A 5.0 μC particle moves through a region containing the...

A $5.0 \mu \mathrm{C}$ particle moves through a region containing the uniform magnetic field $-20 \hat{\mathrm{i}} \mathrm{mT}$ and the uniform electric field $300 \hat{\mathrm{J}} / \mathrm{m} .$ At a certain instant the velocity of the particle is $(17 \mathrm{i}-11 \hat{\mathrm{j}}+7.0 \hat{\mathrm{k}}) \mathrm{km} / \mathrm{s}$. At that instant and in unit-vector nota- tion, what is the net electromagnetic force (the sum of the electric and magnetic forces) on the particle?

A $5.0 \mu \mathrm{C}$ particle moves through a region containing the uniform magnetic field $-20 \hat{\mathrm{i}} \mathrm{mT}$ and the uniform electric field $300 \hat{\mathrm{J}} / \mathrm{m} .$ At a certain instant the velocity of the particle is $(17 \mathrm{i}-11 \hat{\mathrm{j}}+7.0 \hat{\mathrm{k}}) \mathrm{km} / \mathrm{s}$. At that instant and in unit-vector nota-
tion, what is the net electromagnetic force (the sum of the electric and magnetic forces) on the particle?

Key Concepts

Lorentz Force Law

The Lorentz force law describes the total electromagnetic force on a charged particle moving in electric and magnetic fields. The equation F = q(E + v × B) shows that the force is the sum of the electric force (qE) and the magnetic force (q(v × B)), which depend on the charge, the velocity of the particle, and the fields present.

Electric Force

The electric force on a charged particle is determined by the electric field acting on the charge, given by F_e = qE. This force acts in the direction of the electric field if the charge is positive and opposite to it if the charge is negative.

Magnetic Force

The magnetic force on a moving charged particle is given by F_m = q(v × B). This force depends not only on the magnetic field and the charge but also on the velocity of the particle, and its direction is perpendicular to both the velocity and the magnetic field as determined by the right-hand rule.

Cross Product

The vector cross product is a mathematical operation used to compute the magnetic force in the Lorentz force law. It involves multiplying two vectors to obtain a third vector that is perpendicular to the plane of the original vectors, with a magnitude proportional to the sine of the angle between them.

Unit Conversions

Unit conversions are essential when solving problems that involve different scales or units, such as converting microcoulombs to coulombs for charge, or kilometers per second to meters per second for velocity, to ensure consistency within the SI unit system.

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