An electron is accelerated from rest through a potential difference that has a magnitude of 2.70 × 107 V. The mass of the electron is 9.11 × 10-31 kg, and the negative charge of the electron has a magnitude of 1.60 × 10-19 C. (a) What is the relativistic kinetic energy (in joules) of the electron? (b) What is the speed of the electron? Express your answer as a multiple of c, the speed of light in a vacuum.

Question

An electron is accelerated from rest through a potential difference that has a magnitude of 2.70 × 107 V. The mass of the electron is 9.11 × 10-31 kg, and the negative charge of the electron has a magnitude of 1.60 × 10-19 C. (a) What is the relativistic kinetic energy (in joules) of the electron? (b) What is the speed of the electron? Express your answer as a multiple of c, the speed of light in a vacuum.

I was able to solve for Part A = 4.32e-12 J.

But I am unable to get Part B correct.
When I solved I obtained 9.484083425e18c.

Can you please help me solve Part B? Thanks.

Answer 1

Sure, I can help you solve Part B of the problem. To find the speed of the electron, we need to use the relativistic kinetic energy equation and solve for the velocity.

The relativistic kinetic energy of an object (in this case, an electron) is given by the equation:

K = (γ - 1)mc^2

where K is the relativistic kinetic energy, γ is the Lorentz factor, m is the mass of the object, and c is the speed of light in vacuum.

In this case, you have already calculated the kinetic energy as 4.32 × 10^(-12) J.

So, let's rearrange the equation to solve for γ:

γ = (K / (mc^2)) + 1

Substituting the values:

γ = (4.32 × 10^(-12) J) / ((9.11 × 10^(-31) kg) × (3 × 10^8 m/s)^2) + 1

γ ≈ 1.0000000002405 (approximated to 12 decimal places)

Now that we have γ, we can calculate the velocity (v) of the electron using the Lorentz factor:

γ = 1 / sqrt(1 - (v^2 / c^2))

Rearranging the equation, we get:

v^2 / c^2 = 1 - (1 / γ^2)

v^2 / c^2 ≈ 1 - (1 / (1.0000000002405)^2)

v^2 / c^2 ≈ 7.92374511 × 10^(-19)

Taking the square root of both sides, we have:

v / c ≈ sqrt(7.92374511 × 10^(-19))

v / c ≈ 2.815285907 × 10^(-10)

Finally, we multiply the obtained ratio by the speed of light in a vacuum (c) to find the speed of the electron:

v ≈ 2.815285907 × 10^(-10) × 3 × 10^8 m/s

v ≈ 8.445857721 × 10^(-2) m/s

Thus, the speed of the electron is approximately 8.45 × 10^(-2) m/s.

To express this as a multiple of c, we divide the obtained speed by the speed of light in a vacuum:

v / c ≈ (8.45 × 10^(-2) m/s) / (3 × 10^8 m/s)

v / c ≈ 2.82 × 10^(-10)

So, the speed of the electron is approximately 2.82 × 10^(-10) times the speed of light in a vacuum (c).

I hope this helps! Let me know if you have any further questions.