201. Charges \(+2\,\text{C}\) and \(-3\,\text{C}\) are placed at two points where the external potentials are \(10\,\text{V}\) and \(4\,\text{V}\), respectively. The energy of these charges in the external potential is:
ⓐ. \(-32\,\text{J}\)
ⓑ. \(+8\,\text{J}\)
ⓒ. \(-8\,\text{J}\)
ⓓ. \(+32\,\text{J}\)
Correct Answer: \(+8\,\text{J}\)
Explanation: \( \textbf{Given data:} \) \(q_1=+2\,\text{C}\), \(V_1=10\,\text{V}\), \(q_2=-3\,\text{C}\), and \(V_2=4\,\text{V}\).
\( \textbf{Required quantity:} \) Energy in the external potential.
\( \textbf{External-potential energy relation:} \)
\[
U=q_1V_1+q_2V_2
\]
\( \textbf{First contribution:} \)
\[
q_1V_1=(+2\,\text{C})(10\,\text{V})=20\,\text{J}
\]
\( \textbf{Second contribution:} \)
\[
q_2V_2=(-3\,\text{C})(4\,\text{V})=-12\,\text{J}
\]
\( \textbf{Total energy:} \)
\[
U=20\,\text{J}-12\,\text{J}=8\,\text{J}
\]
\( \textbf{Final answer:} \) The energy in the external potential is \(+8\,\text{J}\).
The signs of the charges must be used directly in the products \(qV\).
202. In using \(U=q_1V(\vec{r}_1)+q_2V(\vec{r}_2)+\cdots\) for charges in an external field, the potential \(V\) should be:
ⓐ. Produced only by the same charge whose energy is being calculated
ⓑ. Chosen equal to zero at every charge position
ⓒ. Produced by sources outside the charge system
ⓓ. Treated as a vector component along the force
Correct Answer: Produced by sources outside the charge system
Explanation: The expression \(U=q_1V(\vec{r}_1)+q_2V(\vec{r}_2)+\cdots\) is used for charges placed in a pre-existing external potential. The external potential is produced by sources that are not included as the charges whose energy is being summed in that expression. If the potential of a charge on itself were included, the energy calculation would become physically incorrect. Mutual energy among the placed charges is handled separately by pairwise terms. Potential is still a scalar, so no vector component of \(V\) is needed. The word “external” tells which sources are responsible for the potential values used.
203. Three statements are made about potential energy in electrostatics.
I. For a single charge in an external potential, \(U=qV\).
II. For two point charges, their mutual energy is \(U=\frac{kq_1q_2}{r}\).
III. For many point charges, every distinct pair contribution is counted twice.
ⓐ. I, II, and III
ⓑ. I and II only
ⓒ. I and III only
ⓓ. II and III only
Correct Answer: I and II only
Explanation: Statement I is correct because a charge \(q\) placed at a point of external potential \(V\) has energy \(U=qV\). Statement II is correct for the mutual energy of two point charges separated by \(r\). Statement III is false because each distinct pair contribution is counted once. Counting each pair twice would double the total mutual potential energy. The reason for counting once is that the pair interaction is a single scalar energy contribution. The correct method depends on separating external-potential energy from mutual pair energy.
204. A point charge \(+q\) is placed in a region where the external potential varies as \(V=V_0-\alpha x\), with \(\alpha\gt0\). As the charge moves in the \(+x\)-direction, its potential energy:
ⓐ. Remains constant
ⓑ. Changes as \(x^2\)
ⓒ. Increases linearly
ⓓ. Decreases linearly
Correct Answer: Decreases linearly
Explanation: The potential energy of the charge is \(U=qV\). Substituting the given potential gives \(U=q(V_0-\alpha x)\). Since \(q\gt0\) and \(\alpha\gt0\), the term \(-q\alpha x\) makes \(U\) decrease as \(x\) increases. The variation is linear because \(V\) is linear in \(x\). This is also consistent with the fact that potential decreases along \(+x\). A positive charge loses potential energy when it moves toward lower potential.
205. The potential energy of an electric dipole \(\vec{p}\) in a uniform electric field \(\vec{E}\) is:
ⓐ. \(U=\vec{p}\times\vec{E}\)
ⓑ. \(U=pE\sin\theta\)
ⓒ. \(U=-\vec{p}\cdot\vec{E}\)
ⓓ. \(U=\frac{p}{E}\)
Correct Answer: \(U=-\vec{p}\cdot\vec{E}\)
Explanation: The potential energy of an electric dipole in a uniform electric field is \(U=-\vec{p}\cdot\vec{E}\). In magnitude form, this becomes \(U=-pE\cos\theta\), where \(\theta\) is the angle between \(\vec{p}\) and \(\vec{E}\). The cross product \(\vec{p}\times\vec{E}\) is related to torque, not potential energy. The negative sign shows that the lowest energy occurs when the dipole moment aligns with the field. The expression depends on orientation, not only on the magnitudes \(p\) and \(E\). A dipole in a uniform field has orientation energy even though the net force may be zero.
