what potential difference delta V is required to accelerate protons from the rest to 10% of the speed of light?
To calculate the potential difference (ΔV) required to accelerate protons, we can make use of the kinetic energy equation and the relation between kinetic energy and potential difference.
First, let's calculate the kinetic energy (KE) of a proton moving at 10% of the speed of light. The relativistic kinetic energy equation is:
KE = (γ - 1) * (m0 * c^2)
where
KE is the kinetic energy of the proton,
γ is the Lorentz factor (γ = 1 / sqrt(1 - v^2 / c^2)),
m0 is the rest mass of the proton,
c is the speed of light,
v is the velocity of the proton.
Given that the proton is at rest initially, v = 0.1c (10% of the speed of light).
Now, we can calculate the kinetic energy using the following known values:
m0 (rest mass of the proton) = 1.67 × 10^(-27) kg
c (speed of light) = 3.00 × 10^8 m/s
Plugging in these values, the equation becomes:
KE = (1 / sqrt(1 - 0.1^2)) * (1.67 × 10^(-27) kg) * (3.00 × 10^8 m/s)^2
Calculating this expression will give us the kinetic energy of the proton when it is moving at 10% of the speed of light.
Next, we know that the kinetic energy (KE) is equal to the potential difference (ΔV). So, to find the potential difference required, we substitute KE in the equation with ΔV.
ΔV = KE
Finally, we can solve for ΔV by substituting the value of KE calculated earlier into this equation.