Magnetic Force on a Moving Charged Sphere

Physics Problem: Magnetic Force on a Moving Charged Sphere

Problem Statement

A metal sphere is charged with 25 nC and is thrown with a horizontal velocity of 12 m/s towards the north. Calculate the force on the sphere due to the Earth's magnetic field in a place where the Earth's field is directed east and parallel to the ground.

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Given

• Charge: \( q = 25 \, \text{nC} = 25 \times 10^{-9} \, \text{C} \)
• Velocity: \( \vec{v} = 12 \, \text{m/s} \) (north, \(+\hat{j}\))
• Magnetic field: \( \vec{B} = 50 \, \mu\text{T} = 50 \times 10^{-6} \, \text{T} \) (east, \(+\hat{i}\))

Solution

The magnetic force on a moving charge is given by:

\[ \vec{F} = q \vec{v} \times \vec{B} \]

Direction of the cross product:

\[ \vec{v} \times \vec{B} = \hat{j} \times \hat{i} = -\hat{k} \Rightarrow \text{Force is downward} \]

Magnitude of the force:

\[ F = qvB = 25 \times 10^{-9} \cdot 12 \cdot 50 \times 10^{-6} = 1.5 \times 10^{-11} \, \text{N} \]

Discussion

The force on a charged object moving through a magnetic field depends on the direction of motion and the orientation of the field. In this case, the velocity is north and the magnetic field is east, so the resulting force is perpendicular to both — downward. This is a direct application of the right-hand rule for cross products in vector physics.

Although the force is quite small (\(1.5 \times 10^{-11} \, \text{N}\)), it illustrates how even Earth's weak magnetic field can exert a measurable influence on moving charges.