Problem 52 - University Physics with Modern Physics

Question

A long, straight wire carries a current of 2.50 A. An electron is traveling in the vicinity of the wire. At the instant when the electron is 4.50 \(\mathrm{cm}\) from the wire and traveling with a speed of \(6.00 \times 10^{4} \mathrm{m} / \mathrm{s}\) directly toward the wire, what are the magnitude and direction (relative to the direction of the current) of the force that the magnetic field of the current exerts on the electron?

Step-by-Step Solution

Answer

The magnitude of the force is \( 5.34 \times 10^{-19} \, \text{N} \) directed outward perpendicular to the wire.

Step 1: Identify the Known Values

The current in the wire is given as \( I = 2.50 \, \text{A} \), the distance of the electron from the wire is \( r = 4.50 \, \text{cm} = 0.0450 \, \text{m} \), and the speed of the electron is \( v = 6.00 \times 10^{4} \, \text{m/s} \).

Step 2: Calculate the Magnetic Field

The magnetic field \( B \) created by a long, straight wire at a distance \( r \) is given by the formula: \[ B = \frac{\mu_0 I}{2 \pi r} \] where \( \mu_0 = 4\pi \times 10^{-7} \text{ T} \cdot \text{m/A} \) is the permeability of free space. Substituting the known values: \[ B = \frac{4\pi \times 10^{-7} \times 2.50}{2\pi \times 0.0450} \] Simplifying this, we get: \[ B = \frac{10^{-6} \times 2.50}{0.0450} = \frac{2.50 \times 10^{-6}}{0.0450} \] \[ B \approx 5.56 \times 10^{-5} \, \text{T} \]

Step 3: Find the Force on the Electron

The magnetic force on a moving charged particle is given by \[ F = qvB \sin(\theta) \] where \( q = -1.60 \times 10^{-19} \, \text{C} \) is the charge of the electron, \( v = 6.00 \times 10^{4} \, \text{m/s} \) is its velocity, and \( B \) is the magnetic field. The angle \( \theta \) between velocity and the magnetic field is 90° (since the electron is moving directly toward the wire), hence \( \sin(90^\circ) = 1 \). Substituting the known values: \[ F = (-1.60 \times 10^{-19}) \times 6.00 \times 10^{4} \times 5.56 \times 10^{-5} \] \[ F = -1.60 \times 10^{-19} \times 3.336 \times 10^{0} \] \[ F \approx -5.34 \times 10^{-19} \, \text{N} \] The negative sign indicates the direction of the force is opposite the direction of the magnetic field relative to the current.

Step 4: Determine the Direction of the Force

Using the right-hand rule, point your thumb in the direction of the current and your fingers in the direction of the electron's velocity (towards the wire). Your palm will face into the page if the current is moving rightward, indicating the force due to the magnetic field acting on a positively charged particle. Since the electron is negatively charged, the actual force is opposite, coming out from the page if the current moves rightward.

Key Concepts

Magnetic Field Calculation

To determine the magnetic field around a wire carrying a current, we use the formula derived from Ampère's Law. The magnetic field (\[B\]) at a distance (\[r\]) from a long, straight current-carrying wire is given by the expression:

• \( B = \frac{\mu_0 I}{2 \pi r} \)

• \( \mu_0 = 4\pi \times 10^{-7} \, \text{T} \cdot \text{m/A} \) is the permeability of free space.
• \( I \) is the current through the wire.
• \( r \) is the distance from the wire where the field is being measured.

By substituting these values, we can compute the magnetic field:

• \( B \approx 5.56 \times 10^{-5} \, \text{T} \)

The calculation shows that as you move closer to the wire, the magnetic field gets stronger due to the inverse relationship with distance.

Right-Hand Rule

The right-hand rule is a simple method used to find the direction of the magnetic force on a charged particle moving in a magnetic field. To employ the right-hand rule:

• Point your thumb in the direction of the current.
• Point your fingers in the direction of the electron’s velocity.
• Your palm indicates the direction of the force for a positively charged particle.

For an electron, which is negatively charged, the actual force direction is opposite to that indicated by your palm. In this exercise, when the electron is moving directly towards the wire and the current moves rightward, the palm faces into the page. Thus, for the electron, the force would come out from the page.

Current in Wire

Current in a wire creates a magnetic field that circles the wire. The direction of the magnetic field can be determined by another right-hand rule:

• Point your thumb in the direction of the current (along the wire).
• Your fingers will curl around the wire in the direction of the magnetic field.

When a current of 2.50 A flows through the wire, a magnetic field is created around it. Distinguishing between a current's effect on an electron and the electron's motion within the field is vital. The moving current produces a circular magnetic field, affecting any charged particle, like an electron, within its influence. This interaction ultimately results in a magnetic force acting on the electron.