Betatron, derivation of expressions

[Gloyn]

Hi!

I've got a problem with question 5 on this paper:

http://www.freeexampapers.com/past_papers.php?l=Past_Papers%2FAEA%2FPhysics%2F2006/

Download AEA-PHYS-PP-MayJune-2006-AEA-Paper-1342.pdf.

Starting from b) i), we got that:

ε=Δ∅/Δt → W=εe

Where W is denotes work, ε is e.m.f

Also, for ii) work can be written as:

W=2∏rF

in iv) we use F=Δp/Δt=2erΔB/Δt

as during one cycle mementum changes its direction to the opposite twice. From all that we got:

Δ∅=4∏r^2ΔB

Where 4 comes from? It should be deltaB times area of loop. I tried to check it if 2s does not cancel each other but they don't, where's the mistake?

 

[member 553083]

The mistake is that you have not taken into account the fact that the force is changing direction twice during the cycle. Since the force is changing direction, the area of the loop needs to be considered twice - once when the force is in one direction and again when it is in the other direction. This is why the 4 appears in the equation.