Problem 10 - Physics

Question

A magnetic field has a magnitude of \(1.2 \times 10^{-3} \mathrm{~T}\), and an electric field has a magnitude of \(4.6 \times 10^{3} \mathrm{~N} / \mathrm{C}\). Both fields point in the same direction. A positive \(1.8-\mu \mathrm{C}\) charge moves at a speed of \(3.1 \times 10^{6} \mathrm{~m} / \mathrm{s}\) in a direction that is perpendicular to both fields. Determine the magnitude of the net force that acts on the charge.

Step-by-Step Solution

Answer

The net force on the charge is approximately 10.65 N.

Step 1: Understand the Forces Involved

The problem involves two kinds of forces: Electric force and Magnetic force. The net force is the vector sum of these two forces acting on the charge.

Step 2: Calculate the Electric Force

The electric force (\( F_e \) ) acting on a charge is given by the formula: \( F_e = qE \) where \( q \) is the charge and \( E \) is the electric field strength. Substitute the given values: charge \( q = 1.8 \times 10^{-6} \mathrm{~C} \), and electric field \( E = 4.6 \times 10^3 \mathrm{~N/C} \). This results in: \[ F_e = 1.8 \times 10^{-6} \cdot 4.6 \times 10^3 = 8.28 \mathrm{~N} \]

Step 3: Calculate the Magnetic Force

The magnetic force (\( F_m \) ) acting on a moving charge in a magnetic field is given by: \( F_m = qvB \) where \( v \) is the velocity of the charge and \( B \) is the magnetic field strength. Using the given values: charge \( q = 1.8 \times 10^{-6} \mathrm{~C} \), velocity \( v = 3.1 \times 10^6 \mathrm{~m/s} \), and magnetic field \( B = 1.2 \times 10^{-3} \mathrm{~T} \), compute:\[ F_m = 1.8 \times 10^{-6} \cdot 3.1 \times 10^6 \cdot 1.2 \times 10^{-3} = 6.696 \mathrm{~N} \]

Step 4: Calculate the Net Force

Since the charge is moving perpendicular to both fields and the fields are in the same direction, the electric and magnetic forces are perpendicular, hence use the Pythagorean theorem to find the net force: \( F_{net} = \sqrt{F_e^2 + F_m^2} \). Substitute the forces calculated above: \[ F_{net} = \sqrt{8.28^2 + 6.696^2} = \sqrt{68.5584 + 44.8248} = \sqrt{113.3832} \approx 10.65 \mathrm{~N} \]

Key Concepts

Electric Force

Electric force, a fundamental concept in electromagnetism, acts on a charged particle when it is placed in an electric field. Imagine a field similar to gravity, but instead of pulling everything down, this electric field applies a force depending on whether the charge is positive or negative.

For a positive charge, the electric force direction aligns with the electric field's direction, like in this exercise where both the field and the force are in the same line. Conversely, a negative charge would move in the opposite direction. The strength or magnitude of this force depends on both the charge amount and the field strength itself.

• The formula used to determine electric force is:
  \( F_e = qE \)
• Where \( q \) represents the charge and \( E \) the magnitude of the electric field.
• In the exercise, the electric force is calculated to be \( 8.28 \text{~N} \), using the charge \( 1.8 \times 10^{-6} \text{~C} \) and electric field \( 4.6 \times 10^{3} \text{~N/C} \).

Understanding these principles helps clarify why and how forces act on charges within electric fields, which is crucial for solving many physics problems.

Magnetic Force

Magnetic force plays a pivotal role when a charged particle moves through a magnetic field. In this context, the force generated is perpendicular to both the direction of the magnetic field and that of the moving charge. This perpendicular relationship stems from the nature of magnetic forces, following the right-hand rule often mentioned in physics.

In practical terms,

• The magnetic force formula is expressed as:
  \( F_m = qvB \)
• Where \( q \) is the charge, \( v \) is the velocity of the particle, and \( B \) is the strength of the magnetic field.

During the exercise, the magnetic force was computed to be \( 6.696 \text{~N} \), using the given values of \( q = 1.8 \times 10^{-6} \text{ C} \), \( v = 3.1 \times 10^{6} \text{ m/s} \), and \( B = 1.2 \times 10^{-3} \text{ T} \).

This demonstrates how magnetic fields influence moving charges, resulting in forces that are crucial for applications like electric motors and magnetic resonance imaging.

Net Force

The net force experienced by a charged particle moving through electric and magnetic fields is a combination of the electric force and the magnetic force. Since both forces act perpendicular to each other in our exercise scenario, the net force is calculated using the Pythagorean theorem, which applies to right-angle vector addition.

Simply put, the net force determination requires that:

• We use the formula:
  \( F_{net} = \sqrt{F_e^2 + F_m^2} \)
• Where \( F_e \) and \( F_m \) are the magnitudes of the electric and magnetic forces, respectively.

In this case, with \( F_e = 8.28 \text{~N} \) and \( F_m = 6.696 \text{~N} \), this gives us the total force \( F_{net} \approx 10.65 \text{~N} \).
This final calculation shows how two independent forces can combine to create a resultant force that dictates the movement of the charge, an important consideration in the analysis and design of systems involving electromagnetic forces.