A 150- g ball containing 4.00 ×10^8 excess electrons is dropped into a 125-m vertical shaft. At

step 1 First, we use conservation of energy to find the speed of the ball as it reaches the bottom of t

A 150- g ball containing 4.00 ×10^8 excess electrons is...

A $150-$ g ball containing $4.00 \times 10^{8}$ excess electrons is dropped into a $125-\mathrm{m}$ vertical shaft. At the bottom of the shaft, the ball suddenly enters a uniform horizontal magnetic field that has magnitude 0.250 $\mathrm{T}$ and direction from east to west. If air resistance is negligibly small, find the magnitude and direction of the force that this magnetic field exerts on the ball just as it enters the field.

Key Concepts

Lorentz Force

The Lorentz force describes the force exerted on a charged particle moving in a magnetic field. Its magnitude and direction are determined by the cross product F = q(v × B), meaning the force is perpendicular to both the velocity of the particle and the magnetic field direction.

Motion Under Gravity

Objects in free fall under gravity accelerate from rest and attain a velocity that can be calculated using kinematic equations such as v = ?(2gh), where g is the acceleration due to gravity and h is the height from which the object falls. This velocity is crucial in determining the subsequent magnetic force experienced by the charged object.

Vector Cross Product

The vector cross product is a mathematical operation used to determine a vector that is perpendicular to two given vectors. In the context of magnetic forces, it determines the direction of the force on a moving charge relative to both its velocity and the magnetic field.

Uniform Magnetic Field

A uniform magnetic field has the same magnitude and direction at every point in the region it occupies. This simplifies the analysis since the magnetic force on a moving charged particle remains consistent as long as the particle's speed and charge remain constant.

Charge Quantification

The net charge on an object due to an excess of charged particles is calculated by multiplying the number of extra particles by their individual charge. This net charge is fundamental for determining the magnitude of electromagnetic forces acting on the object.

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