An electron of mass 9.1 ×10^-31 kg moves horizontally toward the north at 3.0 ×10^7 m / s . Dete

step 1 So here with this question the gravitational force exerted by the earth on the electron is given

An electron of mass 9.1 ×10^-31 kg moves horizontally toward...

An electron of mass $9.1 \times 10^{-31} \mathrm{kg}$ moves horizontally toward the north at $3.0 \times 10^{7} \mathrm{m} / \mathrm{s}$ . Determine the magnitude and direction of a $\vec{B}$ field that will exert a magnetic force that balances the gravitational force that Earth exerts on the electron.

Key Concepts

Lorentz Force

This principle states that a charged particle moving in a magnetic field experiences a force given by F = q(v x B). The magnitude and direction of this force depend on the charge, the speed of the particle, the magnetic field strength, and the angle between the velocity and the magnetic field. This concept is central in determining how the magnetic field can be used to counteract other forces acting on the particle.

Magnetic Field Direction and Charge Sign

The direction of the magnetic force arises from the cross product in the Lorentz force law, which involves the velocity and the magnetic field vectors. For negatively charged particles, such as electrons, the force direction is opposite to that predicted by the standard right-hand rule. Understanding how the sign of the charge affects the direction of the magnetic force is crucial in aligning the force correctly to achieve balance.

Gravitational Force

Gravitational force is the force acting on an object due to gravity, commonly calculated as F = mg, where m is the mass and g is the acceleration due to gravity. In contexts where forces are balanced, knowing the gravitational force on the particle is essential as it sets the magnitude of the force that must be counteracted by another force, such as the magnetic force.

Force Equilibrium

This concept involves setting forces equal in magnitude but opposite in direction so that an object remains in a state of constant velocity or rest. In practical terms, to achieve equilibrium, one adjusts parameters (like the magnetic field) so that the net force on the object becomes zero. This principle is used to determine the necessary conditions for balancing gravitational forces with magnetic forces.

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