Question

Answer the following questions about force on a moving charge. An electron \(\left(q=-1.6 \times 10^{-19} \mathrm{C}\right)\) enters a constant \(2-\mathrm{T}\) magnetic field at an angle of \(45^{\circ}\) to the field with a speed of \(2 \times 10^{5} \mathrm{m} / \mathrm{s} .\) Find the magnitude of the force on the electron.

Short answer

Answer: The magnitude of the magnetic force acting on the electron is approximately 4.52 x 10^-14 N.

Step-by-step solution

Recall the magnetic force equation

The equation for the magnetic force acting on a moving charge is given by: $$ F = q|v|B\sin\theta $$ where \(F\) is the magnetic force, \(q\) is the charge, \(v\) is the speed, \(B\) is the magnetic field strength, and \(\theta\) is the angle between the charge's velocity and the magnetic field.

Plug in the given values

We are given the value of charge \(q=-1.6 \times 10^{-19}\, \mathrm{C}\), speed \(|v|=2 \times 10^{5}\, \mathrm{m/s}\), magnetic field strength \(B=2\, \mathrm{T}\), and angle \(\theta=45^{\circ}\). Now, plug these values into the equation: $$ F = (-1.6 \times 10^{-19}\, \mathrm{C})(2 \times 10^5\, \mathrm{m/s})(2\, \mathrm{T})\sin(45^{\circ}) $$

Calculate the force

Now, we simply need to calculate the value of the force using the provided values. Since we are only looking for the magnitude of the force, it doesn't matter that the charge is negative. We just need to focus on the value of the force: $$ F = (1.6 \times 10^{-19}\, \mathrm{C})(2 \times 10^5\, \mathrm{m/s})(2\, \mathrm{T})\sin(45^{\circ}) $$ $$ F = (3.2 \times 10^{-14}\, \mathrm{N} \cdot \mathrm{T} \cdot \mathrm{C})(2\, \mathrm{T})(0.707) $$ $$ F = (6.4 \times 10^{-14}\, \mathrm{N} \cdot \mathrm{T}^2 \cdot \mathrm{C})(0.707) $$ $$ F \approx 4.52 \times 10^{-14}\, \mathrm{N} $$ So, the magnitude of the magnetic force on the electron is approximately \(4.52 \times 10^{-14}\, \mathrm{N}\).

Key concepts

Magnetic Field Strength

When tackling problems involving magnetic fields, it's essential to understand what magnetic field strength represents. Magnetic field strength, commonly denoted by the symbol B, is a measure of the magnetic influence that a magnetic object or electric current exerts on moving charges in its vicinity. It is often referred to as magnetic flux density or magnetic induction. The unit of magnetic field strength is the Tesla (T).

In the context of our problem, the electron is entering a constant magnetic field of 2 T, which is considered a relatively strong magnetic field. This high value indicates that the force acting on the electron due to the field will be significant. The strength of this field significantly governs the trajectory of charged particles, such as electrons, when they pass through it.

Lorentz Force

Magnetic force on a moving charge is a specific instance of a more general phenomenon called the Lorentz force. This force is the combination of electric and magnetic forces on a point charge due to electromagnetic fields. However, in our problem, the electron is only subject to magnetic force since no electric field is mentioned.

The force on a charge moving in a magnetic field is given by the equation F = q|v|B sin(θ), where F is the Lorentz force, q is the charge, |v| is the speed of the particle, B is the magnetic field strength, and θ is the angle between the velocity vector and the magnetic field. This equation tells us how three factors—the charge, speed, and field strength—combine to determine the magnitude of the magnetic force.

Charge Velocity Angle

The angle of the charge's velocity in relation to the magnetic field, denoted by θ, is crucial in calculating the magnitude of the magnetic force. This angle affects how much of the velocity contributes to the force experienced by the charge. If the charge moves parallel or anti-parallel to the magnetic field (θ = 0° or θ = 180°), it experiences no force.

In our problem, the electron enters the field at an angle of 45°, which means that the sin(θ) factor plays a significant role in the resulting force. The sine function has its maximum value of 1 when θ is 90°, and at 45° the value is 0.707. As the sine value decreases, the force's magnitude decreases. Hence, the angle is integral to determining the force exerted on the moving charge.