Magnetic Force on a Moving Charged Rod

Physics Problem: Magnetic Force on a Moving Charged Rod

In a physics experiment, you rub a plastic rod with wool, giving it a negative charge of 15 nC. Calculate the force on the rod due to the Earth's magnetic field if you throw it with a horizontal velocity of 8 m/s towards the east in a location where the Earth's magnetic field is directed south and parallel to the ground.

Given

• Charge: \( q = -15 \, \text{nC} = -15 \times 10^{-9} \, \text{C} \)
• Velocity: \( \vec{v} = 8 \, \text{m/s} \) (east, \(+\hat{i}\))
• Magnetic field: \( \vec{B} = 50 \, \mu\text{T} = 50 \times 10^{-6} \, \text{T} \) (south, \(-\hat{j}\))

Solution

The magnetic force on a moving charge is given by:

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

Direction of the cross product:

\[ \vec{v} \times \vec{B} = \hat{i} \times (-\hat{j}) = -\hat{k} \]

Since the charge is negative, the force is in the opposite direction:

\[ \vec{F} = -q(-\hat{k}) = +\hat{k} \Rightarrow \text{Force is upward} \]

Magnitude of the force:

\[ F = |q| \cdot v \cdot B = 15 \times 10^{-9} \cdot 8 \cdot 50 \times 10^{-6} = 6.0 \times 10^{-12} \, \text{N} \]

Discussion

This problem demonstrates the use of the Lorentz force law to determine the magnetic force on a moving charged object. The direction of the force is determined using the right-hand rule for the cross product of velocity and magnetic field vectors. Since the rod is negatively charged, the direction of the force is opposite to the result of the cross product.

In this case, the rod moves east and the magnetic field points south, so the cross product points downward. Because the charge is negative, the actual force is upward. The magnitude of the force is very small, but it illustrates the fundamental interaction between moving charges and magnetic fields.