A proton, initially traveling in the +x-direction with a speed of 6.00×105m/s , enters a uniform electric field directed vertically upward. After traveling in this field for 4.38×10−7s , the proton’s velocity is directed 45 ∘ above the +x-axis.
To find the electric field strength, we can use the equation for the force experienced by a charged particle in an electric field.
The force experienced by a charged particle in an electric field is given by the equation:
F = qE
Where F is the force, q is the charge of the particle (in this case, the charge of a proton, which is 1.6 × 10^-19 coulombs), and E is the electric field strength.
The force experienced by the proton can also be expressed using Newton's second law of motion:
F = ma
Where m is the mass of the proton (1.67 × 10^-27 kilograms) and a is the acceleration of the proton.
Since the proton is initially traveling in the +x-direction and then moves at an angle of 45 degrees above the +x-axis, we can break down the change in velocity into the x and y components.
The change in x-component of velocity (Δvx) can be found using the equation:
Δvx = vf * cos(theta) - vi
Where vf is the final velocity of the proton in the x-direction, vi is the initial velocity of the proton in the x-direction, and theta is the angle between the final velocity vector and the +x-axis.
From the given information, it is mentioned that the proton's velocity is directed 45 degrees above the +x-axis after traveling in the electric field for a specific time. This means the x-component of the velocity remains unchanged, and only the y-component changes.
The change in y-component of velocity (Δvy) can be found using the equation:
Δvy = vf * sin(theta) - vi
Where vf is the final velocity of the proton in the y-direction.
We know that acceleration can be calculated using the equation:
a = Δv / t
Where Δv is the change in velocity and t is the time elapsed.
Plugging the values into the equation, we get:
Δvx = vf * cos(theta) - vi
Δvy = vf * sin(theta)
By substituting the values into these equations, we can calculate Δvx and Δvy.
Using the formula for acceleration:
Δvx / t = ax
Δvy / t = ay
We can solve for ax and ay.
Next, we can relate the acceleration to the force experienced by the proton.
F = ma = qE
Substituting the values, we obtain:
qE = m(ax^2 + ay^2)
We know the values for q, m, ax, and ay, so we can substitute them into the equation.
By rearranging the equation, we can solve for the electric field strength:
E = (q / m) * sqrt(ax^2 + ay^2)
Plugging in the known values for q, m, ax, and ay, we can calculate the electric field strength.