301. Study the table for a dipole in a uniform electric field and identify the row that needs correction.
| Row | Angle \(\theta\) between \(\vec{p}\) and \(\vec{E}\) | Torque magnitude |
|---|
| P | \(0^\circ\) | \(0\) |
| Q | \(90^\circ\) | \(pE\) |
| R | \(180^\circ\) | \(0\) |
| S | \(30^\circ\) | \(pE\) |
ⓐ. Row P
ⓑ. Row Q
ⓒ. Row R
ⓓ. Row S
Correct Answer: Row S
Explanation: The torque magnitude is \(\tau=pE\sin\theta\). For \(\theta=30^\circ\), \(\sin30^\circ=\frac{1}{2}\), so the torque should be \(\frac{pE}{2}\), not \(pE\). Row P is correct because \(\sin0^\circ=0\). Row Q is correct because \(\sin90^\circ=1\), giving maximum torque \(pE\). Row R is correct because \(\sin180^\circ=0\). The maximum torque occurs at right angles, not at every non-zero angle.
302. A dipole with \(\vec{p}\) initially making an acute angle with a uniform \(\vec{E}\) is released from rest. Its initial tendency is to rotate so that
ⓐ. \(\vec{p}\) becomes more nearly parallel to \(\vec{E}\)
ⓑ. \(\vec{p}\) becomes more nearly anti-parallel to \(\vec{E}\)
ⓒ. the separation between the charges becomes zero
ⓓ. both charges move with the same force direction
Correct Answer: \(\vec{p}\) becomes more nearly parallel to \(\vec{E}\)
Explanation: In a uniform electric field, the dipole experiences a torque \(\vec{\tau}=\vec{p}\times\vec{E}\). This torque tends to align the dipole moment with the electric field. Alignment means reducing the angle \(\theta\) between \(\vec{p}\) and \(\vec{E}\) toward \(0^\circ\). The charges do not need to collapse into each other for this rotation to occur. The forces on the two charges are opposite, not in the same direction. The rotational tendency is toward the lower-energy orientation in which \(\vec{p}\) is parallel to \(\vec{E}\).
303. Assertion: A dipole placed in a uniform electric field can have zero net force but non-zero torque.
Reason: The forces on \(+q\) and \(-q\) are equal and opposite, but they may act along different lines of action.
ⓐ. Assertion is true, but Reason is false
ⓑ. Both Assertion and Reason are true, but Reason does not explain Assertion
ⓒ. Both Assertion and Reason are true, and Reason explains Assertion
ⓓ. Assertion is false, but Reason is true
Correct Answer: Both Assertion and Reason are true, and Reason explains Assertion
Explanation: The Assertion is true for a dipole in a uniform electric field when it is not aligned with the field. The force on \(+q\) is along \(\vec{E}\), while the force on \(-q\) is opposite to \(\vec{E}\). These forces have equal magnitudes, so their vector sum is zero. The Reason is true because equal and opposite forces can still create a couple if their lines of action are different. That couple produces torque and rotational motion. When the dipole is exactly parallel or anti-parallel to the field, the torque also becomes zero.
304. A graph of torque magnitude \(\tau\) versus angle \(\theta\) for a dipole in a fixed uniform electric field is described as starting from zero at \(0^\circ\), reaching a maximum at \(90^\circ\), and returning to zero at \(180^\circ\). This graph matches
ⓐ. \(\tau=pE\theta^2\)
ⓑ. \(\tau=pE\cos\theta\)
ⓒ. \(\tau=\frac{pE}{\sin\theta}\)
ⓓ. \(\tau=pE\sin\theta\)
Correct Answer: \(\tau=pE\sin\theta\)
Explanation: The torque magnitude on a dipole in a uniform electric field is \(\tau=pE\sin\theta\). At \(\theta=0^\circ\), \(\sin0^\circ=0\), so the torque is zero. At \(\theta=90^\circ\), \(\sin90^\circ=1\), so the torque is maximum. At \(\theta=180^\circ\), \(\sin180^\circ=0\), so the torque returns to zero. A cosine relation would be maximum at \(0^\circ\), which does not match the described graph. The sine shape represents the turning effect of the component of the field perpendicular to \(\vec{p}\).
