Ch 22: Electric Charges and Forces

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 22: Electric Charges and Forces

Problem 65c

Chapter 22, Problem 65c

A 10.0 nC charge is located at position (x, y)=(1.0 cm, 2.0 cm). At what (x, y) position(s) is the electric field $open paren 21 comma 600 i hat minus 28 comma 800 j hat close paren$ N/C?

Verified step by step guidance

Step 1: Recall the formula for the electric field due to a point charge:

E = k_e * |q| / r^2

, where

k_e

is Coulomb's constant (

8.99 × 10^9 \, N \, m^2 / C^2

q

is the charge, and

r

is the distance from the charge to the point where the field is being calculated.

Step 2: The electric field is a vector quantity, so it has both magnitude and direction. The given electric field vector is

E = (21,600 \, \(\hat{i}\) - 28,800 \, \(\hat{j}\)) \, N/C

. This means the field has components in the x-direction and y-direction. Use the relationship

E_x = k_e * |q| * (x - x_0) / r^3

and

E_y = k_e * |q| * (y - y_0) / r^3

, where

(x_0, y_0)

is the position of the charge.

Step 3: Calculate the distance

r

from the charge to the point where the electric field is given. Use the formula

r = \(\sqrt{(x - x_0)^2 + (y - y_0)^2}\)

. Substitute

(x_0, y_0) = (1.0 \, \(\text{cm}\), 2.0 \, \(\text{cm}\))

and solve for

r

Step 4: Solve for the coordinates

(x, y)

by equating the given electric field components to the expressions for

E_x

and

E_y

. This will involve solving the system of equations:

21,600 = k_e * |q| * (x - x_0) / r^3

and

-28,800 = k_e * |q| * (y - y_0) / r^3

Step 5: Rearrange the equations to isolate

x

and

y

. Substitute the known values of

k_e

q = 10.0 \, \(\text{nC}\) = 10.0 × 10^{-9} \, C

, and

(x_0, y_0)

. Solve the system of equations to find the coordinates

(x, y)

where the electric field has the given components.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electric Field

The electric field is a vector field that represents the force exerted by an electric charge on other charges in its vicinity. It is defined as the force per unit charge experienced by a positive test charge placed in the field. The direction of the electric field is away from positive charges and towards negative charges, and its magnitude is measured in newtons per coulomb (N/C).

Coulomb's Law

Coulomb's Law describes the electrostatic interaction between charged particles. It states that the force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This law is fundamental for calculating the electric field generated by a point charge, as it helps determine the force experienced by other charges in the field.

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. In the context of electric fields, each charge contributes to the total electric field at a point, and these contributions must be added as vectors, taking into account their magnitudes and directions. This is crucial for solving problems involving multiple charges and determining the net electric field at a specific location.

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Vector Addition By Components

Related Practice

Textbook Question

An electric field $E right arrow equals 200 comma 000 i hat$ N/C causes the point charge in FIGURE P22.68 to hang at an angle. What is θ?

Textbook Question

An electric dipole consists of two opposite charges $\pm q$ separated by a small distance $s$ . The product $p = q s$ is called the dipole moment. Figure P $22.61$ shows an electric dipole perpendicular to an electric field $E$ . Find an expression in terms of $p$ and $E$ for the magnitude of the torque that the electric field exerts on the dipole.

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Textbook Question

You have a lightweight spring whose unstretched length is 4.0 cm. First, you attach one end of the spring to the ceiling and hang a 1.0 g mass from it. This stretches the spring to a length of 5.0 cm. You then attach two small plastic beads to the opposite ends of the spring, lay the spring on a frictionless table, and give each plastic bead the same charge. This stretches the spring to a length of 4.5 cm. What is the magnitude of the charge (in nC) on each bead?

Textbook Question

Three 1.0 nC charges are placed as shown in FIGURE P22.66. Each of these charges creates an electric field E at a point 3.0 cm in front of the middle charge. What are the three fields E₁, E₂, and E₃ created by the three charges? Write your answer for each as a vector in component form.

Textbook Question

The identical small spheres shown in FIGURE P22.64 are charged to +100 nC and −100 nC. They hang as shown in a 100,000 N/C electric field. What is the mass of each sphere?

Textbook Question

Starting from rest, how long does it take an electron to move 1.0 cm in a steady electric field of magnitude 100 N/C?

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A 10.0 nC charge is located at position (x, y)=(1.0 cm, 2.0 cm). At what (x, y) position(s) is the electric field (21,600i^−28,800j^)(21,600\(\hat{i}\)-28,800\(\hat{j}\)) N/C?

Verified video answer for a similar problem:

Key Concepts

Electric Field

Coulomb's Law

Vector Addition

A 10.0 nC charge is located at position (x, y)=(1.0 cm, 2.0 cm). At what (x, y) position(s) is the electric field $open paren 21 comma 600 i hat minus 28 comma 800 j hat close paren$ N/C?