Particle colliders

Question

The question is as follows In the Large Electron Positron collider at CERN, groups of electron and positrons were accelerated along a circular tunnel, so that they collided. Electromagnets were used to keep the particles moving in a circle. Discuss how the strength and direction of the magnetic field would need to be adjusted to: i) keep a particle travelling in a circle while it is increasing its speed ii) bend the paths of both positrons and electrons which are travelling in opposite directions in the same tunnel.

Answer 1

Here I'll use convention of Bold letters to indicate vectors and underline+bold letters to indicate unit vectors

The centripetal force required to keep the particle in a circular orbit is

F_C = m v²/r × r ⋅ ⋅⋅⋅⋅⋅⋅⋅ ⋅⋅ ⋅⋅ ⋅ ⋅⋅⋅⋅⋅ ⋅⋅⋅⋅ ⋅⋅⋅⋅ eq 1

here, m = mass of the particle

v = velocity of the particle

r = radius of the orbit

Lorentz force applied by the magnetic field is

F_L = q v × B ⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅ ⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅ ⋅⋅⋅⋅ ⋅⋅ eq 2

Here to keep the forces in equilibrium, F_L=F_C

If we apply the right hand rule, we will see that magnetic field perpendicular to the plane of orbit creates a Lorentz force directed always to the center.

That makes the angle between v and B 90^o. So, v × B = vB

now if we balance the magnitude of the forces,

qvB = mv²/r

⇒ B = mv/( r q) ⋅ ⋅⋅⋅⋅ ⋅ ⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅ ⋅⋅⋅ eq 3

Now to answer the 1st question

i) keep a particle travelling in a circle while it is increasing its speed

Assuming that we will change only velocity and magnetic field, we differentiate both sides with respect to time,

d/dt B = m/( r q ) × (d/dt v) ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ eq. 4

⇒ d/dt B = constant × (d/dt v)

⇒ d/dt B ∝ d/dt v

so, you will have to increase the magnetic field in the same rate with the velocity

ii) bend the paths of both positrons and electrons which are travelling in opposite directions in the same tunnel.

If we look at eq-4 , we can see that if we replace q with -q and v with -v , there is no change in magnetic field or anything else.

That means, for a particle with same mass but with opposite charge going at the opposite direction will also stay in the orbit.

Answer 2

Hey John,

Answering this question requires understanding of two basic concepts. The first is circular motion; specifically, how a centripetal force causes an angular acceleration to make an object move in a circle. The second is electromagnetism; specifically, how a magnetic field exerts a force on a moving charged particle.

For an object to be traveling in a circle, it must experience a constant angular acceleration- in other words, a change in direction of its velocity so that it doesn't just continue straight. And this angular acceleration must be constant for the particle to follow the curve of a circle. This is often accomplished with a centripetal force- a force on the object that points directly towards the center of the circle at all times. Such a force does not change the linear velocity of the object tangent to the circle; it only causes the velocity in the next moment- when the object is in a different point on the circle- to point its new direction. The linear velocity of the object and the magnitude of the centripetal force will together determine the path the object takes. What that means for this question is that the magnets must exert a centripetal force on the particles in the accelerator, and that varying the force will alter the paths of the particles.

So, how can we create a centripetal force on the particles with magnets? Here's where the right hand rule comes in. It lets us determine the direction of the force a magnetic field exerts on a moving *positively* charged particle. With your right hand, if you point your thumb in the direction of the velocity of the moving charges (v) and your fingers in the direction of the magnetic field (B), your palm will face the direction of the force exerted on the particles (F). The actual equation for this is F = qvBsin(theta). The sin(theta) term means that the force will be 0 when B is parallel or antiparallel to the velocity, and gradually increase as it points more and more perpendicular to it. So for F to point towards the center of the circle, with v tangent to the circle, then B must point somewhere between parallel to the velocity, and directly up. Up will create the strongest force. For negative charges, your thumb points in the opposite direction of the velocity, and since the electrons are meant to go in the opposite direction as the positrons, it works out. Here's a page about the RHR: https://courses.lumenlearning.com/boundless-physics/chapter/magnetic-force-on-a-moving-electric-charge/

Now, I'm a bit confused by part i. A magnetic field can't accelerate charges in the direction of their velocity, the right hand rule shows that. So I assume what's going on is that the particles are first somehow shot into the collider, but at the beginning they're still speeding up, so the centripetal force on them must start off relatively weak and then increase as the particles speed up in order to maintain their path in a circle of a fixed radius. This means the magnetic field could increase in strength, or be rotated from a more parallel orientation to a more perpendicular one.

As for part ii, bending the paths of the particles could be done either by adjusting the magnitude of the centripetal force (i.e. strength of the magnetic field) or by changing its orientation so create a force pointing an entirely different direction.