Point charges q1=−4.5q_1=-4.5 nC and q2=+4.5q_2=+4.5 nC are separated by 3.13.1 mm, forming an electric dipole. The charges are in a uniform electric field whose direction makes an angle of 36.936.9° with the line connecting the charges. What is the magnitude of this field if the torque exerted on the dipole has magnitude 7.2×10−97.2\$$times10^{-9} Nm$$?

Ch 21: Electric Charge and Electric Field

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 21: Electric Charge and Electric Field

Problem 51b

Chapter 21, Problem 51b

Point charges $q_{1} = - 4.5$ nC and $q sub 2 equals positive 4.5$ nC are separated by $3.1$ mm, forming an electric dipole. The charges are in a uniform electric field whose direction makes an angle of $36.9$ ° with the line connecting the charges. What is the magnitude of this field if the torque exerted on the dipole has magnitude $7.2 \times 10^{- 9}$ Nm?

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Understand that the torque $ \tau $ on an electric dipole in a uniform electric field $ E $ is given by the formula $ \tau = pE \sin(\theta) $, where $ p $ is the dipole moment and $ \theta $ is the angle between the dipole moment and the electric field.

Calculate the dipole moment $ p $ using the formula $ p = qd $, where $ q $ is the magnitude of one of the charges and $ d $ is the separation distance between the charges. Here, $ q = 4.5 \times 10^{-9} $ C and $ d = 3.1 \times 10^{-3} $ m.

Substitute the known values into the dipole moment formula: $ p = (4.5 \times 10^{-9} \text{ C})(3.1 \times 10^{-3} \text{ m}) $.

Rearrange the torque formula to solve for the electric field $ E $: $ E = \frac{\tau}{p \sin(\theta)} $.

Substitute the known values into the rearranged formula: $ E = \frac{7.2 \times 10^{-9} \text{ N}\cdot\text{m}}{p \sin(36.9^\circ)} $, using the previously calculated dipole moment $ p $ and the angle $ \theta = 36.9^\circ $.

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Key Concepts

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

Electric Dipole

An electric dipole consists of two equal and opposite point charges separated by a distance. It is characterized by its dipole moment, which is the product of the charge magnitude and the separation distance. Dipoles interact with electric fields, experiencing forces and torques depending on the field's orientation relative to the dipole.

Torque on a Dipole

Torque on an electric dipole in a uniform electric field is given by the cross product of the dipole moment and the electric field vector. The magnitude of the torque is calculated as τ = pE sin(θ), where p is the dipole moment, E is the electric field strength, and θ is the angle between the dipole moment and the field direction.

Uniform Electric Field

A uniform electric field has constant magnitude and direction throughout a region. In such a field, the force on a charge is constant, and the torque on a dipole depends on the angle between the dipole moment and the field. Understanding how dipoles behave in uniform fields is crucial for calculating forces and torques in electrostatic problems.

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Textbook Question

An electric dipole with dipole moment $p$ is in a uniform external electric field $E$ . Show that for the stable orientation in part (b), the dipole's own electric field tends to oppose the external field. Note: Part (b) asked which of the orientations in part (a) is stable, and which is unstable? (Hint: Consider a small rotation away from the equilibrium position and see what happens.) Also, part (a) asked to find the orientations of the dipole for which the torque on the dipole is zero.

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Textbook Question

A point charge $q_{1} = - 4.00$ nC is at the point $x equals 0.600$ m, $y equals 0.800$ m, and a second point charge $q sub 2 equals positive 6.00$ nC is at the point $x equals 0.600$ m, $y equals 0$ . Calculate the magnitude and direction of the net electric field at the origin due to these two point charges.

Textbook Question

A point charge is placed at each corner of a square with side length $a$ . All charges have magnitude $q$ . Two of the charges are positive and two are negative (Fig. E $21.42$ ). What is the direction of the net electric field at the center of the square due to the four charges, and what is its magnitude in terms of $q$ and $a$ ?

Textbook Question

An electric dipole with dipole moment $p$ is in a uniform external electric field $E$ . Find the orientations of the dipole for which the torque on the dipole is zero.

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Textbook Question

An electric dipole with dipole moment $p$ is in a uniform external electric field $E$ . Which of the orientations in part (a) is stable, and which is unstable? (Hint: Consider a small rotation away from the equilibrium position and see what happens.) Note: Part (a) asked to find the orientations of the dipole for which the torque on the dipole is zero.

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Textbook Question

A very long, straight wire has charge per unit length $3.20 \times 10^{- 10}$ C/m. At what distance from the wire is the electric field magnitude equal to $2.50$ N/C?