305. The potential energy of an electric dipole in a uniform electric field is
ⓐ. \(U=pE\sin\theta\)
ⓑ. \(U=\vec{p}\times\vec{E}\)
ⓒ. \(U=-\vec{p}\cdot\vec{E}\)
ⓓ. \(U=\frac{E}{p}\)
Correct Answer: \(U=-\vec{p}\cdot\vec{E}\)
Explanation: The potential energy of a dipole in a uniform electric field is \(U=-\vec{p}\cdot\vec{E}\). In scalar form, it is \(U=-pE\cos\theta\), where \(\theta\) is the angle between \(\vec{p}\) and \(\vec{E}\). The cross product \(\vec{p}\times\vec{E}\) gives torque, not potential energy. The expression \(pE\sin\theta\) gives torque magnitude, not energy. The negative sign in \(U=-pE\cos\theta\) shows that the aligned position has the lowest potential energy. Energy is minimum when \(\vec{p}\) points along \(\vec{E}\).
306. For a dipole in a uniform electric field, the potential energy is minimum when the angle between \(\vec{p}\) and \(\vec{E}\) is
ⓐ. \(90^\circ\)
ⓑ. \(0^\circ\)
ⓒ. \(180^\circ\)
ⓓ. \(270^\circ\)
Correct Answer: \(0^\circ\)
Explanation: The potential energy of a dipole is \(U=-pE\cos\theta\). At \(\theta=0^\circ\), \(\cos0^\circ=1\), so \(U=-pE\), the minimum value. At \(\theta=180^\circ\), \(\cos180^\circ=-1\), so \(U=+pE\), the maximum value. At \(\theta=90^\circ\), the potential energy is \(0\). The dipole naturally tends to rotate toward the minimum-energy orientation. This is the stable alignment where \(\vec{p}\) is parallel to \(\vec{E}\).
307. A dipole of moment \(p=5.0\times10^{-8}\,\text{C m}\) is placed in a uniform electric field \(E=2.0\times10^5\,\text{N C}^{-1}\). Its potential energy when \(\theta=60^\circ\) is
ⓐ. \(-1.0\times10^{-2}\,\text{J}\)
ⓑ. \(+5.0\times10^{-3}\,\text{J}\)
ⓒ. \(-5.0\times10^{-3}\,\text{J}\)
ⓓ. \(0\,\text{J}\)
Correct Answer: \(-5.0\times10^{-3}\,\text{J}\)
Explanation: \( \textbf{Given dipole moment:} \) \(p=5.0\times10^{-8}\,\text{C m}\).
\( \textbf{Electric field:} \) \(E=2.0\times10^5\,\text{N C}^{-1}\).
\( \textbf{Angle:} \) \(\theta=60^\circ\).
\( \textbf{Potential energy relation:} \)
\[
U=-pE\cos\theta
\]
\( \textbf{Product \(pE\):} \)
\[
pE=(5.0\times10^{-8})(2.0\times10^5)=10.0\times10^{-3}=1.0\times10^{-2}\,\text{J}
\]
\( \textbf{Use \(\cos60^\circ=\frac{1}{2}\):} \)
\[
U=-(1.0\times10^{-2})\left(\frac{1}{2}\right)\,\text{J}
\]
\( \textbf{Calculation:} \)
\[
U=-5.0\times10^{-3}\,\text{J}
\]
\( \textbf{Sign meaning:} \) The negative value shows that the dipole is closer to aligned than anti-aligned.
\( \textbf{Final answer:} \) The potential energy is \(-5.0\times10^{-3}\,\text{J}\).
