Problem 1 - College Physics

Question

A charge of 28.0 \(\mathrm{nC}\) is placed in a uniform electric field that is directed vertically upward and that has a magnitude of \(4.00 \times 10^{4} \mathrm{N} / \mathrm{C}\) . What work is done by the electric force when the charge moves (a) 0.450 \(\mathrm{m}\) to the right; (b) 0.670 \(\mathrm{m}\) upward; (c) 2.60 \(\mathrm{m}\) at an angle of \(45.0^{\circ}\) downward from the horizontal?

Step-by-Step Solution

Answer

(a) 0 J, (b) 0.75 J, (c) -2.06 J.

Step 1: Understand the Relationship Between Work, Electric Force, and Electric Field

The work done by an electric force can be calculated by the formula \( W = qEd \cos \theta \), where \( q \) is the charge, \( E \) is the electric field strength, \( d \) is the displacement, and \( \theta \) is the angle between the electric field and the direction of displacement.

Step 2: Work Calculation for 0.450 m to the Right

For part (a), since the movement is to the right while the electric field is vertical, the angle \( \theta \) between the electric field and the direction of displacement is \( 90^{\circ} \). Thus, \( \cos 90^{\circ} = 0 \). The work done is: \[ W = 28.0 \times 10^{-9} \, \text{C} \times 4.00 \times 10^4 \, \text{N/C} \times 0.450 \, \text{m} \times \cos 90^{\circ} \]\[ W = 0 \, \text{J} \].Since the force and displacement are perpendicular, no work is done.

Step 3: Work Calculation for 0.670 m Upward

For part (b), the movement is in the same direction as the electric field (upward). Therefore, \( \theta = 0^{\circ} \). The work done is: \[ W = 28.0 \times 10^{-9} \, \text{C} \times 4.00 \times 10^4 \, \text{N/C} \times 0.670 \, \text{m} \times \cos 0^{\circ} \] \[ W = 28.0 \times 10^{-9} \times 4.00 \times 10^4 \times 0.670 \times 1 \]\[ W = 0.75 \, \text{J} \].

Step 4: Work Calculation for 2.60 m at an Angle of 45.0° Downward

For part (c), the movement is at an angle \( 45.0^{\circ} \) downward, making the angle between the displacement and electric field \( 135^{\circ} \) (since the field is upward). The work done is: \[ W = 28.0 \times 10^{-9} \, \text{C} \times 4.00 \times 10^4 \, \text{N/C} \times 2.60 \, \text{m} \times \cos 135^{\circ} \] \[ \cos 135^{\circ} = -\frac{\sqrt{2}}{2} \], so:\[ W = 28.0 \times 10^{-9} \times 4.00 \times 10^4 \times 2.60 \times -\frac{\sqrt{2}}{2} \]\[ W \approx -2.06 \, \text{J} \].

Key Concepts

Electric Force

Electric force is the attraction or repulsion interaction between any two charged objects. This force is a direct result of the electric field surrounding a charged particle. Think of the electric field as the force field produced by any charged object, which influences other charges present in its vicinity.

In mathematical terms, the electric force (\( F \)) can be calculated by using Coulomb's Law:

• \( F = k \frac{q_1 q_2}{r^2} \)
• where \( F \) is the force
• \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the magnitudes of the charges
• and \( r \) is the distance between the two charges

The direction and magnitude of the electric force depend on the sign and magnitude of the charges involved. Opposite charges attract, while like charges repel each other. This concept is fundamental for understanding how charges behave in an electric field. By understanding the basics of electric force, you can predict the movement of charges and calculate the work done by electric forces when charges move.

Work Done by Electric Force

The work done by an electric force is critical in determining how much energy is needed or released when a charge moves in an electric field. The concept of work in physics relates the force applied to an object and the distance it moves in the direction of the force. In the context of electric fields, we use the formula:

• \( W = qEd \cos \theta \)
• where \( W \) is the work done
• \( q \) is the electric charge
• \( E \) is the electric field strength
• \( d \) is the displacement
• \( \theta \) is the angle between the electric field and the direction of displacement

The cosine term here indicates how much of the applied force effectively contributes to the movement in the intended direction. Therefore, if the movement is perpendicular to the electric field (\( \theta = 90^{\circ} \)), no work is done because all the force is "wasted" by pulling without moving in the intended direction. Conversely, if movement aligns with the force (\( \theta = 0^{\circ} \)), the maximum work is done.

Angle of Displacement

The angle of displacement is crucial when calculating the work done by electric force because it affects how effectively a force contributes to moving a charge. When a charge moves at an angle in an electric field, only a component of the force assists with this movement. This is why the angle \( \theta \) in the formula \( W = qEd \cos \theta \) is vital.

• If \( \theta = 0^{\circ} \), the movement is fully aligned with the electric field, so the cosine part becomes 1, maximizing the work done.
• If \( \theta = 90^{\circ} \), the charge moves perpendicular to the electric field, resulting in no work getting done.
• For angles greater than \( 90^{\circ} \), the work can actually become negative, meaning that the force is applied in such a way that it opposes the displacement.

Understanding how the angle affects the displacement ensures you can calculate the exact amount of work being performed by electric forces in various scenarios. This understanding is critical in designing systems that require specific movements or energy calculations such as motors and electrical networks.

Electric Charge Calculation

Electric charge calculation is fundamental when working with electric forces and fields. Understanding how to determine the values of charge is crucial for various applications in physics and engineering. A charge (\( q \)) quantifies the electricity held by an object, and its effect is observed through the electric field it generates.

• Charges are typically measured in coulombs (C), but in many practical applications, smaller units like nanocoulombs (nC) or microcoulombs (µC) are used, as they deal with relatively small quantities.
• Charge quantification helps in using formulas such as \( F = Eq \) to determine the force applied on charges in an electric field.
• In practical scenarios, electric charge calculations enable the analysis and design of circuits, understanding capacitors, and examining electrostatic scenarios and phenomena.

By effectively calculating the charge, you will better comprehend the interaction within an electric field and predict the related physical changes. This knowledge aids in addressing various problems involving the movement of charges and energy transformations.