A proton moving with velocity \(\vec{v}_{1}=3.6 \times 10^{4} \hat{\jmath} \mathrm{m} / \mathrm{s}\) experiences a magnetic force of \(7.4 \times 10^{-16} \hat{\imath} \mathrm{N}\). A second proton moving on the \(x\) -axis experiences a magnetic force of \(2.8 \times 10^{-16} \hat{\jmath} \mathrm{N} .\) Find the magnitude and direction of the magnetic field (assumed uniform), and the velocity of the second proton.

The term magnetic field is a fundamental concept in physics, specifically in electromagnetism. It represents a vector field that permeates space around magnetic materials and moving electric charges. Imagining the magnetic field as lines can be helpful; these lines emerge from the north pole of a magnet and terminate at the south pole.

Magnetic fields have both a direction and a magnitude, and they exert forces on moving charges or other magnetic materials within the field. The force experienced by a moving charge is known as the Lorentz force, which is a key component in understanding how charged particles like protons are affected when they move through a magnetic field.

In the context of the given exercise, the proton's movement and the resulting force allow us to deduce properties of the magnetic field present. Magnetic fields are measured in units of Tesla (T) and this field can be visualized as the perpendicular force exerted on a moving charged particle within the field.

The charge of a proton is a crucial piece of information when dealing with electromagnetism. Protons carry a positive electric charge, which is usually denoted as +1 elementary charge. In the International System of Units (SI), the elementary charge is a constant and its value is approximately equal to \( 1.6 \times 10^{-19} \text{coulombs} \).

This charge plays a vital role in calculating the effects of electromagnetic forces on a proton. As demonstrated in the exercise, the charge of a proton is used with the Lorentz force equation to compute the magnetic field that is acting upon it, as well as to determine the proton's resultant acceleration or deviation when in motion within that field.

The right-hand rule is a mnemonic utilized to understand the direction of various vector quantities in electromagnetism, particularly the direction of the magnetic force on a charged particle moving in a magnetic field. To apply the right-hand rule for determining the direction of the Lorentz force, you must:

• Point your fingers in the direction of the velocity (\(\vec{v}\)) of the positive charge,
• Orient your palm towards the direction of the magnetic field (\(\vec{B}\)), and
• Extend your thumb — it will then point in the direction of the Lorentz force (\(\vec{F}\)).

It is important to use your right hand for this rule as using your left hand will give you the opposite direction. In our exercise, the right-hand rule helps determine the direction of the magnetic field based on the known direction of the force and the velocity of the proton.