A proton is moving horizontally halfway between two parallel plates that are separated by 0.60 cm. The electric field due to the plates has magnitude 720,000 N/C between the plates away from the edges. If the plates are 5.6 cm long, find the minimum speed of the proton if it just misses the lower plate as it emerges from the field.

v = 4,244,699 m/s = (4.245 × 10⁶) m/s

Explanation:

The electric force on the proton is given by

F = qE

where q = charge on the proton = (1.602 × 10⁻¹⁹) C

E = Electric field = 720,000 N/C

F = (1.602 × 10⁻¹⁹ × 720000)

F = (1.153 × 10⁻¹³) N

But this force will accelerate the proton in this magnetic field in a form of trajectory motion.

We can obtain the acceleration using Newton's first law of motion relation

F = ma

m = mass of a proton = (1.673 × 10⁻²⁷) kg

a = (F/m)

a = (1.153 × 10⁻¹³)/(1.673 × 10⁻²⁷)

a = 68,944,411,237,298 m/s²

a = (6.894 × 10¹³) m/s²

This acceleration directs the proton from the positive plate to the negative plate, covering a distance of y = 0.006 m (the distance between the plates)

Using Equations of motion, we can obtain the time taken for the proton to move from the rest at the positive plate to the negative one.

u = initial velocity of the proton = 0 m/s

y = vertical distance covered by the proton = 0.006 m

a = acceleration of the proton in this direction = (6.894 × 10¹³) m/s²

t = time taken for the proton to complete this distance = ?

y = ut + (1/2) at²

0.006 = 0 + [(1/2)×(6.894 × 10¹³)×t²]

0.006 = (3.447 × 10¹³) t²

t² = (0.006)/(3.447 × 10¹³)

t² = 1.741 × 10⁻¹⁶

t = (1.32 × 10⁻⁸) s

Then we can then calculate the minimum speed to navigate the entire length of the plates without hitting the plates.

v = ?

x = 0.056 n

t = (1.32 × 10⁻⁸)

v = (x/t)

v = (0.056)/(1.32 × 10⁻⁸)

v = 4,244,699 m/s = (4.245 × 10⁶) m/s

Hope this Helps!!!

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