a) Minimum speed of the proton:
b) Angle of the velocity:
Explanation:
a)
The proton experiences a vertical force due to the electric field, given by:
where
is the proton charge
is the magnitude of the electric field
The vertical acceleration of the proton is therefore
where
is its mass
Therefore, the vertical position of the proton at time t is
where we assumed that the initial vertical velocity is zero (because the proton is fired horizontally) and the initial vertical position, halfway between the two plates, is the origin.
The horizontal motion of the proton instead is uniform, so the horizontal position is given by
where is the initial speed. This equation can be rewritten as
And substituting into the eq. for y,
Solving for the initial speed,
The proton just misses one of the plate when
x = 5.60 cm = 0.056 m (length of the plates)
y = 0.25 cm = 0.0025 m (half the distance between the plates)
Therefore, we find the initial speed:
b)
In order to find the angle, we just need to analyze the horizontal and vertical component of the final velocity of the proton.
The horizontal velocity is constant so it is:
The vertical velocity is given by:
where:
(initial vertical velocity is zero)
(acceleration)
y = 0.0025 m (vertical displacement)
Solving for ,
Therefore, the final angle of the velocity with respect to the horizontal is:
And since the electric field is downward (the proton just misses the lower field), it means that the angle is below the horizontal:
Learn more about electric fields:
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