Electricity and Magnetism
# Magnetic Fields

(a) deflect upward, with its speed remaining constant;

(b) continue to move in the horizontal direction with constant velocity;

(c) move in a circular orbit and become trapped by the field;

(d) deflect downward, moving in a semicircular path with constant speed, and exit the field moving to the left; or

(e) deflect out of the plane of the paper?

To which of the above questions is the answer yes?

Choose coordinates \(x\) and \(y\) on a horizontal 2-d graphene sheet. We will inject electrons into the sheet from the origin \(O\) such that they enter the upper half-plane (\(y>0\)) but the direction is otherwise totally random. The speed of electrons in graphene is \(v_F\), and the mass of each electron is \(m\).

We now put an electron detector along the \(x\)-axis (\(y=0\)) and apply an external magnetic field \(B_0\) perpendicular to the plane (parallel with the \(z\)-axis). At what value of \(x\) **in meters** will the detector receive the greatest signal?

Assume that every electron that goes to the lower half plane (\(y<0\)) will disappear (for example, the lower half plane is made of a special material which absorbs electrons).

**Details and assumptions**

To make the problem numerically simpler, use the following values for your calculation:

\(v_F=1~\mbox{m/s}\)

\(m=1~\mbox{kg}\)

\(B_0=1~\mbox{T}\)

The electron charge is \(e=1~\mbox{C}\).

When a charged particle is traveling through a uniform magnetic field, which of the following statements are true for that magnetic field?

(a) It increases the kinetic energy of the particle.

(b) It exerts a force on the particle along the direction of its motion.

(c) It exerts a force on the particle that is parallel to the field.

(d) It doesn't change the magnitude of the momentum of the particle.

(e) It exerts a force that is perpendicular to the direction of motion.

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