Betatron, derivation of expressions

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SUMMARY

The discussion focuses on the derivation of expressions related to electromotive force (e.m.f) and work in the context of a physics problem from the AEA Physics past paper. The key equations discussed include ε = Δ∅/Δt and W = εe, with W also expressed as W = 2∏rF. The confusion arises from the factor of 4 in the equation Δ∅ = 4∏r²ΔB, which is clarified as accounting for the direction change of force during the cycle, effectively doubling the area consideration in the loop.

PREREQUISITES
  • Understanding of electromotive force (e.m.f) and its applications
  • Familiarity with the concept of work in physics
  • Knowledge of momentum and its relation to force
  • Basic grasp of calculus, particularly derivatives
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  • Study the principles of electromagnetism, focusing on Faraday's Law of Induction
  • Explore the derivation of work-energy principles in physics
  • Investigate the relationship between force, momentum, and area in electromagnetic systems
  • Review advanced calculus techniques relevant to physics applications
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Students preparing for physics examinations, educators teaching electromagnetism, and anyone interested in the mathematical derivation of physical laws related to work and e.m.f.

Gloyn
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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?
 
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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.