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)

Answers 2

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\times10^{-19}\times \left[\begin{array}{ccc}i&j&k\\-3.68&6.35&-1.91\\-3.05&5.28&-1.09\end{array}\right]\times105

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 =\sqrt{(7963.2\times10^{-19}) ^{2} +(4561.2\times10^{-19}) ^{2} +(-151.2\times10^{-19}) ^{2} }

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 = 24\times10^{-19}\times105\times\sqrt{9.1^{2}+(-2.16^{2})+(-5.61^{2}) }

F = 27483.9 x 10⁻¹⁹ N

AI generated Answer

i component of F: -1.01 x 10-13 N j component of F: -5.02 x 10-13 N k component of F: 9.07 x 10-13 N Magnitude of F: 1.17 x 10-12 N New magnitude of F: 2.02 x 10-12 N Explanation: The force vector on each particle, F, is given by the equation F = qvB, where q is the charge of the particle (24.0 x 10-19 C), v is its velocity ([(-3.68)i + (6.35)j + (-1.91)k ] x 105 m/s), and B is the uniform magnetic field ([(-3.05)i + (5.28)j + (-1.09)k ] T). Plugging in the appropriate values, F = (24.0 x 10-19 C)([(-3.68)i + (6.35)j + (-1.91)k ] x 105 m/s)([(-3.05)i + (5.28)j + (-1.09)k ] T). Solving for the i component of F, we get F = (-1.01 x 10-13) N. Similarly, for the j component of F, we get F = (-5.02 x 10-13) N, and for the k component of F, we get F = (9.07 x 10-13) N. The magnitude of F can be calculated by using the equation |F| = sqrt(Fxi^2 + Fyj^2 + Fzk^2), which, in this case, gives us |F| = sqrt((-1.01 x 10-13 N)^2 + (-5.02 x 10-13 N)^2 + (9.07 x 10-13 N)^2) = 1.17 x 10-12 N. When a uniform electric field of [ (5.94)i + (-3.97)j + (-5.55)k ] x 105 N/C is added to the system, the new force vector on each particle is given by F' = F + Eq, where F is the original force vector and E is the electric field. Thus, the new magnitude of F can be calculated by using the equation |F'| = sqrt((Fxi + Exq)^2 + (Fyj + Eyq)^2 + (Fzk + Ezq)^2), which gives us |F'| = sqrt((-1.01 x 10-13 N + (5.94 x 10-5 N/C)(24.0 x 10-19 C))^2 + (-5.02 x 10-13 N + (-3.97 x 10-5 N/C)(24.0 x 10-19 C))^2 + (9.07 x 10-13 N + (-5.55 x 10-5 N/C)(24.0 x 10-19 C))^2) = 2.02 x 10-12 N.
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