308. Match the dipole orientation in a uniform electric field with the correct potential energy.
| Orientation | Potential energy |
|---|
| P. \(\theta=0^\circ\) | 1. \(+pE\) |
| Q. \(\theta=90^\circ\) | 2. \(-pE\) |
| R. \(\theta=180^\circ\) | 3. \(0\) |
ⓐ. P-1, Q-2, R-3
ⓑ. P-2, Q-3, R-1
ⓒ. P-3, Q-1, R-2
ⓓ. P-2, Q-1, R-3
Correct Answer: P-2, Q-3, R-1
Explanation: The potential energy is \(U=-pE\cos\theta\). For \(\theta=0^\circ\), \(\cos0^\circ=1\), so \(U=-pE\). For \(\theta=90^\circ\), \(\cos90^\circ=0\), so \(U=0\). For \(\theta=180^\circ\), \(\cos180^\circ=-1\), so \(U=+pE\). Therefore, the correct matching is P-2, Q-3, R-1. The aligned position is the lowest-energy state, while the anti-aligned position is the highest-energy state.
309. A graph of \(U\) versus \(\theta\) for a dipole in a uniform field is based on \(U=-pE\cos\theta\). The graph has a maximum at
ⓐ. \(\theta=0^\circ\)
ⓑ. \(\theta=60^\circ\)
ⓒ. \(\theta=180^\circ\)
ⓓ. \(\theta=90^\circ\)
Correct Answer: \(\theta=180^\circ\)
Explanation: The potential energy is \(U=-pE\cos\theta\). At \(\theta=180^\circ\), \(\cos180^\circ=-1\), so \(U=+pE\). This is the maximum value of \(U\). At \(\theta=0^\circ\), the energy is \(-pE\), which is the minimum value. At \(\theta=90^\circ\), the energy is \(0\). The energy graph therefore places the anti-parallel orientation at the top and the parallel orientation at the bottom.
310. A dipole initially aligned with a uniform electric field is rotated slowly to \(\theta=90^\circ\). If its dipole moment is \(p\) and the field is \(E\), the increase in potential energy is
ⓐ. \(\frac{pE}{2}\)
ⓑ. \(pE\)
ⓒ. \(2pE\)
ⓓ. \(0\)
Correct Answer: \(pE\)
Explanation: \( \textbf{Potential energy formula:} \)
\[
U=-pE\cos\theta
\]
\( \textbf{Initial angle:} \) \(\theta_i=0^\circ\).
\( \textbf{Initial energy:} \)
\[
U_i=-pE\cos0^\circ=-pE
\]
\( \textbf{Final angle:} \) \(\theta_f=90^\circ\).
\( \textbf{Final energy:} \)
\[
U_f=-pE\cos90^\circ=0
\]
\( \textbf{Change in potential energy:} \)
\[
\Delta U=U_f-U_i
\]
\( \textbf{Substitution:} \)
\[
\Delta U=0-(-pE)
\]
\( \textbf{Result:} \)
\[
\Delta U=pE
\]
\( \textbf{Final answer:} \) The potential energy increases by \(pE\).
311. The orientation \(\theta=0^\circ\) for a dipole in a uniform electric field is called stable equilibrium because
ⓐ. the net force becomes non-zero only there
ⓑ. the dipole has maximum potential energy there
ⓒ. the torque is maximum there
ⓓ. a small disturbance produces restoring torque
Correct Answer: a small disturbance produces restoring torque
Explanation: At \(\theta=0^\circ\), the dipole moment is parallel to the electric field. The torque is zero at that exact orientation because \(\tau=pE\sin0^\circ=0\). The potential energy is minimum there, since \(U=-pE\). If the dipole is slightly disturbed, the torque tends to bring it back toward alignment. This is the meaning of stable equilibrium. Maximum torque occurs at \(\theta=90^\circ\), not at the stable aligned position.
