Find the final speed gained by a proton when it is accelerated from rest through a potential difference of 100 volts. Also find its final kinetic energy in electron volt
To find the final speed gained by a proton when it is accelerated through a potential difference, we can use the formula for the kinetic energy gained, as well as the charge and mass of a proton.
The formula for the kinetic energy gained by a charged particle accelerated through a potential difference is given by:
K.E. = qV
where K.E. is the kinetic energy gained, q is the charge of the particle, and V is the potential difference.
Now, let's substitute the given values into the formula:
q = charge of a proton = 1.6 x 10^-19 Coulombs (C)
V = potential difference = 100 volts (V)
K.E. = (1.6 x 10^-19 C) x (100 V)
Calculating this expression, we find:
K.E. = 1.6 x 10^-17 Joules (J)
To convert this kinetic energy from Joules to electron volts (eV), we can use the conversion factor:
1 eV = 1.6 x 10^-19 J
Now, let's substitute the value of K.E. in Joules into the conversion formula to find the kinetic energy in electron volts:
K.E. (eV) = (1.6 x 10^-17 J) / (1.6 x 10^-19 J/eV)
Simplifying this expression, we get:
K.E. (eV) = 100 eV
So, the final kinetic energy of the proton is 100 electron volts (eV).
To find the final speed gained by the proton, we can use the formula for kinetic energy:
K.E. = (1/2)mv^2
where K.E. is the kinetic energy gained, m is the mass of the proton, and v is the final velocity.
Since the proton is accelerated from rest, its initial kinetic energy is zero. Using this information, we can rewrite the formula as:
K.E. = ΔK.E. = (1/2)mv^2
Where ΔK.E. is the change in kinetic energy.
We already calculated the change in kinetic energy as 1.6 x 10^-17 J. Now, let's substitute this value and the mass of a proton into the formula to solve for the final velocity:
1.6 x 10^-17 J = (1/2)(mass of proton)(v^2)
The mass of a proton is approximately 1.67 x 10^-27 kg. Substituting this value into the equation, we have:
1.6 x 10^-17 J = (1/2)(1.67 x 10^-27 kg)(v^2)
Simplifying this expression, we find:
v^2 = (2 x 1.6 x 10^-17 J) / (1.67 x 10^-27 kg)
v^2 = 1.92 x 10^10 m^2/s^2
Taking the square root of both sides, we get:
v ≈ 1.39 x 10^5 m/s
Therefore, the final speed gained by the proton is approximately 1.39 x 10^5 meters per second.