Chapter 23: Problem 15
A charge of 28.0 nC is placed in a uniform electric field that is directed vertically upward and has a magnitude of 4.00 \(\times 10^4\) V\(/\)m. What work is done by the electric force when the charge moves (a) 0.450 m to the right; (b) 0.670 m upward; (c) 2.60 m at an angle of 45.0\(^\circ\) downward from the horizontal?
Short Answer
Step by step solution
Understanding the Problem
Equation for Work Done by Electric Field
Calculating Work for 0.450 m to the Right
Calculating Work for 0.670 m Upward
Calculating Work for 2.60 m at 45 Degrees Downward
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Work Done by Electric Force
- \( W = qEd \cos \theta \)
- Here, \( W \) is the work done, \( q \) is the charge, \( E \) is the electric field strength, \( d \) is the distance moved, and \( \theta \) is the angle between the direction of the field and the direction of motion.
This depends on the displacement's direction relative to the field. If the charge moves parallel to the electric field direction, maximum work is done.
If it moves perpendicular, no work is done.
Uniform Electric Field
It is often represented by equally spaced parallel lines indicating direction and magnitude. The electric field force on a charge in a uniform electric field is constant, which simplifies calculation of the work done. In this exercise, the field is directed vertically upward, meaning any vertical motion of a charge either aligns or opposes this field.
This uniformity allows us to use the simple work formula:
- \( W = qEd \cos \theta \)
- If the charge is positioned upward, it experiences a force along the vertical direction.
- Horizontal movements do not affect the work done by the field.
Angle Between Field and Motion
It modifies the impact of the field on the charge's movement, as seen in the cosine term of the formula \( W = qEd \cos \theta \).
An angle of 0° signifies the motion is in the same direction as the field, and the cosine of 0° is 1 thus maximum work is done by the force.
Conversely, an angle of 90° means motion is perpendicular to the field, resulting in no work (since \( \cos 90° = 0 \)).
When the angle is greater than 90°, such as with 135°, the motion opposes the field to some degree, resulting in negative work done by the electric force, as the force moves opposite to the motion direction. Understanding this angle can help predict how different directions and magnitudes of motion affect energy changes on the charge.