312. The orientation \(\theta=180^\circ\) for a dipole in a uniform electric field is unstable because
ⓐ. displacement drives it away from anti-alignment
ⓑ. the potential energy is minimum there
ⓒ. the torque first vanishes, then returns it to anti-alignment
ⓓ. the dipole has no charges in that orientation
Correct Answer: displacement drives it away from anti-alignment
Explanation: At \(\theta=180^\circ\), the dipole moment is anti-parallel to the electric field. The torque is zero exactly at this orientation because \(\sin180^\circ=0\). However, the potential energy is maximum, \(U=+pE\). A slight rotation away from this position produces a torque that tends to increase the departure from anti-alignment. That is why the position is unstable equilibrium. The dipole still has its two charges; only its orientation relative to \(\vec{E}\) has changed.
313. Study the statements about a dipole in a uniform electric field.
I. The net force on the dipole is zero.
II. The torque magnitude is \(pE\sin\theta\).
III. The potential energy is \(-pE\cos\theta\).
IV. The stable equilibrium occurs at \(\theta=180^\circ\).
The supported statements are
ⓐ. II and IV only
ⓑ. I, II and III
ⓒ. I and IV only
ⓓ. I, II, III and IV
Correct Answer: I, II and III
Explanation: Statement I is true because the forces on \(+q\) and \(-q\) in a uniform field are equal and opposite. Statement II is true because the torque magnitude is \(\tau=pE\sin\theta\). Statement III is true because the potential energy is \(U=-pE\cos\theta\). Statement IV is false because \(\theta=180^\circ\) is the maximum-energy anti-aligned orientation and is unstable. Stable equilibrium occurs at \(\theta=0^\circ\). The uniform field produces rotation tendency but not net translational force.
314. A dipole is placed in a non-uniform electric field. Compared with the uniform-field case, the dipole may experience a net force because
ⓐ. the two charges become equal in sign
ⓑ. the dipole moment becomes zero automatically
ⓒ. electric field stops being a vector
ⓓ. charges may feel unequal field magnitudes
Correct Answer: charges may feel unequal field magnitudes
Explanation: In a uniform electric field, the field magnitude is the same at both charges of a dipole. The forces on \(+q\) and \(-q\) are then equal in magnitude and opposite in direction, giving zero net force. In a non-uniform field, the two charges may lie in regions of different field strength. The force magnitudes on the two charges may no longer be equal. The dipole can then experience a net force as well as a torque. The dipole moment does not vanish, and the charges do not become equal in sign; the change is in the spatial variation of \(\vec{E}\).
315. A dipole is placed in a non-uniform electric field so that the positive charge lies in a slightly stronger field region than the negative charge. The dipole may translate because
ⓐ. the dipole moment becomes zero in a non-uniform field
ⓑ. the two charge forces need not be equal
ⓒ. the two charges become like charges
ⓓ. electric field stops exerting force on negative charge
Correct Answer: the two charge forces need not be equal
Explanation: In a uniform electric field, both charges of a dipole experience equal field magnitudes, so the two forces have equal magnitudes and opposite directions. In a non-uniform electric field, the field magnitude can be different at the locations of \(+q\) and \(-q\). Then the force magnitudes \(qE\) on the two charges need not be equal. Their vector sum can therefore be non-zero, producing translational motion. The dipole moment does not vanish just because the field is non-uniform. The charges remain equal and opposite; the new feature is that \(\vec{E}\) varies from point to point.
316. A dipole is placed in two different regions.
| Case | Field condition | Possible effect on dipole |
|---|
| P | Uniform electric field | Zero net force, possible torque |
| Q | Non-uniform electric field | Possible net force and possible torque |
| R | Uniform electric field and \(\vec{p}\parallel\vec{E}\) | Zero torque |
| S | Non-uniform electric field | Net force must always be zero |
The row that needs correction is
ⓐ. Row S
ⓑ. Row Q
ⓒ. Row R
ⓓ. Row P
Correct Answer: Row S
Explanation: Row S needs correction because a non-uniform electric field can exert a non-zero net force on a dipole. In a non-uniform field, the two charges of the dipole may experience different field magnitudes. Their forces may then fail to cancel completely. Row P is valid because a uniform field gives equal and opposite forces on the two charges, although a torque may remain. Row Q is valid because a non-uniform field can produce both translation and rotation. Row R is valid because \(\tau=pE\sin\theta\), and \(\theta=0^\circ\) gives zero torque.
