can rearrange this equation to solve for the energy gained:
E = K = (1/2)mv^2
Substituting the value of the velocity calculated earlier, we can find the energy gained by the electron.
Now we can plug in the energy gained by the electron and the charge of the electron into the original equation to solve for the voltage:
E = qV
Solving for V:
V = E / q
Now, let's calculate the voltage used to accelerate the electron.