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Chapter 17: Problem 35

The electric field caused by a certain point charge has a magnitude of \(6.50 \times 10^{3} \mathrm{~N} / \mathrm{C}\) at a distance of \(0.100 \mathrm{~m}\) from the charge. What is the magnitude of the charge?

Short Answer

Expert verified
The magnitude of the charge is approximately \(7.23 \times 10^{-9} \mathrm{C}\).

Step by step solution

01

Understand the Formula

We will use Coulomb's Law to find the magnitude of the charge. The formula for the electric field due to a point charge is given by: \[ E = \frac{k \cdot |q|}{r^2} \]where \(E\) is the electric field magnitude (\(6.50 \times 10^3 \, \mathrm{N/C}\)), \(k\) is Coulomb's constant (\(8.99 \times 10^9 \, \mathrm{N}\cdot \mathrm{m}^2/\mathrm{C}^2\)), \(|q|\) is the magnitude of charge we need to find, and \(r\) is the distance (\(0.100 \, \mathrm{m}\)).
02

Rearrange the Formula

We need to solve for \(|q|\). Rearrange the formula to get:\[ |q| = \frac{E \cdot r^2}{k} \]
03

Substitute the Known Values

Substitute the values into the rearranged formula:\[ |q| = \frac{6.50 \times 10^3 \, \mathrm{N/C} \times (0.100 \, \mathrm{m})^2}{8.99 \times 10^9 \, \mathrm{N} \cdot \mathrm{m}^2/\mathrm{C}^2} \]
04

Calculate the Magnitude of the Charge

Carry out the calculation:\[ |q| = \frac{6.50 \times 10^3 \times 0.01}{8.99 \times 10^9} \approx 7.23 \times 10^{-9} \, \mathrm{C} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coulomb's Law
Coulomb's Law is a fundamental principle in electromagnetism that describes the force between two charged objects. This force is directly proportional to the product of the magnitudes of the charges, and inversely proportional to the square of the distance between them. The mathematical expression for Coulomb's Law is:

\[ F = k \frac{|q_1 \, q_2|}{r^2} \]
Where:
  • \( F \) is the force between the charges,
  • \( q_1 \) and \( q_2 \) are the values of the charges,
  • \( r \) is the distance between the charges,
  • \( k \) is Coulomb's constant.
In simpler terms, the larger the charges and the closer they are, the stronger the force will be. Coulomb's Law not only predicts the magnitude of this force but also indicates that it is a **field** force, meaning it acts at a distance.
Point Charge
A point charge is an idealized model that represents a charge that is concentrated at a single point in space. It's a useful concept for simplifying complex calculations since it allows us to ignore the charge's actual size or shape.

When dealing with point charges, we can focus on how they affect electric fields and forces around them. For example:
  • When a positive point charge influences another charge, it repels other positive charges and attracts negative ones.
  • The effect of a point charge on its surroundings can be easily calculated using formulas derived from Coulomb's Law, as the effects are radially symmetrical.
This simplified model is commonly used in physics and engineering, particularly when dealing with tiny charges like those found in atomic-scale particles.
Electric Field Magnitude
The term 'electric field magnitude' refers to the strength of an electric field at a specific point in space. It tells us how much force a positive test charge would feel at that point.

Electric fields are represented by the symbol \( E \) and have the units newtons per coulomb (N/C). The magnitude of an electric field created by a point charge is given by the equation:
\[ E = \frac{k \cdot |q|}{r^2} \]Where:
  • \( E \) is the electric field magnitude,
  • \( k \) is Coulomb's constant,
  • \( |q| \) is the magnitude of the charge creating the field,
  • \( r \) is the distance from the charge.
By using this formula, you can find how much the space around a point charge is "electric" at any given distance, which is crucial for understanding how charges interact through fields in a region.
Coulomb's Constant
Coulomb's constant is a critical value in calculations involving electric forces and fields. Denoted by the symbol \( k \), its value is approximately \( 8.99 \times 10^9 \, \mathrm{N \cdot m^2/C^2} \).

This constant arises from the properties of the vacuum and the nature of electric force interactions. It sets the scale for the interaction strength between charges. Here's why Coulomb's constant is essential:
  • Helps define the proportional relationship between force, charge, and distance in Coulomb's Law.
  • Ensures that units are consistent when calculating electric forces in newtons and fields in N/C.
Understanding \( k \) allows us to predict how likely two charges will actually pull or push on each other at a given distance, making it a cornerstone of electric force calculations.

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Most popular questions from this chapter

A \(9.60 \mu \mathrm{C}\) point charge is at the center of a cube with sides of length \(0.500 \mathrm{~m}\). (a) What is the electric flux through one of the six faces of the cube? (b) How would your answer to part (a) change if the sides were \(0.250 \mathrm{~m}\) long? Explain.

Which of the following is true about \(\vec{E}\) inside a negatively charged sphere as described here? A. It points from the center of the sphere to the surface and is largest at the center. B. It points from the surface to the center of the sphere and is largest at the surface. C. It is zero. D. It is constant but is not zero.

Three point charges are arranged along the \(x\) axis. Charge \(q_{1}=-4.50 \mathrm{nC} \quad\) is \(\quad\) located \(\quad\) at \(\quad x=0.200 \mathrm{~m}, \quad\) and \(\quad\) charge \(q_{2}=+2.50 \mathrm{nC}\) is at \(x=-0.300 \mathrm{~m}\). A positive point charge \(q_{3}\) is located at the origin. (a) What must the value of \(q_{3}\) be for the net force on this point charge to have magnitude \(4.00 \mu \mathrm{N}\) ? (b) What is the direction of the net force on \(q_{3} ?\) (c) Where along the \(x\) axis can \(q_{3}\) be placed and the net force on it be zero, other than the trivial answers of \(x=+\infty\) and \(x=-\infty ?\)

A negative charge of \(-0.550 \mu \mathrm{C}\) exerts an upward \(0.200 \mathrm{~N}\) force on an unknown charge \(0.300 \mathrm{~m}\) directly below it. (a) What is the unknown charge (magnitude and sign)? (b) What are the magnitude and direction of the force that the unknown charge exerts on the \(-0.550 \mu \mathrm{C}\) charge?

A positively charged rubber rod is moved close to a neutral copper ball that is resting on a nonconducting sheet of plastic. (a) Sketch the distribution of charges on the ball. (b) With the rod still close to the ball, a metal wire is briefly connected from the ball to the earth and then removed. After the rubber rod is also removed, sketch the distribution of charges (if any) on the copper ball.
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