317. Consider the following statements about a dipole in an external electric field.
I. In a uniform electric field, the net force on the dipole is zero.
II. In a non-uniform electric field, the dipole may experience a net force.
III. A dipole in a non-uniform field can still experience torque.
IV. A non-uniform field changes the two charges of a dipole into equal positive charges.
The supported statements are
ⓐ. I, II, III and IV
ⓑ. I and IV only
ⓒ. II and IV only
ⓓ. I, II and III
Correct Answer: I, II and III
Explanation: Statement I is valid because the forces on \(+q\) and \(-q\) in a uniform field are equal and opposite. Statement II is valid because non-uniformity can make the two force magnitudes unequal. Statement III is also valid because the two forces may still produce a turning effect depending on the dipole orientation. Statement IV is false because the charges of a dipole remain \(+q\) and \(-q\). Non-uniformity changes how the field varies in space, not the identity of the charges. The distinction is between field variation and charge conversion.
318. Assertion: A dipole placed in a non-uniform electric field may have a non-zero net force.
Reason: The electric field at the positions of \(+q\) and \(-q\) may have different magnitudes.
ⓐ. Both Assertion and Reason are true, and Reason explains Assertion
ⓑ. Both Assertion and Reason are true, but Reason does not explain Assertion
ⓒ. Assertion is true, but Reason is false
ⓓ. Assertion is false, but Reason is true
Correct Answer: Both Assertion and Reason are true, and Reason explains Assertion
Explanation: The Assertion is true because a non-uniform electric field can make the forces on the two charges unequal. The Reason is also true because the two charges of a dipole are separated in space. If \(\vec{E}\) changes from one point to another, \(+q\) and \(-q\) need not experience the same field magnitude. Unequal opposite-directed forces can leave a resultant force on the dipole. This explains why the dipole can translate in addition to rotating. The uniform-field result of zero net force is a special case, not a universal rule for all fields.
319. Electric flux through a surface gives a measure of
ⓐ. the mass of a charged particle
ⓑ. how much charge is stored inside every conductor
ⓒ. how much electric field passes through it
ⓓ. the speed of light in vacuum
Correct Answer: how much electric field passes through it
Explanation: Electric flux describes the amount of electric field passing through a surface in a directional sense. It depends on the electric field, the area of the surface, and the orientation of the surface. A large surface placed normal to a strong field has larger flux than the same surface placed edge-on. Flux is not the charge stored in a conductor, although Gauss’s law later connects closed-surface flux with enclosed charge. It is also unrelated to mass or the speed of light in electrostatics. The key idea is field-through-surface, with orientation included.
320. The area vector \(\vec{A}\) of a plane surface is drawn
ⓐ. normal to the surface with magnitude equal to the area
ⓑ. along the surface with magnitude equal to the perimeter
ⓒ. opposite to every electric field by definition
ⓓ. only when the surface is spherical
Correct Answer: normal to the surface with magnitude equal to the area
Explanation: For a plane surface, the area vector \(\vec{A}\) is perpendicular to the surface. Its magnitude is the area \(A\) of the surface. The direction of \(\vec{A}\) is a chosen normal direction for an open surface, while for a closed surface the outward normal is used. The area vector is not drawn along the surface plane. It is also not defined as always opposite to \(\vec{E}\). This normal direction is why the angle in flux is measured between \(\vec{E}\) and \(\vec{A}\), not between \(\vec{E}\) and the surface plane.
