MHB Multiple choice question about electric fields and magnetic fields

[AztecChaze]

Consider a region where a 25-volt-per-meter electric field and a 15-millitesla magnetic field exist and are along the same direction. If the electron is in the said region, is moving at a direction 20 degrees counter-clockwise from the direction of the magnetic field, and is experiencing a total force of 5 × 10-18 Newtons, determine the speed of the electron. Assume the velocity vector, the electric field, and the magnetic field are all lying on the same plane.

a.) 10.46 km/s
b.) 1.32 km/s
c.) 3.65 km/s
d.) 5.20 km/s

 

[Euge]

By the Lorentz law, the electromagnetic force is given by $\mathbf{F} = -e(\mathbf{E} + \mathbf{v} \times \mathbf{B})$, where $e$ is the elementary charge, equal to $1.6 \times 10^{-19} C$. Since the electric and magnetic fields point in the same direction, we may suppose that they are in the $x$-direction. Then $\mathbf{v} \times \mathbf{B}$ points in the negative z-direction and has magnitude $vB\sin(20^\circ)$. Therefore $\mathbf{F} = -e(E\,\mathbf{\hat{x}} - vB\sin(20^\circ)\,\mathbf{\hat{z}})$. Taking the square of the magnitude on both sides of the vector equation yields $F^2 = e^2[E^2 + (vB \sin 20^\circ)^2]$. Solving for $v$ results in $$v = \frac1{B\sin20^\circ}\sqrt{\left(\frac{F}{e}\right)^2 - E^2}$$ Plug in the given values of $F$, $E$ and $B$ into this formula to determine the answer.