A beam of particles, each with a charge of 24.0 x 10-19 C, moves at [ (-3.68)i + (6.35)j + (-1.91)k ] x 105 m/s through a uniform magnetic field of [ (-3.05)i + (5.28)j + (-1.09)k ] T. Let F represent the force vector on each particle.

What is the i component of F?
(include units with answer)

What is the j component of F?
(include units with answer)

What is the k component of F?
(include units with answer)

What is the magnitude of F?
(include units with answer)

A uniform electric field of [ (5.94)i + (-3.97)j + (-5.55)k ] x 105 N/C is added to the system. What is the new magnitude of F?
(include units with answer)

The i component of F is 7963.2 x 10⁻¹⁹ N.

The j component of F is 4561.2 x 10⁻¹⁹ N.

The k component of F is -151.2 x 10⁻¹⁹ N.

Magnitude of force is 9178.05 x 10⁻¹⁹ N.

Magnitude of force in presence of electric field is 27483.91 x 10⁻¹⁹ N.

Explanation:

Lorentz Force is defined as the force experience by charge particle in presence of electric and magnetic field. It is given by :

F = q ( v x B + E )     .....(1)

Here F is force, v is velocity of particle , B is magnetic field and E is electric field all are in vector notation and q is charge of the particle.

In first case, given :

E = 0

q = 24 x 10⁻¹⁹ C

v = [(-3.68)i + (6.35)j + (-1.91)k] x 105 m/s

B = [(-3.05)i + (5.28)j + (-1.09)k] T

Force, F = q x (v x B )

Substitute the suitable values in the above equation.

F = 24 x 10⁻¹⁹ x { [6.35 x (-1.09) - (-1.91) x (5.28)] i - [(-3.68) x (-1.09) - (-1.91) x (-3.05)] j + [(-3.68) x (5.28) - 6.35 x (-3.05) ] k } x 105

F = 24 x 10⁻¹⁹ x 105 x [(3.16)i + (1.81)j + (-0.06)k] N   ....(2)

The i component of F is 7963.2 x 10⁻¹⁹ N.

The j component of F is 4561.2 x 10⁻¹⁹ N.

The k component of F is -151.2 x 10⁻¹⁹ N.

Magnitude of force, F =

F = 9178.05 x 10⁻¹⁹ N

In this case , E is also present and is equal to [(5.94)i + (-3.97)j + (-5.55)k] x 105 N/C. From equation (1) and (2) ;

F = 24 x 10⁻¹⁹ x [(3.16)i + (1.81)j + (-0.06)k + (5.94)i + (-3.97)j + (-5.55)k] x 105

F = 24 x 10⁻¹⁹ x 105 x [(9.1)i + (-2.16)j + (-5.61)k]

Magnitude of force, F =

F = 27483.9 x 10⁻¹⁹ N

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