An electron enters a region of uniform magnetic field of magnitude

PHYSICS · Multi-part · A Level · Physics · Fields and their consequences · Fields and their consequences

Problem

B=1.2\,\text{T}

. At the instant it crosses the boundary into the field, the electron has a velocity of magnitude

v=3.0\times10^{6}\,\text{m\,s}^{-1}

that makes an angle of

40^{\circ}

with the magnetic field direction. Take the mass of the electron to be

m_{e}=9.11\times10^{-31}\,\text{kg}

and the magnitude of its charge to be

e=1.60\times10^{-19}\,\text{C}

. (a) Derive an expression for the magnitude of the magnetic force on the electron in terms of

e

v

B

, and the angle

\alpha

between the velocity and the magnetic field. (b) Using the given numerical values, calculate the magnitude of the magnetic force on the electron. (c) Explain why the component of the electron’s velocity parallel to the magnetic field remains constant, while the perpendicular component changes direction. (d) Show that the electron follows a helical path and derive an expression for the pitch

p

(the distance between successive turns of the helix) in terms of

v

\alpha

m_{e}

e

, and

B

. State clearly any intermediate relationships you use. (e) Calculate the numerical value of the pitch for the given data.

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