A particle with positive charge q = 2.08 10-18 C moves with a velocity v = (4î + 4ĵ − ) m/s through a region where both a uniform magnetic field and a uniform electric field exist.(a)Calculate the total force on the moving particle, taking B = (4î + 3ĵ + ) T and E = (3î − ĵ − 2) V/m. (Give your answers in N for each component.)Fx = NFy = NFz = N(b)What angle does the force vector make with the positive x-axis? (Give your answer in degrees counterclockwise from the +x-axis.) ° counterclockwise from the +x-axis(c)What If? For what vector electric field would the total force on the particle be zero? (Give your answers in V/m for each component.)Ex = V/mEy = V/mEz = V/m

The force on a charged particle moving in a magnetic field is given by the Lorentz force law, which states that F = q(E + v × B). Here, E is the electric field, v is the velocity of the particle, and B is the magnetic field. The cross product v × B gives the component of the velocity perpendicular to the magnetic field, multiplied by the strength of the magnetic field.

(a) We can calculate the total force on the particle by plugging in the given values into the Lorentz force law:

F = q(E + v × B) = 2.08 * 10^-18 C * ((3î - ĵ - 2k) + (4î + 4ĵ - k) × (4î + 3ĵ + k)) = 2.08 * 10^-18 C * ((3î - ĵ - 2k) + (16î + 12ĵ - 4k)) = 2.08 * 10^-18 C * (19î + 11ĵ - 6k)

So, Fx = 19 * 2.08 * 10^-18 N = 3.95 * 10^-17 N Fy = 11 * 2.08 * 10^-18 N = 2.29 * 10^-17 N Fz = -6 * 2.08 * 10^-18 N = -1.25 * 10^-17 N

(b) The angle θ that the force vector makes with the positive x-axis can be calculated using the dot product formula:

cos θ = F . î / |F| |î| = (3.95 * 10^-17 N * 1) / (sqrt((3.95 * 10^-17 N)^2 + (2.29 * 10^-17 N)^2 + (-1.25 * 10^-17 N)^2) * 1) = 3.95 * 10^-17 N / sqrt((3.95 * 10^-17 N)^2 + (2.29 * 10^-17 N)^2 + (-1.25 * 10^-17 N)^2)

Solving for θ gives us the angle in radians. To convert to degrees, multiply by 180/π.

(c) For the total force on the particle to be zero, the electric field E must be such that E = -v × B. Solving for E gives:

E = -(4î + 4ĵ - k) × (4î + 3ĵ + k) = -16î - 12ĵ + 4k

So, Ex = -16 V/m, Ey = -12 V/m, and Ez = 4 V/m.