301. A dipole is placed in a uniform electric field with \(\vec{p}\) antiparallel to \(\vec{E}\). Which statement is correct?
ⓐ. The torque is zero, but the orientation is unstable.
ⓑ. The torque is maximum, and the orientation is stable.
ⓒ. The torque is \(pE\), and the orientation is stable.
ⓓ. The torque is \(2pE\), and the orientation is unstable.
Correct Answer: The torque is zero, but the orientation is unstable.
Explanation: When \(\vec{p}\) is antiparallel to \(\vec{E}\), the angle between them is \(180^\circ\). The torque magnitude on a dipole is \(\tau=pE\sin\theta\). Since \(\sin180^\circ=0\), the torque is zero in the exact antiparallel position. However, this position is not stable. If the dipole is displaced slightly from \(180^\circ\), the torque tends to rotate it further away from the antiparallel position and toward alignment with the field. Thus, the antiparallel orientation is a zero-torque but unstable equilibrium position.
302. Which pair correctly describes a dipole in a uniform electric field when \(\vec{p}\) is neither parallel nor antiparallel to \(\vec{E}\)?
ⓐ. Net force is zero, but torque is non-zero.
ⓑ. Net force is non-zero, but torque is zero.
ⓒ. Both net force and torque are always zero.
ⓓ. Both charges experience forces in the same direction.
Correct Answer: Net force is zero, but torque is non-zero.
Explanation: In a uniform electric field, the field strength is the same at the positive and negative charges of the dipole. The force on \(+q\) is along \(\vec{E}\), while the force on \(-q\) is opposite to \(\vec{E}\). These forces have equal magnitudes and opposite directions, so the net translational force on the dipole is zero. When the dipole is not parallel or antiparallel to the field, the two forces act along different lines of action. Such a pair of forces forms a couple and produces torque. Therefore, the dipole can rotate even though it has no net force.
303. A dipole of moment \(p\) is aligned with a uniform electric field \(E\). Which pair gives the net force and torque on it?
ⓐ. Net force \(=pE\), torque \(=0\)
ⓑ. Net force \(=0\), torque \(=0\)
ⓒ. Net force \(=0\), torque \(=pE\)
ⓓ. Net force \(=2pE\), torque \(=pE\)
Correct Answer: Net force \(=0\), torque \(=0\)
Explanation: A uniform electric field exerts equal and opposite forces on the two charges of a dipole. Therefore, the vector sum of the two forces is zero. When the dipole moment is aligned with the field, the angle between \(\vec{p}\) and \(\vec{E}\) is \(0^\circ\). The torque magnitude is \(\tau=pE\sin\theta\). Substituting \(\theta=0^\circ\) gives \(\tau=0\). Hence, an aligned dipole in a uniform field has zero net force and zero torque.
304. A dipole is placed in a uniform electric field directed toward the right. Which force description is correct?
ⓐ. The force on \(+q\) is toward the right, and the force on \(-q\) is toward the left.
ⓑ. The force on \(+q\) is toward the left, and the force on \(-q\) is toward the right.
ⓒ. Both \(+q\) and \(-q\) experience force toward the right.
ⓓ. Both \(+q\) and \(-q\) experience zero force.
Correct Answer: The force on \(+q\) is toward the right, and the force on \(-q\) is toward the left.
Explanation: The force on a charge in an electric field is given by \(\vec{F}=q\vec{E}\). A positive charge experiences force in the same direction as the electric field. Since the field is toward the right, the force on \(+q\) is toward the right. A negative charge experiences force opposite to the electric field. Therefore, the force on \(-q\) is toward the left. These equal and opposite forces explain why a dipole in a uniform field has zero net force but can still experience torque if the forces act along different lines.
305. For a dipole of moment \(p\) in a uniform electric field \(E\), at which angle is the torque magnitude equal to \(\frac{pE}{2}\) for an acute angle?
ⓐ. \(15^\circ\)
ⓑ. \(30^\circ\)
ⓒ. \(45^\circ\)
ⓓ. \(60^\circ\)
Correct Answer: \(30^\circ\)
Explanation: \( \textbf{Torque formula:} \) The torque magnitude on a dipole in a uniform electric field is \(\tau=pE\sin\theta\).
\( \textbf{Given condition:} \) \(\tau=\frac{pE}{2}\).
\( \textbf{Substitution:} \) \(pE\sin\theta=\frac{pE}{2}\).
