Magnetic Force on a Moving Charged Ball

Physics Problem: Magnetic Force on a Moving Charged Ball

Problem Statement

During a physics demonstration, a rubber ball is given a positive charge of 30 nC. Calculate the force on the ball due to the Earth's magnetic field if it is thrown with a horizontal velocity of 5 m/s towards the south in a location where the Earth's magnetic field is directed west and parallel to the ground.

Given

• Charge: \( q = 30 \, \text{nC} = 30 \times 10^{-9} \, \text{C} \)
• Velocity: \( \vec{v} = 5 \, \text{m/s} \) (south, \( -\hat{j} \))
• Magnetic field: \( \vec{B} = 50 \, \mu\text{T} = 50 \times 10^{-6} \, \text{T} \) (west, \( -\hat{i} \))

Solution

The magnetic force on a moving charge is given by:

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

Direction of the cross product:

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

Magnitude of the force:

\[ F = qvB = 30 \times 10^{-9} \cdot 5 \cdot 50 \times 10^{-6} = 7.5 \times 10^{-12} \, \text{N} \]

Discussion

This problem illustrates the magnetic force acting on a positively charged object moving through Earth's magnetic field. The direction of the force is determined using the right-hand rule: the velocity vector points south, the magnetic field vector points west, and the resulting force is upward.

The magnitude of the force is small, but it demonstrates the interaction between motion, charge, and magnetic fields. This is a classic application of the Lorentz force law in electromagnetism.