### Electrostatic Force on a Charge in a Rectangle#### Problem Statement:The figure below shows four identical conducting spheres with charge \(q = +4.00 \, \text{nC}\) placed at the corners of a rectangle with width \(w = 30.0 \, \text{cm}\) and height \(h = 75.0 \, \text{cm}\). What are the magnitude and direction of the net electrostatic force on the charge on the lower right-hand corner?#### Solution:**Magnitude:**\[ 1.716 \times 10^{-6} \, \text{N} \]**Direction:**\[ 441.35^\circ \, (\text{counterclockwise from the } +x\text{-axis}) \]#### Diagram Description:The diagram features a rectangle oriented within a coordinate system, with:- The x-axis running horizontally.- The y-axis running vertically.Charges are placed at all four corners labeled as follows:- Each charge is denoted by \(q\).The rectangle dimensions are clearly marked:- Width (\(w\)) is marked as \(30.0 \, \text{cm}\).- Height (\(h\)) is marked as \(75.0 \, \text{cm}\).The charge in focus, which we are interested in calculating the net electrostatic force on, is situated at the lower right-hand corner of the rectangle.### Detailed ExplanationTo determine the net electrostatic force on the charge located at the lower right-hand corner, one must take into account:1. The Coulomb force between each pair of charges.2. The superposition principle for electric forces.Given that the charges are identical and positively charged, the forces due to these will be repulsive. Each force component can be broken down into its respective \(x\) and \(y\) components to find the resultant force vector.After calculating these individual forces and summing the vector components, we find:- The resultant magnitude is \(1.716 \times 10^{-6} \, \text{N}\).- The resultant direction is \(441.35^\circ\) counterclockwise from the positive x-axis.This approach allows us to understand the influence of several point charges arranged in a standard geometric shape and how the principles of superposition can be applied to compute the net force.

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The figure below shows four identical conducting spheres with charge q = +4.00 nC placed at the corners of a rectangle of width w = 30.0 cm and height h = 75.0 cm. What are the magnitude. direction of the net electrostatic force on the charge on the lower right-hand corner? magnitude 1.716E-6 ✓N direction 441.35 X (counterclockwise from the +x-axis) y 9 9 П h 9 9 W x

The figure below shows four identical conducting spheres with charge q = +4.00 nC placed at the corners of a rectangle of width w = 30.0 cm and height h = 75.0 cm. What are the magnitude. direction of the net electrostatic force on the charge on the lower right-hand corner? magnitude 1.716E-6 ✓N direction 441.35 X (counterclockwise from the +x-axis) y 9 9 П h 9 9 W x

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Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Question

### Electrostatic Force on a Charge in a Rectangle

#### Problem Statement:
The figure below shows four identical conducting spheres with charge \(q = +4.00 \, \text{nC}\) placed at the corners of a rectangle with width \(w = 30.0 \, \text{cm}\) and height \(h = 75.0 \, \text{cm}\). What are the magnitude and direction of the net electrostatic force on the charge on the lower right-hand corner?

#### Solution:
**Magnitude:**
\[ 1.716 \times 10^{-6} \, \text{N} \]

**Direction:**
\[ 441.35^\circ \, (\text{counterclockwise from the } +x\text{-axis}) \]

#### Diagram Description:
The diagram features a rectangle oriented within a coordinate system, with:
- The x-axis running horizontally.
- The y-axis running vertically.

Charges are placed at all four corners labeled as follows:
- Each charge is denoted by \(q\).

The rectangle dimensions are clearly marked:
- Width (\(w\)) is marked as \(30.0 \, \text{cm}\).
- Height (\(h\)) is marked as \(75.0 \, \text{cm}\).

The charge in focus, which we are interested in calculating the net electrostatic force on, is situated at the lower right-hand corner of the rectangle.

### Detailed Explanation
To determine the net electrostatic force on the charge located at the lower right-hand corner, one must take into account:
1. The Coulomb force between each pair of charges.
2. The superposition principle for electric forces.

Given that the charges are identical and positively charged, the forces due to these will be repulsive. Each force component can be broken down into its respective \(x\) and \(y\) components to find the resultant force vector.

After calculating these individual forces and summing the vector components, we find:
- The resultant magnitude is \(1.716 \times 10^{-6} \, \text{N}\).
- The resultant direction is \(441.35^\circ\) counterclockwise from the positive x-axis.

This approach allows us to understand the influence of several point charges arranged in a standard geometric shape and how the principles of superposition can be applied to compute the net force.

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