\( \textbf{Cancel common factor:} \) For non-zero \(p\) and \(E\), \(\sin\theta=\frac{1}{2}\).
\( \textbf{Acute-angle value:} \) \(\sin30^\circ=\frac{1}{2}\).
\( \textbf{Final result:} \) The acute angle is \(30^\circ\).
306. A dipole has moment \(2.0\times10^{-8}\,\text{C m}\) in a uniform field \(5.0\times10^4\,\text{N C}^{-1}\). What is the maximum torque on it?
ⓐ. \(1.0\times10^{-3}\,\text{N m}\)
ⓑ. \(2.0\times10^{-3}\,\text{N m}\)
ⓒ. \(5.0\times10^{-4}\,\text{N m}\)
ⓓ. \(0\,\text{N m}\)
Correct Answer: \(1.0\times10^{-3}\,\text{N m}\)
Explanation: \( \textbf{Given values:} \) \(p=2.0\times10^{-8}\,\text{C m}\) and \(E=5.0\times10^4\,\text{N C}^{-1}\).
\( \textbf{Torque relation:} \) \(\tau=pE\sin\theta\).
\( \textbf{Maximum condition:} \) Torque is maximum when \(\sin\theta=1\), which occurs at \(\theta=90^\circ\).
\( \textbf{Maximum torque:} \) \(\tau_{\max}=pE\).
\( \textbf{Substitution:} \) \(\tau_{\max}=(2.0\times10^{-8})(5.0\times10^4)\,\text{N m}\).
\( \textbf{Simplification:} \) \(\tau_{\max}=10.0\times10^{-4}\,\text{N m}=1.0\times10^{-3}\,\text{N m}\).
\( \textbf{Final result:} \) The maximum torque is \(1.0\times10^{-3}\,\text{N m}\).
307. A dipole in a uniform electric field experiences no net force but may experience torque. Why?
ⓐ. The forces on \(+q\) and \(-q\) are unequal and opposite.
ⓑ. Equal and opposite forces act at different points.
ⓒ. The force on \(+q\) is zero, while the force on \(-q\) is non-zero.
ⓓ. The field acts only on the dipole moment and not on charges.
Correct Answer: Equal and opposite forces act at different points.
Explanation: In a uniform electric field, the positive and negative charges of a dipole experience forces of equal magnitude because the field strength is the same at both charges. The directions are opposite because the charges have opposite signs. Therefore, the net translational force on the dipole is zero. However, the two forces act at different points of the dipole. Equal and opposite forces with different lines of action can form a couple. This couple produces torque and tends to rotate the dipole. Hence, a uniform electric field can rotate a dipole without giving it a net force.
308. Which orientation of a dipole in a uniform electric field is stable?
ⓐ. \(\vec{p}\) perpendicular to \(\vec{E}\)
ⓑ. \(\vec{p}\) opposite to \(\vec{E}\)
ⓒ. \(\vec{p}\) parallel to \(\vec{E}\)
ⓓ. \(\vec{p}\) at \(120^\circ\) to \(\vec{E}\)
Correct Answer: \(\vec{p}\) parallel to \(\vec{E}\)
Explanation: A dipole in a uniform electric field experiences torque \(\tau=pE\sin\theta\). At \(\theta=0^\circ\), \(\vec{p}\) is parallel to \(\vec{E}\), so the torque is zero. If the dipole is displaced slightly from this aligned position, the field torque tends to reduce the angle and bring the dipole back toward alignment. That restoring tendency makes the parallel orientation stable. At \(\theta=180^\circ\), the torque is also zero, but a small displacement makes the dipole rotate away from that position. Therefore, the stable orientation is \(\vec{p}\) parallel to \(\vec{E}\).
309. For a dipole in a uniform electric field, at which orientations is the torque zero?
ⓐ. Only \(\theta=90^\circ\)
ⓑ. Only \(\theta=45^\circ\)
ⓒ. \(\theta=0^\circ\) and \(\theta=180^\circ\)
ⓓ. \(\theta=60^\circ\) and \(\theta=120^\circ\)
Correct Answer: \(\theta=0^\circ\) and \(\theta=180^\circ\)
Explanation: \( \textbf{Torque relation:} \) The torque magnitude on a dipole is \(\tau=pE\sin\theta\).
\( \textbf{Zero-torque condition:} \) For non-zero \(p\) and \(E\), torque becomes zero when \(\sin\theta=0\).
