In this part you have to determine electron velocity v for a certain acceleratin

Question

In this part you have to determine electron velocity v for a certain accelerating voltage V_a, by two different methods: by knowing accelerating voltage (you may use the pre-lab homework problem #2) by measuring the orbit radius r (using the equation that you derived in Part 1, Theory) For each method, write the following in your lab-report: Write a verbal description. Include the quantities you will measure. Write down the mathematical equations you will use. List all assumptions you make for solving the problem. List the possible sources of experimental uncertainty and how you could minimize them. Use a mirrored scale to avoid parallax error as shown at the picture: reflections of your eye and the edge of the beam should be on the same line and seen at the same point as the edge of the beam. Perform the experiment. Make sure that you keep V_a constant during the experiment. Make sure that you take steps to minimize experimental uncertainties and the effect of the assumptions. Record your measurements in an appropriate format. List sources of experimental uncertainty. Estimate the uncertainty in your result. How would you minimize uncertainties? Calculate the velocity, based on your procedure and measurements. Include the experimental uncertainty in each value of velocity that you determine. After you have done both experiments, compare the two outcomes. Are they within your experimental uncertainty of each other? If these two values are different, which one is larger? If you have time repeat the experiment with a different value of the magnetic field. Do you obtain a similar result? Compare your results with results of other groups. Do you think there is a systematic error? Does one method always yield a larger value than the other? Think what could be the reason? Think about the assumptions that you made. Can they affect your experiment creating a systematic error? Explain how the outcome of the experiment depends on this assumption, i.e. if the assumption increases or decreases one of the measured values.

Explanation / Answer

Case a) Accelerating voltage V is known

now, we know that electron is accelerated by a potential V would have gained a kinetic energy of qV
and we also know that KE = 0.5mv^2 [ m is mass of electron]
so, 0.5mv^2 = qV
v = sqroot(2qV/m)

quantities needed to be measured = V ( accelerating potential) [ q ( charge on electron and m mass on electron can be used directly from the known values)]

The assumption used are:
energy losses to nay other source like friction or stray magnetic/electric fields are 0

Case b) orbit radius is measured
orbit radius = r
strength of magnetic field = B
mass of particle = m
velocity = v
centripital force acting on a particle inscribing circular motion inside a constant magnetic field B is mv^2/r
and magnetic force responsible for this is qvB
equating both
qvB = mv^2/r
v = qBr/m

quantities to be measured : r ( radisu of curvature of electron in magnetic field), B ( strength of magnetic field), (m and q can be used form the known values)

Assumptions:
Magnetic field B is perpendicular to the plane of motion of the electron
Magfnetic field has a constant magnitude
The electron always remains in a magnetic field and never comes out

Uncertainity
in case a) dv/v = sqroot(2q/m)dV/2sqroot(V)*sqroot(2qV/m = dV/2V
in case b) dv/v = (rdB + Bdr)/Br = dB/B + dr/r

so uncertainities can be more in case 2 as there are more quantities that are error prone and the error depends on more factors
moreover in case a a factor of 1/2 reduces the error by half , and no such factor exists in the other case

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