Final answer:
To accelerate an electron to gain 96 keV energy in a 1.20×10⁶ V/m electric field, it must travel through 0.500 m. To increase the electron's energy by 50.0 GeV, it would have to be accelerated over a distance of 26.04167 km.
Step-by-step explanation:
Calculating Electron Energy and Distance in an Electric Field
For an electron accelerated in a uniform electric field, the energy acquired by the electron can be calculated using the relationship between energy (eV), electric field (V/m), and distance (m). Given a field strength of 1.20×10⁶ V/m, the energy imparted to the electron as it travels through a distance of 0.500 m is found by using this formula:
Energy (eV) = Electric Field Strength (V/m) × Distance (m) × Charge of an electron (e)
The charge of an electron, e, is 1.60×10⁻¹⁹ C. Therefore, the energy gained in electronvolts (eV) is 1.20×10⁶ V/m × 0.500 m × 1.60×10⁻¹⁹ C which equals 96,000 eV, or 96 keV when converted to kilo-electron volts.
To find out over what distance it needs to be accelerated to increase its energy by 50.0 GeV, we again use the same relationship:
Distance (m) = Energy (eV) / (Electric Field Strength (V/m) × Charge of an electron (e))
Given that 50.0 GeV is equivalent to 5.0×10⁹ eV, and using the electric field strength of 1.20×10⁶ V/m, the distance needed would be 5.0×10⁹ eV / (1.20×10⁶ V/m × 1.60×10⁻¹⁹ C), which equals 26.04167 km.