\( \textbf{Relevant angles:} \) In the range from \(0^\circ\) to \(180^\circ\), this occurs at \(0^\circ\) and \(180^\circ\).
\( \textbf{Physical meaning:} \) These are the aligned and antiparallel orientations.
\( \textbf{Comparison:} \) At \(90^\circ\), the torque is maximum, not zero.
\( \textbf{Final result:} \) The torque is zero at \(\theta=0^\circ\) and \(\theta=180^\circ\).
310. A dipole of moment \(3.0\times10^{-8}\,\text{C m}\) is placed in a uniform electric field \(4.0\times10^5\,\text{N C}^{-1}\). What is the maximum torque on it?
ⓐ. \(1.2\times10^{-2}\,\text{N m}\)
ⓑ. \(7.5\times10^{-14}\,\text{N m}\)
ⓒ. \(3.0\times10^{-8}\,\text{N m}\)
ⓓ. \(4.0\times10^5\,\text{N m}\)
Correct Answer: \(1.2\times10^{-2}\,\text{N m}\)
Explanation: \( \textbf{Given values:} \) \(p=3.0\times10^{-8}\,\text{C m}\) and \(E=4.0\times10^5\,\text{N C}^{-1}\).
\( \textbf{Torque formula:} \) \(\tau=pE\sin\theta\).
\( \textbf{Maximum condition:} \) Torque is maximum when \(\sin\theta=1\), which occurs at \(\theta=90^\circ\).
\( \textbf{Maximum torque:} \) \(\tau_{\max}=pE\).
\( \textbf{Substitution:} \) \(\tau_{\max}=(3.0\times10^{-8})(4.0\times10^5)\,\text{N m}\).
\( \textbf{Coefficient calculation:} \) \(3.0\times4.0=12.0\).
\( \textbf{Power calculation:} \) \(10^{-8}\times10^5=10^{-3}\).
\( \textbf{Final simplification:} \) \(\tau_{\max}=12.0\times10^{-3}\,\text{N m}=1.2\times10^{-2}\,\text{N m}\).
\( \textbf{Final result:} \) The maximum torque is \(1.2\times10^{-2}\,\text{N m}\).
311. Which pair correctly identifies stable and unstable equilibrium of a dipole in a uniform electric field?
ⓐ. Stable at \(90^\circ\), unstable at \(0^\circ\)
ⓑ. Stable at \(0^\circ\), unstable at \(180^\circ\)
ⓒ. Stable at \(180^\circ\), unstable at \(0^\circ\)
ⓓ. Stable at \(270^\circ\), unstable at \(90^\circ\)
Correct Answer: Stable at \(0^\circ\), unstable at \(180^\circ\)
Explanation: The angle \(\theta\) is measured between \(\vec{p}\) and \(\vec{E}\). At \(\theta=0^\circ\), the dipole is aligned with the field and the torque is zero. A small displacement from this position produces a restoring torque that brings the dipole back toward alignment, so it is stable. At \(\theta=180^\circ\), the dipole is antiparallel to the field and the torque is also zero at the exact position. However, a small displacement produces torque that carries the dipole farther away from the antiparallel position. Therefore, \(0^\circ\) is stable and \(180^\circ\) is unstable.
312. A dipole in a uniform electric field is given a small angular displacement from the aligned position. What does the field torque tend to do?
ⓐ. Increase the displacement away from alignment
ⓑ. Keep the dipole fixed at the displaced angle
ⓒ. Bring the dipole back toward alignment
ⓓ. Make the net charge of the dipole non-zero
Correct Answer: Bring the dipole back toward alignment
Explanation: In the aligned position, \(\vec{p}\) is parallel to \(\vec{E}\), so \(\theta=0^\circ\). The torque magnitude is \(\tau=pE\sin\theta\), so the torque is zero at exact alignment. If the dipole is displaced slightly, a torque appears. The direction of this torque is such that it tends to reduce the angle between \(\vec{p}\) and \(\vec{E}\). This brings the dipole back toward the aligned position. The torque changes the orientation of the dipole, not its net charge. Hence, the aligned position is stable.
313. A dipole has moment \(5.0\times10^{-8}\,\text{C m}\) and is placed in a uniform field \(1.2\times10^5\,\text{N C}^{-1}\). What is the torque magnitude when \(\theta=60^\circ\)? Use \(\sin60^\circ\approx0.866\).
