An electron is shot into a uniform magnetic field B = (.5 x + .8 y)T with an initial velocity of v = (4 x + 3 y) × 106m/s. What is the force on the electron?
To find the force on the electron, we can use the formula for the force experienced by a charged particle moving in a magnetic field:
F = q * (v x B)
Where:
F is the force experienced by the charged particle,
q is the charge of the particle (in this case, the charge of an electron which is -1.6 × 10^-19 coulombs),
v is the velocity vector of the particle, and
B is the magnetic field vector.
To compute the cross product (v x B), we can use determinants:
v x B = |i j k|
|4 3 0|
|0.5 0.8 0|
The cross product of v and B gives us a vector in the direction perpendicular to both v and B. Performing the determinant calculation, we get:
v x B = (3 * 0 - 0.8 * 0) * i + (0.5 * 0 - 4 * 0) * j + (4 * 0.8 - 3 * 0.5) * k
= 0 * i + 0 * j + 2.2 * k
= 2.2 * k
Therefore, the cross product (v x B) results in a vector with a magnitude of 2.2 in the z-direction.
Now, we can calculate the force:
F = q * (v x B)
= -1.6 × 10^-19 C * 2.2 * k
Using the given value for the charge of an electron, we can evaluate the force:
F = -2.72 × 10^-19 kN
Therefore, the force experienced by the electron in the uniform magnetic field is -2.72 × 10^-19 kilonewtons in the z-direction.