- #1
DeIdeal
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I might've figured this out by myself already, check the EDITs.
Induction: ℰ = -dΦ/dt <=> ∫ E·dr = -d/dt ( ∫ B·dA )
Work done by a force: W = ∫ F·dr
(I should probably use these three as well, but I don't see why, check my comments)
Newton's second law: F = ma
a = v² / r u_r
Lorentz force: F = qv × B
As the increase in B is linear, magnetic flux density can be written as a function of time so that
B(t) = -(B_max/T) t k
Where the direction of B comes from Lorentz Force / Right-hand-rule, as the force of the magnetic field must point towards the center of the circle.
The electron aqcuires its energy only from the induced electric field since F_B does no work (as it's perpendicular to the velocity).
From Faraday-Henry law
-dΦ/dt = ∫ E·dr
-d( ∫ B·dA )/dt = 1/q ∫ F·dr // B||dA, B<0, A>0
-qA d(-|B|)/dt = W
E = q πr² B_max/T
(It seems I made a mistake with the direction of B after all, or should it be q = |q| here?)
...
And that seems all too simple. Still, I don't see where I made the mistake, so I'd appreciate if someone were to point me to the right direction.
We were actually given advice on how to approach the problem. It included deriving an equation for the velocity of the electron. This can obviously be done from the Lorentz force equation and F_B = m a_n, but I see no reason for doing this. Nor would I know how to actually use v(t). Should I do something like P = dW/dt = F_E v?
The units are correct, q πr² B_max/T gives Joules, but the equation doesn't seem reasonable. It has no mass dependency which seems a bit unusual considering a betatron is a particle accelerator.
EDIT: Wait, is it ∫ B · dA = B(t) A_circle that's incorrect? Should it be something like, umm,
dA = 2 π r dr = 2 π r v(t) dt ? And then what, integrate B(t)v(t) over T and do d/dT or something like that?
EDIT2: Changing ∫ B · dA to 2 π r ∫ B(t)v(t) dt gave me an equation that seems more reasonable (it has a mass dependency, correct units and q² so it's positive):
E = 2/3 q²/m B_max² π r²
Unless someone points out any mistakes with my reasoning, I think I'm going with this.
Homework Statement
"Charged particles are accelerated in a betatron in a varying magnetic field so that the trajectory of the particles is a circle with a set radius and the varying field is perpendicular to the circle. Determine the energy acquired by an electron during one acceleration period (one full circle) when the period lasts for time T and magnetic flux density increases from zero to its maximum value B_max linearly during one acceleration period."Homework Equations
Induction: ℰ = -dΦ/dt <=> ∫ E·dr = -d/dt ( ∫ B·dA )
Work done by a force: W = ∫ F·dr
(I should probably use these three as well, but I don't see why, check my comments)
Newton's second law: F = ma
a = v² / r u_r
Lorentz force: F = qv × B
The Attempt at a Solution
I chose a coordinate system where the positive direction for u_φ is counterclockwise and positive k[/B] points upwards.As the increase in B is linear, magnetic flux density can be written as a function of time so that
B(t) = -(B_max/T) t k
Where the direction of B comes from Lorentz Force / Right-hand-rule, as the force of the magnetic field must point towards the center of the circle.
The electron aqcuires its energy only from the induced electric field since F_B does no work (as it's perpendicular to the velocity).
From Faraday-Henry law
-dΦ/dt = ∫ E·dr
-d( ∫ B·dA )/dt = 1/q ∫ F·dr // B||dA, B<0, A>0
-qA d(-|B|)/dt = W
E = q πr² B_max/T
(It seems I made a mistake with the direction of B after all, or should it be q = |q| here?)
...
And that seems all too simple. Still, I don't see where I made the mistake, so I'd appreciate if someone were to point me to the right direction.
We were actually given advice on how to approach the problem. It included deriving an equation for the velocity of the electron. This can obviously be done from the Lorentz force equation and F_B = m a_n, but I see no reason for doing this. Nor would I know how to actually use v(t). Should I do something like P = dW/dt = F_E v?
The units are correct, q πr² B_max/T gives Joules, but the equation doesn't seem reasonable. It has no mass dependency which seems a bit unusual considering a betatron is a particle accelerator.
EDIT: Wait, is it ∫ B · dA = B(t) A_circle that's incorrect? Should it be something like, umm,
dA = 2 π r dr = 2 π r v(t) dt ? And then what, integrate B(t)v(t) over T and do d/dT or something like that?
EDIT2: Changing ∫ B · dA to 2 π r ∫ B(t)v(t) dt gave me an equation that seems more reasonable (it has a mass dependency, correct units and q² so it's positive):
E = 2/3 q²/m B_max² π r²
Unless someone points out any mistakes with my reasoning, I think I'm going with this.
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