91Ó°ÊÓ

A magnetic field is a vector field that represents the magnetic influence of electric charges in relative motion and magnetic materials. It is denoted by the symbol \( \vec{B} \) and interacts with moving charges to exert a force.

In the context of the exercise, the magnetic field is generated by a long, straight current-carrying wire. The magnitude of this field at a distance \( d \) from the wire can be calculated using Ampère's law: \[ B = \frac{\mu_0 I}{2 \pi d} \] where \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, \text{T}\cdot\text{m/A} \)), \( I \) is the current, and \( d \) is the distance from the wire. The direction of this field is circular around the wire.

• The field circles around the wire forming closed loops.
• Its strength decreases as you move farther from the wire.

A current-carrying wire is a conductor through which electric current flows. The flow of electric charges within the wire creates a magnetic field in the surrounding space. This is due to the inherent nature of moving charges generating magnetic fields.
In this scenario, we have a long straight wire carrying a current of \( 350 \, \text{mA} \). This produces a magnetic field around the wire. It's important to visualize the field as circular loops forming around the wire.

• The field's direction can be determined using the right-hand rule.
• The magnitude is influenced by the current's strength and the distance from the wire.

These properties make current-carrying wires useful in generating magnetic fields for a wide array of applications including electromagnets and inductors.

The right-hand rule is a helpful mnemonic for determining the direction of the magnetic field relative to a current or the force relative to a magnetic field and current. When dealing with a current-carrying wire, the rule is applied as follows:

• Point your thumb in the direction of the electric current.
• Your fingers will naturally curl in the direction of the magnetic field created by the current.

In our exercise, the current in the wire is upward (in the positive \( \hat{j} \) direction). Using the right-hand rule, you will find that the magnetic field forms concentric circles around the wire. At the position of the proton, the field direction is towards the negative \( \hat{i} \) direction (or the negative x-axis). This understanding is crucial in solving magnetic force problems.

Protons, fundamental particles of the atomic nucleus, carry a positive charge. The charge of a proton is exactly \( 1.6 \times 10^{-19} \) Coulombs. Understanding the charge of a proton is essential when calculating the magnetic force it experiences.

The magnetic force \( \vec{F} \) exerted on a proton moving with velocity \( \vec{v} \) in a magnetic field \( \vec{B} \) is given by: \[ \vec{F} = q(\vec{v} \times \vec{B}) \] where \( q \) represents the charge of the proton. Knowing this allows us to compute the resulting force from magnetic interactions.

• The proton has a positive charge.
• This charge interacts with magnetic fields to experience forces.

The cross product is a mathematical operation that finds a vector perpendicular to two given vectors. In physics, it is used to calculate the magnetic force on a charged particle moving in a magnetic field. The cross product \( \vec{v} \times \vec{B} \) where \( \vec{v} \) is velocity and \( \vec{B} \) is magnetic field, informs us about the direction and magnitude of the force:

• It utilizes a determinant format for calculation.
• The result is perpendicular both to velocity and the magnetic field vectors.

In the exercise, the proton's velocity is along the negative y-axis and the magnetic field is along the negative x-axis. By solving the determinant, you determine the direction of the force along the positive z-axis.