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 = 21.2810¹²J1.6710²kg

v = 2.5610¹²J1.6710²kg

v = 1.53310¹²²m²s²

v = 1.53310¹m²s²

or

v = 15.3310¹m²s²

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

= 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 = 3.91510ms¹1.6710²kg1.610¹C2.5T


r = 6.53810²kgms¹410¹CT


r = 1.634 x 10⁻¹ m


or


r = 0.1634 m -----------------------Ans. (2)



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