How much work must be done to increase the speed of an electron (a) from 0.18c to 0.19c and (b) from 0.98c to 0.99c? Note that the speed increase is 0.01c in both cases.
Total energy is
E(rest) + E(kin)
= m*c^2/sqrt[1 - (v/c)^2]
Calculate that at each speed and take the difference.
(a) m*c^2/sqrt[1 - .19^2]
- m*c^2/sqrt[1 - .18^2]
= m*c^2[1.01855 - 1.01660]
= m*c^2* 1.95*10^-3
I got 1*10^-16 but what's the unit for my answer? Because the book says 1.0 ke V is the answer for part (a). How do I get there?
You should have got 1.6*10^(-16) as your answer - Which is the value of 1keV.
To calculate the work done to increase the speed of an electron, we need to use the formula for the work-energy principle. The work done (W) is equal to the change in kinetic energy (ΔKE) of the electron.
The formula for kinetic energy (KE) is:
KE = (1/2) * m * v^2,
where m is the mass of the electron and v is its velocity. Since the mass of an electron is constant, the change in kinetic energy is directly proportional to the square of the change in velocity.
(a) To calculate the work done to increase the speed of the electron from 0.18c to 0.19c, we need to find ΔKE. Since the velocity increases by 0.01c, we have:
ΔKE = (1/2) * m * (0.19c)^2 - (1/2) * m * (0.18c)^2.
(b) To calculate the work done to increase the speed of the electron from 0.98c to 0.99c, we follow the same steps. The velocity again increases by 0.01c, so we have:
ΔKE = (1/2) * m * (0.99c)^2 - (1/2) * m * (0.98c)^2.
To find the values of ΔKE, you need to know the mass of the electron (m) and the speed of light (c). In SI units, the mass of an electron is approximately 9.11 × 10^−31 kg, and the speed of light is approximately 3 × 10^8 meters per second (m/s).