Chapter 21: Problem 71
Three identical point charges \(q\) are placed at each of three corners of a square of side \(L\). Find the magnitude and direction of the net force on a point charge \(-3q\) placed (a) at the center of the square and (b) at the vacant corner of the square. In each case, draw a free-body diagram showing the forces exerted on the \(-3q\) charge by each of the other three charges.
Short Answer
Step by step solution
Understand the Problem Setup for Part (a)
Determine Forces Exerted on \(-3q\) in Part (a)
Calculate Net Force on \(-3q\) in Part (a)
Understand Problem Setup for Part (b)
Assess Forces on \(-3q\) in Part (b)
Calculate Net Force on \(-3q\) in Part (b)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Net Force
This means considering both the size and direction of each force. For example, if three forces are applied to a point charge at the center of a square, and they are equal in magnitude and opposite in direction, their net force will cancel out to zero. This is a common result when forces are symmetrically arranged around a point.
Net force determines the acceleration and ultimate motion of the object. If the net force is zero, the object will remain at rest or move with constant velocity, according to Newton's first law. However, if there's a net force, the object will accelerate in the direction of this resultant force. Understanding net force helps us predict how charges and particles will behave under multiple interactions.
Point Charge
In our exercise, point charges are used at each corner of a square and at the center (or vacant corner). The charge values are represented by symbols such as \(q\) and \(-3q\). Point charges make it easier to apply mathematical equations to solve for forces between them. They allow for easy usage of the inverse square law in Coulomb's description, where the force magnitude depends on the distance between charges:
- The force is directly proportional to the product of the two charges.
- The force is inversely proportional to the square of the distance between them.
Vector Addition
When calculating forces such as in the provided square problem, each force from a point charge must be added vectorially. Essentially, you assess both horizontal and vertical components. If done correctly, the result is a single vector representing the net force.
For example, the exercise demonstrates how forces at 90-degree angles can be added using vector addition, revealing their combined effect on a point charge. The Pythagorean theorem often helps to calculate the magnitude of the resultant vector:
- Add the square of each perpendicular component.
- Take the square root of the sum to find the resultant vector's magnitude.