Problem 58 - University Physics with Modern Physics

Question

A long, straight wire carries a current of 8.60 A. An electron is traveling in the vicinity of the wire. At the instant when the electron is 4.50 cm from the wire and traveling at a speed of 6.00 \(\times\) 10\(^4\) m/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 magnetic force on the electron is approximately \(3.67 \times 10^{-19} N\) outward from the wire.

Step 1: Determine the Magnetic Field

The magnetic field around a long, straight wire carrying current can be calculated using Ampere's Law. This is given by the formula \[ B = \frac{{\mu_0 \cdot I}}{{2\pi \cdot r}} \]where \(B\) is the magnetic field, \(\mu_0 = 4\pi \times 10^{-7} \ T\cdot m/A\) is the permeability of free space, \(I = 8.60 \ A\) is the current, and \(r = 0.045\ m\) is the distance of the electron from the wire. Plugging in values, we get\[ B = \frac{{4\pi \times 10^{-7} T\cdot m/A \cdot 8.60 A}}{{2\pi \cdot 0.045 m}} \approx 3.82 \times 10^{-5} \ T \].

Step 2: Use the Lorentz Force Equation

The force on a charged particle due to a magnetic field is given by the Lorentz force equation \[ F = q \cdot v \cdot B \cdot \sin(\theta) \]where \(F\) is the force, \(q = -1.6 \times 10^{-19} \ C\) is the charge of the electron, \(v = 6.00 \times 10^4 m/s\) is the velocity of the electron, \(B = 3.82 \times 10^{-5} T\) is the magnetic field, and \(\theta = 90^\circ\) since the velocity is directly towards the wire, implying the field is perpendicular to the motion. Therefore, \(\sin(\theta) = 1\). The equation simplifies to \[ F = -1.6 \times 10^{-19} C \cdot 6.00 \times 10^4 m/s \cdot 3.82 \times 10^{-5} T \approx -3.67 \times 10^{-19} N \].

Step 3: Determine the Direction of Force

As the electron moves perpendicularly towards the wire, the right-hand rule states that if you point your thumb in the direction of current and your fingers in the direction of the electron's velocity, then the force's direction (palm) is outward. However, for an electron, reverse the force's direction because of its negative charge. Hence, the force is directed outward, away from the wire.

Key Concepts

Ampere's Law

Ampere's Law is a useful tool for calculating the magnetic field created by an electrical current. It states that the magnetic field in a loop is proportional to the current flowing through the loop. For long, straight wires, this law simplifies. You use the formula: \[ B = \frac{{\mu_0 \cdot I}}{{2\pi \cdot r}} \] Here, \(B\) represents the magnetic field, \(\mu_0\) is the permeability of free space, \(I\) is the current, and \(r\) is the distance from the wire. This makes it easy to calculate the magnetic field at a given point around the wire. The magnetic field decreases with distance, inversely proportional to \(r\). Always ensure units are consistent to avoid errors. For problems involving electricity and magnetism, converting measurements to meters and amperes simplifies calculations. Using these principles, you find the magnetic field strength around wires and effects on nearby charges.

Lorentz Force

The Lorentz Force is crucial for understanding how charged particles behave in magnetic fields. It determines how the force acting on a charged particle depends not only on the magnetic field but also on the particle's velocity. It's given by: \[ F = q \cdot v \cdot B \cdot \sin(\theta) \] - \(F\) is the force on the particle. - \(q\) is the charge of the particle, negative for electrons. - \(v\) is the particle's velocity. - \(B\) is the magnetic field. - \(\theta\) is the angle between the velocity and the magnetic field. When this angle is 90 degrees, \(\sin(\theta)\) is 1, meaning the velocity is perpendicular to the field. The Lorentz force tells you both the magnitude and direction of the force. Remember, a negative charge like an electron will move opposite to the direction calculated for a positive charge. Thoroughly understanding the Lorentz Force helps predict the movement of electrons in magnetic fields, crucial for applications like electric motors and generators.

Right-Hand Rule

The Right-Hand Rule is a simple method to determine the direction of a magnetic force on a charged particle. To use it: 1. **Point your thumb** in the direction of the current (positive charge movement). 2. **Extend your fingers** in the direction of velocity. 3. **The force's direction** will be where your palm is facing. This method works directly for positive charges. However, for electrons or negative charges, the force's direction is opposite to what the right-hand rule indicates. This useful tool helps visualize vector directions in three dimensions, making it easier to grasp the interactions within electromagnetic fields. It’s crucial to mastering concepts in physics, especially in scenarios with charged particles in magnetic fields. Always be mindful to reverse the direction for negative charges to ensure accuracy in real-world physics applications.