# Charge to Mass Ratio of the Electron

Thomas Markovich and John Mazzou

Departments of Physics University of Houston Houston, TX 77204-5006 (Dated: September 23, 2010) We sought to reproduce the experiment ﬁrst preformed by KT Bainbridge to determine the charge to mass ratio of the electron. In this paper, we derived the relationship between this ratio and measurable quantities, detailed our experimental setup, with in depth and speciﬁc circuit diagrams. We determined the mass to charge ratio to be 6.54341±.00474661e7[Cg − 1] with a percent diﬀerence of 63%. Because of the large error, we explore possible sources of error.

I.

INTRODUCTION

a perfect circle. Thus: m v2 = qvB r (II.3)

The ratio of the mass of an electron to its charge is a fundamental quantity that has eﬀects in ﬁelds as varied as electromagnetism and quantum chemistry and important to verify personally. Furthermore, replicating this experiment provided the authors with invaluable laboratory experience with wiring advanced circuits, practice with data analysis, and practice using digital instruments. To this end, in this experiment, we sought to measure this ratio to a reasonable accuracy using the experimental apparatus ﬁrst designed by K.T. Bainbridge. The remainder of this paper will be organized as follows. In section II, we will provide a brief introduction to the requisite electromagnetic theory, replete with equations for error analysis. In Section III, we will provide a brief outline of our experimental setup, with circuit diagrams; Section IV will contain our results, discussion, and analysis; and in Section V, we will provide the reader with our conclusions from the experiment.

where m is the mass of the electron and r is the radius of orbit. If the charged particle is accelerated through an electric potential of V, then the velocity of the charged mass is then simply v= 2qV . m (II.4)

Then, if we substitute Equation (II.3) into Equation (II.4), we get that q 2V = 2 2. m r B (II.5)

II. A.

THEORY

Electromagnetics

Our derivation, however, is not complete, as we cannot directly measure our magnetic ﬁeld. However, because we are using a Helmholtz coil assembly to provide the magnetic ﬁeld, we can determine B in terms of measurable quantities. More speciﬁcally, B is given by √ 64µ0 N I B= √ (II.6) 125R where N is the number of coils, I is the current through the coils, R is the radius of the coils, and µ0 is the vacuum permeability [1]. When we substitute Equation (II.6) into (II.5) to get that e 125 R2 V = . m 32 µ0 N r2 I 2 We will use this equation to calculate the ratio of weighted mean values for V, I, and r. e m

Moving charges are subject to a force under the inﬂuence of a magnetic ﬁeld as determined by the Lorentz Force Equation given below F =q E+v×B , (II.1)

where F is the force, E is the electric ﬁeld vector, v is the velocity, and B is the magnetic ﬁeld vector. From the above equation, it is immediately obvious that if the applied magnetic ﬁeld is perpendicular to the direction of motion of the charge, the force causes the particle to move in a circular path [1]. Furthermore, Equation (II.1) reduces to F = qvB. (II.2)

(II.7) using

B.

Error Analysis

We know from mechanics that the centripetal force and the Lorentz force balance each other if the particle forms

Our experiment requires us to take multiple current measurements at diﬀerent accelerating potentials. We assumed that the error in the accelerating potential was simply that of the digital multimeter and the error in

2 the current measurements was the standard deviation of the mean. As suggested in the lab manual, we took ﬁve current measurements for the middle peg for each accelerating potential with the assumption that the standard deviation of the mean would be roughly the same for all the pegs in each accelerating potential[2]. To calculate the standard deviation of the mean, we used the formula below [3] 1 n(n − 1) n 2

Just as...

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