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 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610당신이 사용하는 게 아니라요?교과서 변경

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Young & Freedman Calc 14th Edition

Ch 21: Electric Charge and Electric Field

문제 53b

21장, 문제 53b

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?

검증된 단계별 안내

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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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

관련 실천

교과서 질문

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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교과서 질문

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.

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교과서 질문

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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교과서 질문

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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교과서 질문

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?

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교과서 질문

A charge of $negative 6.50$ nC is spread uniformly over the surface of one face of a nonconducting disk of radius $1.25$ cm. Why is the field in part (a) stronger than the field in part (b)? Why is the field in part (c) the strongest of the three fields? Note: Part (a) asked to find the magnitude and direction of the electric field this disk produces at a point $P$ on the axis of the disk a distance of $2.00$ cm from its center. Part (b) asked to find the magnitude and direction of the electric field at point $P$ , supposing that the charge were all pushed away from the center and distributed uniformly on the outer rim of the disk. Part (c) asked to find the magnitude and direction of the electric field at point $P$ if the charge is all brought to the center of the disk.

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