ⓐ. \(6.0\times10^{-3}\,\text{N m}\)
ⓑ. \(5.2\times10^{-3}\,\text{N m}\)
ⓒ. \(3.0\times10^{-3}\,\text{N m}\)
ⓓ. \(1.2\times10^{-2}\,\text{N m}\)
Correct Answer: \(5.2\times10^{-3}\,\text{N m}\)
Explanation: \( \textbf{Given values:} \) \(p=5.0\times10^{-8}\,\text{C m}\), \(E=1.2\times10^5\,\text{N C}^{-1}\), and \(\theta=60^\circ\).
\( \textbf{Torque formula:} \) \(\tau=pE\sin\theta\).
\( \textbf{Product of \(p\) and \(E\):} \) \(pE=(5.0\times10^{-8})(1.2\times10^5)=6.0\times10^{-3}\,\text{N m}\).
\( \textbf{Angle factor:} \) \(\sin60^\circ\approx0.866\).
\( \textbf{Substitution:} \) \(\tau=(6.0\times10^{-3})(0.866)\,\text{N m}\).
\( \textbf{Final calculation:} \) \(\tau\approx5.2\times10^{-3}\,\text{N m}\).
\( \textbf{Final result:} \) The torque magnitude is approximately \(5.2\times10^{-3}\,\text{N m}\).
314. A dipole is placed in a non-uniform electric field. Which effect can occur that does not occur for the same dipole in a uniform electric field?
ⓐ. The dipole moment becomes zero automatically.
ⓑ. The two charges stop experiencing electric force.
ⓒ. The net charge of the dipole becomes non-zero.
ⓓ. The dipole may experience a net force.
Correct Answer: The dipole may experience a net force.
Explanation: In a uniform electric field, the positive and negative charges of a dipole experience equal and opposite forces. These forces give zero net translational force but can produce torque because they act at different points. In a non-uniform electric field, the field strength at \(+q\) and \(-q\) may be different. Then the forces on the two charges need not have equal magnitudes. As a result, their vector sum may not be zero. The dipole can therefore experience a net force in addition to torque. The net charge of the dipole remains zero because the charges are still \(+q\) and \(-q\).
315. A dipole of moment \(p\) is placed perpendicular to a uniform electric field \(E\). Which pair gives its net force and torque magnitude?
ⓐ. Net force \(=0\), \(\tau=pE\)
ⓑ. Net force \(=pE\), \(\tau=0\)
ⓒ. Net force \(=2pE\), \(\tau=pE\)
ⓓ. Net force \(=0\), \(\tau=0\)
Correct Answer: Net force \(=0\), \(\tau=pE\)
Explanation: \( \textbf{Uniform-field force:} \) The forces on \(+q\) and \(-q\) in a uniform field are equal in magnitude and opposite in direction.
\( \textbf{Net force:} \) Their vector sum is therefore zero.
\( \textbf{Perpendicular condition:} \) Perpendicular means \(\theta=90^\circ\).
\( \textbf{Torque formula:} \) \(\tau=pE\sin\theta\).
\( \textbf{Substitution:} \) \(\tau=pE\sin90^\circ=pE\).
\( \textbf{Final result:} \) The net force is zero and the torque magnitude is \(pE\).
316. Which statement correctly compares a dipole in a uniform electric field with a dipole in a non-uniform electric field?
ⓐ. In both fields, the dipole must have zero torque.
ⓑ. In a uniform field the net force is zero, while in a non-uniform field a net force may occur.
ⓒ. In a uniform field the dipole moment becomes zero, while in a non-uniform field it remains non-zero.
ⓓ. In a non-uniform field the charges of the dipole must become equal in sign.
Correct Answer: In a uniform field the net force is zero, while in a non-uniform field a net force may occur.
Explanation: In a uniform electric field, the two charges of a dipole experience equal and opposite forces. This makes the net translational force zero, although a torque may still act if the dipole is not aligned with the field. In a non-uniform electric field, the field strength at the two charges can be different. The two forces may then have unequal magnitudes, so their vector sum need not be zero. The dipole may experience a net force as well as torque. The charges of the dipole remain equal in magnitude and opposite in sign in both cases. Therefore, the key difference is the possibility of net force in a non-uniform field.
317. A dipole initially makes an angle \(\theta\) with a uniform electric field, where \(0^\circ<\theta<180^\circ\). What is the general tendency of the torque due to the field?
ⓐ. It tends to align \(\vec{p}\) with \(\vec{E}\).
