Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

Answers without the blur.

Just sign up for free and you're in.

Short Answer

Figure 22-38a shows two charged particles fixed in place on an x-axis with separation L. The ratio $q_{1} / q_{2}$ of their charge magnitudes is . Figure 22-38b shows the x component $E_{n e t}, X$ of their net electric field along the x-axis just to the right of particle 2. The x-axis scale is set by $x_{s} = 30.0 c m$ . (a) At what value of $x > 0 is E_{n e t}, x$ is maximum? (b) If particle 2 has charge $- q_{2} = - 3 e$ , what is the value of that maximum?

The value of x at which $E_{net}, X$ is maximum is 34 cm
The value of that maximum if particle 2 has a charge -3e is $2.2 \times 10^{- 8} N / C$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: The given data

The two charged particles are on the x-axis with separation, L .
The value of the ratio, $\frac{q_{1}}{q_{2}} = 4$
The x-axis scale is set as: $x_{s} = 30.0 cm$
Particle 2 has a charge of $- q_{2} = - 3 e$

Step 2: Understanding the concept of electric field

Using the concept of the electric field at a given point, we can get the value of an individual electric field by a charge. Again for the maximum value of the net field, we can differentiate the electric field equation for getting the value of x. Now, substituting the value of x, we can get the value of the required electric field.

Formulae:

The magnitude of the electric field, $E = \frac{q}{4 {πε}_{0} R^{2}} \hat{R}$ (1)

where R = The distance of field point from the charge, and q = charge of the particle

According to the superposition principle, the electric field at a point due to more than one charge,

$\vec{E} = \sum_{i - 1}^{n} \vec{E_{i}} = \sum_{i - 1}^{n} \frac{q_{i}}{4 {πε}_{0} r_{i}^{2}} {\hat{r}}_{i}$ (2)

Step 3: a) Calculation of the value of Enetx for being maximum

For it to be possible for the net field to vanish at some x > 0, the two individual fields (caused by $q_{1}$ and $q_{2}$ ) must point in opposite directions for x > 0. They are therefore oppositely charged considering their positions. Further, since the net field points more strongly leftward for the small positive x (where it is very close to $q_{2}$ ), then we conclude that localid="1657282744540" $q_{2}$ is the negative-valued charge. Thus, $q_{1}$ is a positive-valued charge.

From the given ratio, we can now considered for getting a maximum electric field that

$q_{1} = 4 e and q_{2} = e$

Thus using equation (1), we can find the individual fields, and substituting this into equation (2), we can get the net electric as:

$E_{n e t} = E_{1} + E_{2} = \frac{4}{4 {πε}_{0} {(L + x)}^{2}} - \frac{e}{4 {πε}_{0} {(x)}^{2}} . . . . . . . . . . . . . . . . . . . (3)$

Setting $E_{net} = 0$ at (see graph) x = 20 cm , the graph immediately leads to To get the maximum value of the electric field, we can differentiate the above equation for x, and equating it to zero, we can get the value of x as:

$\frac{d}{dx} [\frac{4 e}{4 {πε}_{0} {(L + x)}^{2}} - \frac{e}{4 {πε}_{0} {(x)}^{2}}] = 0 x = (\frac{2}{3} \sqrt[3]{2} + \frac{1}{3} \sqrt[3]{4} + \frac{1}{3}) L x = 1.70 (20 cm) x = 34 cm$

Hence, the value of x is 34 cm

Step 4: b) Calculation of that maximum electric field

Substituting the given values in equation (3), we can the maximum electric field for the charge of particle 2, as follows:

$E = \frac{4 e}{4 {πε}_{0} (L + x)} - \frac{3 e}{4 {πε}_{0} {(x)}^{2}} = 9 \times 10^{9} N . m^{2} . C^{- 2} \times 1.6 \times 10^{- 19} C [(\frac{4}{{(0.34 m + 0.20 m)}^{2}}) - (\frac{3}{{(0.20 m)}^{2}})] = 2.2 \times 10^{- 8} N / C$

Hence, the value of the electric field is $2.2 \times 10^{- 8} N / C$

Find Study Materials for

Create Study Materials

Select your language

Fundamentals Of Physics

Answers without the blur.

Short Answer

Step by Step Solution

Step 1: The given data

Step 2: Understanding the concept of electric field

Step 3: a) Calculation of the value of Enetx for being maximum

Step 4: b) Calculation of that maximum electric field

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Select your language

Fundamentals Of Physics

Answers without the blur.

Short Answer

Step by Step Solution

Step 1: The given data

Step 2: Understanding the concept of electric field

Step 3: a) Calculation of the value of Enetx for being maximum

Step 4: b) Calculation of that maximum electric field

Most popular questions for Physics Textbooks

Want to see more solutions like these?

Recommended explanations on Physics Textbooks

Work Energy and Power

Radiation

Astrophysics

Physics of Motion

Scientific Method Physics

Fluids

Torque and Rotational Motion

Nuclear Physics

Fields in Physics

Measurements

Space Physics

Electricity

Physical Quantities and Units

Medical Physics

Electrostatics

Further Mechanics and Thermal Physics

Mechanics and Materials

Engineering Physics

Energy Physics

Force

Particle Model of Matter

Waves Physics

Magnetism

Modern Physics

Atoms and Radioactivity

Dynamics

Electricity and Magnetism

Conservation of Energy and Momentum

Fundamentals of Physics

Kinematics Physics

Circular Motion and Gravitation

Linear Momentum

Rotational Dynamics

Oscillations

Translational Dynamics

Geometrical and Physical Optics

Thermodynamics

Magnetism and Electromagnetic Induction

Electromagnetism

Electric Charge Field and Potential

Turning Points in Physics