Final answer:
The potential difference an electron must pass through to be accelerated from a given initial to a final velocity can be determined by calculating the change in kinetic energy and equating it to the work done by the electric field. This is done by using the mass and charge of the electron and applying the principle of energy conservation.
Step-by-step explanation:
To find the potential difference that an electron must pass through to be accelerated from a velocity of 4.50×106 m/s to 8.00×106 m/s, we use the principle of energy conservation. Specifically, the change in kinetic energy (ΔKE) of the electron must be equal to the work done on the electron by the electric field, which is the charge of the electron times the potential difference (ΔV).
ΔKE = ½ m(v2f - v2i),
where m is the mass of the electron, vf is the final velocity and vi is the initial velocity.
Since an electron accelerated through a potential difference of 1 V gains an energy of 1 eV, we can equate the change in kinetic energy in joules to the energy in electron volts to find the potential difference.
Firstly, calculate the change in the kinetic energy (in joules):
ΔKE = ½ (9.11 × 10-31 kg) ((8.00 × 106 m/s)2 - (4.50 × 106 m/s)2)
Note that 1 eV = 1.60×10-19 J, so the energy in electron volts can be obtained by dividing the change in kinetic energy by the charge of the electron:
ΔV = ΔKE / e,
where e is the absolute value of the electron's charge (1.60 × 10-19 C).
Once ΔKE is found in joules, divide by e to find the potential difference. The answer should be in volts (V).