Two point charges travel in straight

PHYSICS · Multi-part · AP · Physics 2: Algebra-based · Magnetism and Electromagnetism · Magnetism and Electromagnetism

Problem

Two point charges travel in straight, parallel lines separated by a perpendicular distance

d

. At time

t=0

, both charges are in a plane that is a distance

r

from the origin O. • Charge A has magnitude

q_{A}

and speed

v_{A}

. Its velocity is exactly perpendicular to the line joining Charge A to O at that instant. • Charge B has magnitude

q_{B}

and moves at an angle of

30^{\circ}

relative to the line joining Charge B to O, measured between its velocity vector and that line. Both charges are moving in the same general direction ("to the right" in the laboratory frame). The observation point O is fixed in space. (a) Derive symbolic expressions for the magnitudes

B_{A}

and

B_{B}

of the magnetic fields at O due to Charge A and Charge B respectively, expressed in terms of the given symbols (

q_{A},\;q_{B},\;v_{A},\;v_{B},\;r,

and numerical constants). (b) Using your expressions from part (a), derive an equation for the speed

v_{B}

that would make

B_{B}=B_{A}

. Show each algebraic step. (c) Evaluate your expression from part (b) for the case

q_{A}=q_{B}

v_{A}=1.5\times10^{6}\,\text{m}\,\text{s}^{-1}

, and

r=0.050\,\text{m}

. Report

v_{B}

with appropriate units and explain whether this speed is physically reasonable for an electron in a typical laboratory setting.

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