Answer:
The time required for the proton to re-emerge into the field-free region is 5.47 x 10⁻⁸ s
Step-by-step explanation:
given;
mass of proton, m = 1.67 × 10⁻²⁷ kg
charge of proton, q = 1.6 × 10⁻¹⁹ C
speed of proton, v = 1 × 10⁶ m/s
uniform magnetic field, B = 1.2 T
Force on the proton due to uniform magnetic field = qvB
Force on the proton in the circular region = ma = mv²/r
The time required for the proton to re-emerge into the field-free region, is the same as the time required for the proton to return to its starting position. During this period, the proton has made one complete revolution.
v = 2πr/t
t = 2πr/v
where;
t is the required for the proton to re-emerge into the field-free region
r is the radius of the circular region
mv²/r = qvB
r = mv²/qvB
r = mv/qB
r = (1.67 × 10⁻²⁷ x 1 × 10⁶) / (1.6 × 10⁻¹⁹ x 1.2)
r = 8.698 x 10⁻³ m
t = 2πr/v
t = (2π x 8.698 x 10⁻³) / (1 × 10⁶)
t = 5.47 x 10⁻⁸ s
Therefore, the time required for the proton to re-emerge into the field-free region is 5.47 x 10⁻⁸ s