Cyclotron-where am i going wrong still??

Question

ok so here's the question, followed by what i thought was the way to do the problem. hopefully someone can steer me back on track

An electron moves on a circular orbit in a uniform magnetic field. The strength of the field is 2.50×10-4 tesla. The kinetic energy of the electron is 104.0 eV. (1 eV = 1 electron volt = 1.6 ×10-19 joules.) Calculate the radius of the orbit.

(electron parameters: the charge is -e where e = 1.602×10-19 C; the mass is m = 9.11×10-31 kg.)

ok i solved for velocity by doing K=.5mv^2. Then tried

doing mv^2/r=Bqv (using strength of field for B, electron

charge for q, mass for m, velocity for v) and got

2.42E-16 which apparently is wrong

can anyone help determine where i'm going wrong?

Answer 1

The centripetal force P = Bqv = mv^2/r = F, the centrifugal force.  Solve for r = mv^2/Bqv and E = 1/2 mv^2; so that v^2 = 2E/m.

Then r = m(2E)/m//Bqv = 2E/Bq sqrt(2E/m) = sqrt(2) sqrt(Em)/Bq = sqrt(2*(104*1.6E-19)*(9.11E-31))/((2.5E-... = 0.13765464 meter.

Ah...Li is correct, if v > 1/4 c, where relativistic mass M > 1.03 m rest mass, you need to fold in relativistic effects...I forgot to check...my bad.  But if v < 1/4 c, the error in radius will be small and, depending what you need it for, insignificant.

PS:  Abe is correct, use the variables and reduce them.  Then you can plug in the numbers; you'll find that will reduce numerical errors quite a bit.

Answer 2

If the electron is moving at relativistic speeds, the Newtonian kinetic energy approximation (KE = 1/2*m*v^2) is not accurate.  You need to use the relativistic kinetic energy equation:

KE = m*c^2 - m0*c^2

where m0 is the rest mass of the particle, m is the mass of the particle while moving, and c is the speed of light.

The particle's moving mass is:

m = m0*1/sqrt(1 - v^2/c^2)

where v is the particle's velocity

Answer 3

Don't know where you got that answer. But you should always work problems out symbolically first because these problems are especially prone to calculation errors. Do all calculation only at the end.

Solve for v:     KE = kinetic energy = 1/2 mv^2

                       v = sqrt(2 KE/m)

Then we have the centripetal force being supplied by the magnetic lorentz force:

                     mv^2/r = qvB

                     mv/r = qB

                     r = mv/qB

                     r = m / qB * sqrt(2 KE / m)

                     r = 1 / qB * sqrt(2 KE m)

q = 1.6e-19 C

B = 2.5e-4 Tesla

KE = 104eV * 1.6e-19 J/ev = 1.66e-17 J

m= 9.11e-31 kg

When you plug these numbers in, you get r = 0.137 m