Question

An electron moves within a uniform magnetic field. Its instantaneous velocity is v = vy j + vz k, where vy =5m/sandvz =3m/s. ThefieldisgivenbyB=Bxi+ByjwhereBx =2TandBy =4T. Thechargeonthe electron is −1.6 × 10−19 C. What force F is exerted on the electron by the magnetic field?

Answer 1

F = 3.2 x 10⁻¹⁹ x (6i - 3j + 5k) N

Step-by-step explanation:

Magnetic fields exert forces on moving charge. The force F exerted by a magnetic field B on a charge q moving with velocity v in an electric field is called the magnetic Lorentz force. It is given by

F = qv × B

The magnitude of the Lorentz force F is F = qvB sinθ, where θ is the smallest angle between the directions of the vectors v and magnetic field B. When v and B are perpendicular to each other, sinθ = 1 then F has its maximum possible magnitude F = qv x B

Given:

Instantaneous velocity v = vyj + vzk but vy =5m/s and vz =3m/s.

v = 5j + 3k m/s

Field B = Bxj + Byj but Bx =2T and By =4T

B = 2i + 4j T

From magnetic Lorentz force F = q v x B

F = −1.6 × 10⁻¹⁹ x [ (5j + 3k) X (2i + 4j) ]

solving vector multiplication of V X B

`V X B =\left[\begin{array}{ccc}i&-j&k\\0&5&3\\2&4&0\end{array}\right]`

`=\left[\begin{array}{ccc}5&3\\4&0\\\end{array}\right] i - \left[\begin{array}{ccc}0&3\\2&0\\\end{array}\right] j + \left[\begin{array}{ccc}0&5\\2&4\\\end{array}\right] k`

=
`(0 - 12)i -(0-6)j+ (0-10)k`

V X B =
`-12i + 6j - 10k`

F = −1.6 × 10⁻¹⁹ x (-12i + 6j -10k)

Expanding the bracket and simplifying to the lowest term;

F = 3.2 x 10⁻¹⁹ x (6i - 3j + 5k) N