91Ó°ÊÓ

When dealing with electric fields and charges, the electric force can be calculated using the formula \( F_E = qE \). Here, \( q \) represents the charge, measured in coulombs, while \( E \) stands for the electric field strength, measured in newtons per coulomb (N/C). It helps us understand how a charge behaves under the influence of an electric field. In our exercise:

• The charge \( q \) is \( 1.8 \times 10^{-6} \, \mathrm{C} \).
• The electric field \( E \) is \( 4.6 \times 10^3 \, \mathrm{N/C} \).
• The calculated electric force, \( F_E \), is concluded by multiplying these two values: \( F_E = (1.8 \times 10^{-6})(4.6 \times 10^3) = 8.28 \, \mathrm{N} \).

It's crucial to grasp that this force acts directly along the direction of the electric field and is unaffected by the magnetic field. The electric force is a result of the interaction between the charge and electric field's direction, dictating the acceleration of the charge.

The magnetic force on a moving charge can be determined using the equation \( F_B = qvB \sin \theta \). This formula elucidates how a charge is affected when moving within a magnetic field. Let's break down these components:

• \( q \) is the charge magnitude (\( 1.8 \times 10^{-6} \, \mathrm{C} \)).
• \( v \) is the velocity of the charge (\( 3.1 \times 10^6 \, \mathrm{m/s} \)).
• \( B \) represents the magnetic field strength (\( 1.2 \times 10^{-3} \, \mathrm{T} \)).
• \( \theta \) is the angle between the velocity vector and the magnetic field, which is \( 90^\circ \) given they are perpendicular, so \( \sin 90^\circ = 1 \).

Thus, the magnetic force calculation uses: \( F_B = (1.8 \times 10^{-6})(3.1 \times 10^6)(1.2 \times 10^{-3}) = 6.696 \, \mathrm{N} \). It's essential to remember that this force acts perpendicular to both the velocity of the charge and the magnetic field direction, potentially altering the charge's trajectory.

In physics, forces can be combined to determine a net force using vector addition. When forces act in different directions, as in this exercise, the net force \( F_{net} \) on an object is the vector sum of the individual forces, electric and magnetic in our case. Crucially, the electric and magnetic forces here are orthogonal, meaning they are at right angles to each other. This allows us to employ the Pythagorean theorem to compute the resultant magnitude:

• Electric force \( F_E = 8.28 \, \mathrm{N} \).
• Magnetic force \( F_B = 6.696 \, \mathrm{N} \).

Given their perpendicular arrangement, the vector sum is \[ F_{net} = \sqrt{F_E^2 + F_B^2} \], which calculates to \[ \sqrt{8.28^2 + 6.696^2} \approx 10.66 \, \mathrm{N} \]. This net force represents how strongly the charge is influenced combined effect of both fields. The direction of this force is crucially a result of the vector combination and is not aligned with either individual force's direction on its own.