ⓑ. It tends to make \(\vec{p}\) permanently perpendicular to \(\vec{E}\).
ⓒ. It tends to reverse both charges of the dipole.
ⓓ. It tends to make the net force non-zero in the uniform field.
Correct Answer: It tends to align \(\vec{p}\) with \(\vec{E}\).
Explanation: A dipole in a uniform electric field experiences torque \(\vec{\tau}=\vec{p}\times\vec{E}\). This torque acts only on the orientation of the dipole. For a freely rotating dipole at a general angle, the rotational effect tends to bring the dipole moment toward the direction of the field. The aligned position \(\theta=0^\circ\) is stable, while the antiparallel position \(\theta=180^\circ\) is unstable. The torque does not reverse the charges or change the net charge of the dipole. In a uniform field, the net translational force remains zero. Therefore, the field torque tends to align \(\vec{p}\) with \(\vec{E}\).
318. What is electric flux through a surface most directly associated with?
ⓐ. Amount of mass enclosed by the surface
ⓑ. Amount of charge crossing per unit time
ⓒ. Amount of work done in moving charge
ⓓ. Field passing through the surface
Correct Answer: Field passing through the surface
Explanation: Electric flux is a measure of how much electric field passes through a given surface. In field-line language, it is associated with the number of electric field lines crossing the surface. If more field lines pass normally through the surface, the flux is larger. If the surface is parallel to the field lines, few or no lines cross it, so the flux is smaller or zero. Flux depends on electric field, surface area, and orientation. It is not a measure of mass or time rate of charge transfer. Therefore, electric flux is most directly linked with field lines crossing a surface.
319. For a uniform electric field \(\vec{E}\) and a plane area vector \(\vec{A}\), which expression gives electric flux?
ⓐ. \(\Phi_E=EA\sin\theta\)
ⓑ. \(\Phi_E=\frac{E}{A}\cos\theta\)
ⓒ. \(\Phi_E=\vec{E}\cdot\vec{A}\)
ⓓ. \(\Phi_E=\vec{E}\times\vec{A}\)
Correct Answer: \(\Phi_E=\vec{E}\cdot\vec{A}\)
Explanation: \( \textbf{Flux idea:} \) Electric flux measures the effective electric field passing through a surface.
\( \textbf{Area vector:} \) The vector \(\vec{A}\) has magnitude equal to area and direction normal to the surface.
\( \textbf{Dot product form:} \) For a uniform field and a plane surface, \(\Phi_E=\vec{E}\cdot\vec{A}\).
\( \textbf{Magnitude form:} \) This becomes \(\Phi_E=EA\cos\theta\), where \(\theta\) is the angle between \(\vec{E}\) and \(\vec{A}\).
\( \textbf{Why dot product:} \) Flux is a scalar and depends on the component of field normal to the surface.
\( \textbf{Final result:} \) The electric flux is \(\Phi_E=\vec{E}\cdot\vec{A}\).
320. A uniform electric field \(E=500\,\text{N C}^{-1}\) is perpendicular to a plane surface of area \(0.20\,\text{m}^2\). What is the electric flux through the surface?
ⓐ. \(100\,\text{N m}^2\text{C}^{-1}\)
ⓑ. \(2500\,\text{N m}^2\text{C}^{-1}\)
ⓒ. \(0.40\,\text{N m}^2\text{C}^{-1}\)
ⓓ. \(500\,\text{N m}^2\text{C}^{-1}\)
Correct Answer: \(100\,\text{N m}^2\text{C}^{-1}\)
Explanation: \( \textbf{Given field:} \) \(E=500\,\text{N C}^{-1}\).
\( \textbf{Given area:} \) \(A=0.20\,\text{m}^2\).
\( \textbf{Orientation:} \) The field is perpendicular to the surface, so it is parallel to the area vector.
\( \textbf{Angle with area vector:} \) \(\theta=0^\circ\).
\( \textbf{Flux formula:} \) \(\Phi_E=EA\cos\theta\).
\( \textbf{Substitution:} \) \(\Phi_E=(500)(0.20)\cos0^\circ\,\text{N m}^2\text{C}^{-1}\).
\( \textbf{Simplification:} \) \(\Phi_E=100\,\text{N m}^2\text{C}^{-1}\).
\( \textbf{Final result:} \) The electric flux is \(100\,\text{N m}^2\text{C}^{-1}\).
