An electron is accelerated from rest through a potential difference that has a magnitude of 2.90 × 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.
relativistic KE=Voltage*charge
to find the speed,
relastivistic KE=restmass*c*gamma
gamma=(1/sqrt(1-(v/c)^2) -1)
so take relavitic KE, divide by restmass, then divide by speed of light.
result=gamma
Now ignore the -1 in the gamma, look at it later. If v is approaching c, it can be ignored.
square both sides (ignoring the -1)
solve for v in terms of c. If v/c is within .5 or greater, use that v, it is pretty accurate. If v/c is greater than .5, you have to consider the -1 in gamma. You can do that by expanding the gamma with the binomial expansion, and there are programs to do that, and I am told calculators to do it. I would not relish doing more than a few terms by hand.
To find the relativistic kinetic energy of the electron, we can use the formula:
K = (γ - 1)mc^2
where K is the relativistic kinetic energy, γ is the Lorentz factor, m is the mass of the electron, and c is the speed of light in a vacuum.
(a) First, let's calculate the Lorentz factor:
γ = 1 / √(1 - v^2/c^2)
where v is the velocity of the electron.
To find the velocity of the electron, we can use the formula for the potential difference:
V = qV
where V is the potential difference, q is the charge, and V is the voltage.
Rearranging the equation, we get:
V = Kq
Solving for K, we find:
K = V / q
Plugging in the values, we have:
K = (2.90 × 10^7 V) / (1.60 × 10^-19 C)
K = 1.81 × 10^26
Now, let's calculate the Lorentz factor:
γ = 1 / √(1 - (v^2 / c^2))
Rearranging the equation, we have:
v^2 / c^2 = 1 - 1 / γ^2
Simplifying the equation, we get:
v^2 / c^2 = 1 - 1 / (1.81 × 10^26)^2
v^2 / c^2 = 1 - 1 / 3.28 × 10^52
v^2 / c^2 = 0.99999999999 (approximately)
Taking the square root of both sides, we get:
v / c = √(0.99999999999)
v / c = 0.999999999995 (approximately)
(b) The speed of the electron is approximately equal to the speed of light in a vacuum:
v ≈ c
So, the speed of the electron is approximately c.
To answer this question, we need to use the concepts of potential difference and relativistic kinetic energy.
(a) The relativistic kinetic energy (K) of an object can be calculated using the formula:
K = (γ - 1) * mc^2
Where γ (gamma) is the Lorentz factor, m is the mass of the object, and c is the speed of light in a vacuum.
First, we need to calculate the Lorentz factor (γ). It can be calculated using the formula:
γ = 1 / √(1 - v^2/c^2)
Where v is the velocity of the object.
Since the electron is initially at rest (v = 0), the Lorentz factor simplifies to γ = 1.
Now, we can calculate the relativistic kinetic energy (K) using the formula:
K = (γ - 1) * mc^2
Given that the mass of the electron (m) is 9.11 × 10^-31 kg and the speed of light (c) is approximately 3.00 × 10^8 m/s, we can substitute these values into the formula to calculate K.
(b) To calculate the speed of the electron, we can use the formula for kinetic energy:
K = (1/2)mv^2
Rearranging the formula, we have:
v^2 = (2K) / m
Solving for v, we take the square root of both sides:
v = √((2K) / m)
Given the value of K from part (a) and the mass of the electron (m), we can substitute these values into the formula to calculate v.
Finally, to express the speed (v) as a multiple of c, we need to divide it by the speed of light (c):
v/c = √((2K) / m) / c
Now, we can follow these steps to calculate the relativistic kinetic energy and the speed of the electron.