Two point charges move through space

PHYSICS · Multi-part · AP · Physics C: Electricity and Magnetism · Magnetic Fields and Electromagnetism · Magnetic Fields and Electromagnetism

Problem

Two point charges move through space, each producing a magnetic field measured at a common observation point O. • Charge 1 has magnitude

q=1.0\,\mu\text{C}

and moves at speed

v_{1}=2.0\times10^{5}\,\text{m}\,\text{s}^{-1}

. • The shortest distance from Charge 1 to point O at the instant of measurement is

d_{1}=0.040\,\text{m}

, and its velocity is perpendicular to the line joining the charge to O. • Charge 2 has magnitude

2q

and travels in a direction perpendicular to the line joining it to O. The shortest distance from Charge 2 to point O is

d_{2}=0.060\,\text{m}

. Its speed

v_{2}

is unknown. In a laboratory adjustment, the experimenter wishes the two charges to produce magnetic fields of equal magnitude at point O. (a) Derive an expression for the ratio

\dfrac{B_{2}}{B_{1}}

of the magnetic field magnitudes at point O in terms of

q,\;v_{1},\;v_{2},\;d_{1},\text{ and }d_{2}

. Do not substitute numerical values yet. (b) Use your result from part (a) to determine the speed

v_{2}

that makes

B_{2}=B_{1}

. Show all steps and include units with your answer. (c) Suppose the experimenter could halve the distance

d_{2}

while keeping

v_{2}

at the value found in part (b). Would

B_{2}

now be larger than, smaller than, or equal to

B_{1}

? Explain quantitatively.

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