An 8 MeV proton enters perpendicularly into a uniform magnetic field of 2.5 T. (a) Find the force on the electron (b) What will be the radius of the path of the proton? (Answer: 1.6 x 10⁻¹¹ N, 0.17 m)
Given:
K.E. of Proton = 8 MeV = 8 x 10⁶ x 1.6 x 10⁻¹⁹ J = 1.28 x 10⁻¹² J
Angle = θ = 90°
Magnetic Field = B = 2.5 T
Force = F = 2 x 10⁻¹² N
[ ∴ Charge of proton = q = 1.6 x 10⁻¹⁹ C]
[ ∴ Mass of proton = m = 1.67 x 10⁻²⁷ kg]
To Find:
(a) Magnetic Force on Electron = F = ?
(b) Radius of the path of Proton = r = ?
Solution:
(a) Magnetic Force on Electron = F = ?
Since we have the magnetic force formula:
F = q ( v x B )
F = q v B sinθ --------------------(1)
To find the magnitude of the magnetic force, we have to first find the velocity of the Proton. Keeping the given quantity we have a formula for K.E.
K.E. = 12 mv²
or
v² = 2K.E.m
Or
v = √2K.E.m
by putting the corresponding values
v = √2x1.28x10⁻¹²J1.67x10⁻²⁷kg
v = √2.56x10⁻¹²J1.67x10⁻²⁷kg
v = √1.533x10⁻¹²⁺²⁷m²s⁻²
v = √1.533x10¹⁵m²s⁻²
or
v = √15.33x10¹⁴m²s⁻²
v = 3.915 x 10⁷ m s⁻¹ --------------(2)
Now putting the value of velocity v of proton eqn (2) and other corresponding values in equation (1)
F = q v B sinθ
F = 1.6 x 10⁻¹⁹ J x 3.915 x 10⁷ m s⁻¹ x 2.5 T sin90°
{ sin90° = 1}
F = 15.666 x 10⁻¹⁹⁺⁷ N
F = 15.666 x 10⁻¹² N
or
F = 1.5666 x 10⁻¹¹ N ------------------ Ans. (1)
-----------------------------------------------
(b) Radius of the path of Proton = r = ?
We have a formula for the charge (e ) to mass (m) ratio:
em = vBr
or
r = vmeB
by putting values
r = 3.915x10⁷ms⁻¹x1.67x10⁻²⁷kg1.6x10⁻¹⁹Cx2.5T
r = 6.538x10⁻²⁰kgms⁻¹4x10⁻¹⁹CT
r = 1.634 x 10⁻¹ m
or
r = 0.1634 m -----------------------Ans. (2)
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