Problem 68 - Fundamentals of Physics

Question

A long straight wire carries a current of \(50 \mathrm{~A} .\) An electron, traveling at \(1.0 \times 10^{7} \mathrm{~m} / \mathrm{s}\), is \(5.0 \mathrm{~cm}\) from the wire. What is the magnitude of the magnetic force on the electron if the electron velocity is directed (a) toward the wire, (b) parallel to the wire in the direction of the current, and (c) perpendicular to the two directions defined by (a) and (b)?

Step-by-Step Solution

Answer

For (a) and (b), the force is 0. For (c), the force is \(3.2 \times 10^{-16} \text{ N}\).

Step 1: Identify magnetic field using Ampère's law

For a long straight wire carrying a current, we use Ampère's Law to find the magnetic field. The magnitude of the magnetic field at a distance \( r \) from a wire carrying current \( I \) is given by the equation:\[ B = \frac{\mu_0 I}{2\pi r} \]where \( \mu_0 \) is the permeability of free space: \( \mu_0 = 4\pi \times 10^{-7} \text{ Tm/A} \).Insert the values:\[ B = \frac{4\pi \times 10^{-7} \times 50}{2\pi \times 0.05} \]Solving this gives:\[ B = 2 \times 10^{-4} \text{ T} \].

Step 2: Calculate magnetic force (a) Velocity toward the wire

The magnetic force on a moving charge is given by the equation:\[ F = qvB \sin \theta \]where \( q \) is the charge of the electron (\( q = -1.6 \times 10^{-19} \text{ C} \)), \( v \) is the velocity, \( B \) is the magnetic field, and \( \theta \) is the angle between \( v \) and \( B \).For the velocity directed toward the wire, the angle \( \theta = 0^\circ \) or \( 180^\circ \) (since the magnetic field circles the wire, but for a direct radius the sine is zero), so \( \sin \theta = 0 \). Thus, the force is:\[ F = qvB \sin 0^\circ = 0 \].

Step 3: Calculate magnetic force (b) Velocity parallel to the wire

For the velocity parallel to the wire, the angle \( \theta = 0^\circ \) (parallel to the current), hence \( \sin \theta = 0 \). Thus, the force is:\[ F = qvB \sin 0^\circ = 0 \].

Step 4: Calculate magnetic force (c) Velocity perpendicular to both directions

When the electron's velocity is perpendicular to both the wire and the previous configurations, \( \theta = 90^\circ \), thus \( \sin 90^\circ = 1 \). The force magnitude is then:\[ F = qvB \]Insert the given values:\[ F = (-1.6 \times 10^{-19} \text{ C})(1.0 \times 10^{7} \text{ m/s})(2 \times 10^{-4} \text{ T}) \]This gives:\[ F = -3.2 \times 10^{-16} \text{ N} \](Since force is a vector, the negative sign shows direction opposite to the assumed positive direction).

Step 5: Conclusion

For configurations (a) and (b), the magnetic force is \( F = 0 \). For configuration (c), the magnitude of the magnetic force is \( 3.2 \times 10^{-16} \text{ N} \) directed opposite to the assumed positive direction.

Key Concepts

Ampère's Law

Ampère's Law is a fundamental principle in electromagnetism that relates the magnetic field around a current-carrying conductor to the electric current flowing through it. It is similar to Gauss's Law in electrostatics. Ampère's Law is mathematically expressed as:
\[\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc}\]where the left side is the line integral of the magnetic field \( \mathbf{B} \) around a closed loop, \( d\mathbf{l} \) is a differential length element along the path, \( \mu_0 \) is the permeability of free space, and \( I_{enc} \) is the total current enclosed by the path.
For a straight, long wire, Ampère’s Law can simplify to calculate the magnetic field at a distance \( r \) from the wire:
\[B = \frac{\mu_0 I}{2\pi r}\]This equation shows how the magnetic field strength decreases with increasing distance from the wire.

• The magnetic field is proportional to the current \( I \).
• It is inversely proportional to the distance \( r \).

Current-Carrying Wire

A current-carrying wire produces a magnetic field around it. This is due to the motion of electric charges, specifically the electrons moving along the wire.
The direction of the magnetic field produced can be determined by the right-hand rule:

• Point your thumb in the direction of the current flow.
• Your fingers curl in the direction of the magnetic field lines.

For a wire with current flowing upwards, the field circulates in a counterclockwise direction when viewed from above.
This magnetic field influences other charged particles or current-carrying wires nearby. It is essential in forming electromagnets and is a fundamental part of how motors and generators operate.

Electromagnetic Theory

Electromagnetic theory explains how electric current and magnetic fields interact. It is a central topic in physics and underlies the technology that powers much of our modern life.
This theory consists of a few key points:

• Electric currents produce magnetic fields, which are described by Ampère's Law.
• Changing magnetic fields can induce electric currents, as described by Faraday's Law.
• The interaction between magnetic fields and charges moving at certain angles results in forces, described by the Lorentz force law: \( F = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \), where \( F \) is the force, \( q \) is the charge, \( \mathbf{E} \) is the electric field, \( \mathbf{v} \) is the velocity of the charge, and \( \mathbf{B} \) is the magnetic field.

This theory not only explains how electric currents can create magnets, but also how moving charged particles, like electrons, experience forces that affect their motion when in a magnetic field. It encompasses both the generation of electromagnetic waves (like light) and the functionality of everyday devices such as transformers, inductors, and radio transmitters.