### Video Transcript

Two particles have the same linear
momentum, but particle π΄ has four times the charge particle π΅. If both particles move in a plane
perpendicular to a uniform magnetic field, what is the ratio π
sub π΄ to π
sub π΅
of the radii of their circular orbits?

To start off, letβs consider the
scenario of whatβs happening. We have two particles, we can call
the blue one particle π΄ and the pink one particle π΅, entering a magnetic field of
uniform strength that weβve called capital π΅. Weβre told in the problem statement
that these particles have the same linear momentum, which means that if we take the
product of their mass and their velocity we get the same number. This means that if we write that π
linear momentum is equal to π times π£, then we can say that π sub π΄, the linear
momentum of particle π΄, is equal to π sub π΅, that of particle π΅.

But not only do the particles have
linear momentum, they also are charged particles; they have a nonelectric charge to
them. That means that as the particles
enter this uniform magnetic field, theyβll experience a magnetic force which will
push them in a circular path. Thereβs a very tidy equation for
the radius of that circular path that a charged particle follows in the uniform
magnetic field. The radius of that circular arc is
known to equal the linear momentum of the particle π times π£ divided by its charge
times the strength of the magnetic field that itβs in. What we want to solve for is the
ratio of those radii for particle π΄ and particle π΅. As we do this, we saw that the
linear momentum the two particles is equal, and weβre also told in the problem
statement that the charge of particle π΄ is equal to four times the charge of
particle π΅.

Keeping these two relationships in
mind, letβs now write expressions for the circular radii of particle π΄ and particle
π΅ separately. The circular radius of particle π΄
is equal to π sub π΄, its linear momentum, divided by its charge times capital π΅,
the magnetic field that this particle is in. And then the radius of particle π΅
is equal to π sub π΅ divided by π sub π΅ times capital π΅, which is not the
particle π΅ but rather the magnetic field. Now that we have these expressions,
letβs divide them one by another to find the ratio π
sub π΄ to π
sub π΅. That equation then for π
sub π΄ to
π
sub π΅ is equal to this expression. And we see right away that the
magnetic field, capital π΅, cancels from these terms; it divides out.

At this point we can start to use
the information given to us in the problem statement. For example, π sub π΄, the linear
momentum of particle π΄, is equal to that of particle π΅. This means that we can replace π
sub π΅ in our denominator with π sub π΄. And note that we can also just as
well have replaced π sub π΄ with π sub π΅. Regardless of how we do that
substitution, we see that this term now also cancels from our expression. We can then simplify the right-hand
side of this expression so that it becomes π sub π΅ over π sub π΄. Thatβs what π
sub π΄ over π
sub
π΅ is equal to. But now we can make yet another
substitution because π sub π΄ we know is equal to four times π sub π΅. And with that substitution made, we
see that the factor of π sub π΅ cancels out. Weβre left then with a purely
numerical ratio: one to four. That then is our answer. Thatβs the ratio of the radius of
particle π΄ to the radius of particle π΅ in this magnetic field.