A charged particle with charge q is moving with speed v perpendicular to a uniform magnetic field. A second identical charged particle is moving with speed 2v perpendiculuar to the same magnetic field. The time to complete one full circular revolution for the first particle is T1. The time to complete one full circular revolution for the particle moving with speed 2v is

f. Unsure why.

To find the time to complete one full circular revolution for the particle moving with speed 2v, let's use the equation for the period of a charged particle in a magnetic field.

The equation for the period (T) of a charged particle moving in a circular path with radius r and with speed v in a magnetic field B is given by:

T = 2πm / (qB)

Where:
T is the period (time to complete one full circular revolution)
π is a mathematical constant approximately equal to 3.14159
m is the mass of the charged particle
q is the charge of the particle
B is the strength of the magnetic field

In this case, the two charged particles have the same charge (q) and identical masses (m). The only difference is their speeds, with one particle moving with speed v and the other particle moving with speed 2v.

Since the charge and the mass are the same for both particles, the only factor affecting the period is the speed of the particle (v in this case). Therefore, we can directly compare the periods of the two particles by using the following relationship:

T1 / T2 = v1 / v2

Where:
T1 is the period of the first particle (with speed v)
T2 is the period of the second particle (with speed 2v)
v1 is the speed of the first particle
v2 is the speed of the second particle

Since we know the period T1 for the first particle, we can rearrange the equation to solve for T2:

T2 = (T1 * v2) / v1

Substituting the values given:
v1 = v and v2 = 2v

T2 = (T1 * 2v) / v

Simplifying the expression:
T2 = 2T1

Therefore, the time to complete one full circular revolution (T2) for the second particle moving with speed 2v is twice the time (T1) for the first particle.