. A proton is released from rest at point A in a uniform electric field that has a magnitude of . The proton undergoes a displacement to point B in the direction of . Calculate the speed of the proton after completing the displacement.
To calculate the speed of the proton after completing the displacement in the uniform electric field, we need to use the concept of work and energy.
Here are the steps to solve this problem:
Step 1: Determine the electric force acting on the proton.
The electric force (F) acting on a charged particle in a uniform electric field is given by the equation: F = q * E,
where q is the charge of the particle and E is the magnitude of the electric field.
In this problem, the charge of the proton (q) is a fundamental constant, which is q = +1.6 x 10^-19 C (Coulombs), and the magnitude of the electric field (E) is given.
Step 2: Determine the work done on the proton by the electric field.
The work done (W) on the proton as it moves from point A to point B in the direction of the displacement is given by the equation: W = F * d,
where d is the displacement vector.
Since the electric field and the displacement are parallel, the angle between them is 0 degrees, which means cos(0) = 1. Therefore, the work done can be simplified to: W = F * d * cos(0).
Step 3: Calculate the speed of the proton.
The work done on the proton is equal to the change in kinetic energy (KE) of the proton, as per the work-energy principle. Therefore, we can write the equation: W = KE_final - KE_initial.
Since the proton is released from rest at point A, its initial kinetic energy (KE_initial) is zero.
So, the equation becomes: W = KE_final - 0.
Since we know the work done (W), we can find the final kinetic energy (KE_final).
The kinetic energy of the proton is given by the equation: KE = (1/2) * m * v^2,
where m is the mass of the proton and v is its velocity (speed).
The mass of the proton (m) is a fundamental constant, which is m = 1.67 x 10^-27 kg.
Solving for v, we can rearrange the equation to: v = √(2 * KE / m).
Finally, substitute the value of KE_final into the equation to find the speed of the proton after completing the displacement.