206. A dipole in a uniform electric field has \(\theta=0^\circ\), where \(\theta\) is the angle between \(\vec{p}\) and \(\vec{E}\). Its potential energy is:
ⓐ. \(0\)
ⓑ. \(\frac{pE}{2}\)
ⓒ. \(-pE\)
ⓓ. \(+pE\)
Correct Answer: \(-pE\)
Explanation: The dipole potential energy is \(U=-pE\cos\theta\). At \(\theta=0^\circ\), \(\cos0^\circ=1\). Therefore, \(U=-pE\). This is the minimum energy orientation of the dipole in a uniform field. In this orientation, the dipole moment is aligned with the electric field. The \(\theta=180^\circ\) orientation would give \(+pE\), which is the maximum energy orientation. The sign is decided by the dot product and the negative sign in the energy expression.
207. The row that correctly describes the orientation of a dipole in a uniform electric field is:
| Row | Angle \(\theta\) | Potential energy \(U=-pE\cos\theta\) | Nature |
|---|
| P | \(0^\circ\) | \(-pE\) | Stable equilibrium |
| Q | \(90^\circ\) | \(-pE\) | Stable equilibrium |
| R | \(180^\circ\) | \(-pE\) | Stable equilibrium |
| S | \(0^\circ\) | \(+pE\) | Unstable equilibrium |
ⓐ. Row Q
ⓑ. Row P
ⓒ. Row R
ⓓ. Row S
Correct Answer: Row P
Explanation: At \(\theta=0^\circ\), the dipole moment is aligned with the electric field. The energy is \(U=-pE\cos0^\circ=-pE\), which is the lowest possible value. This orientation is stable because a small angular displacement produces a torque tending to restore the dipole toward alignment. At \(\theta=90^\circ\), \(U=0\), not \(-pE\). At \(\theta=180^\circ\), \(U=+pE\), and the equilibrium is unstable. The energy minimum identifies the stable orientation.
208. A dipole with moment \(p=5.0\times10^{-8}\,\text{C m}\) is placed in a uniform electric field \(E=2.0\times10^5\,\text{N C}^{-1}\). If \(\theta=60^\circ\), its potential energy is:
ⓐ. \(+5.0\times10^{-3}\,\text{J}\)
ⓑ. \(-1.0\times10^{-2}\,\text{J}\)
ⓒ. \(+1.0\times10^{-2}\,\text{J}\)
ⓓ. \(-5.0\times10^{-3}\,\text{J}\)
Correct Answer: \(-5.0\times10^{-3}\,\text{J}\)
Explanation: \( \textbf{Given data:} \) \(p=5.0\times10^{-8}\,\text{C m}\), \(E=2.0\times10^5\,\text{N C}^{-1}\), and \(\theta=60^\circ\).
\( \textbf{Required quantity:} \) Dipole potential energy \(U\).
\( \textbf{Formula:} \)
\[
U=-pE\cos\theta
\]
\( \textbf{Angular value:} \)
\[
\cos60^\circ=\frac{1}{2}
\]
\( \textbf{Product} \) \(pE\):
\[
pE=(5.0\times10^{-8})(2.0\times10^5)=1.0\times10^{-2}\,\text{J}
\]
\( \textbf{Apply angular factor and sign:} \)
\[
U=-(1.0\times10^{-2})\left(\frac{1}{2}\right)
\]
\( \textbf{Calculation:} \)
\[
U=-5.0\times10^{-3}\,\text{J}
\]
\( \textbf{Final answer:} \) The potential energy is \(-5.0\times10^{-3}\,\text{J}\).
The negative sign appears because the angle is less than \(90^\circ\), so \(\cos\theta\) is positive.
209. A graph of \(U\) against \(\theta\) for a dipole in a uniform field follows \(U=-pE\cos\theta\). The graph has:
ⓐ. Minimum at \(0^\circ\) and maximum at \(180^\circ\)
ⓑ. Zero value at \(0^\circ\) and \(180^\circ\)
ⓒ. Maximum at \(0^\circ\) and minimum at \(180^\circ\)
ⓓ. Same value for every angle
Correct Answer: Minimum at \(0^\circ\) and maximum at \(180^\circ\)
Explanation: The energy is \(U=-pE\cos\theta\). At \(\theta=0^\circ\), \(\cos\theta=1\), so \(U=-pE\), the minimum value. At \(\theta=180^\circ\), \(\cos\theta=-1\), so \(U=+pE\), the maximum value. At \(\theta=90^\circ\), \(U=0\). The graph is not constant because the dipole energy depends on orientation. The stable position corresponds to the energy minimum, not the maximum.
210. The work required by an external agent to rotate a dipole slowly from \(\theta=0^\circ\) to \(\theta=180^\circ\) in a uniform field is:
ⓐ. \(2pE\)
ⓑ. \(0\)
ⓒ. \(pE\)
ⓓ. \(-2pE\)
Correct Answer: \(2pE\)
Explanation: \( \textbf{Initial orientation:} \) \(\theta_i=0^\circ\).
\[
U_i=-pE\cos0^\circ=-pE
\]
\( \textbf{Final orientation:} \) \(\theta_f=180^\circ\).
\[
U_f=-pE\cos180^\circ=+pE
\]
\( \textbf{Slow rotation condition:} \) External work equals change in potential energy.
\[
W_{\text{ext}}=\Delta U=U_f-U_i
\]
\( \textbf{Substitution:} \)
\[
W_{\text{ext}}=pE-(-pE)
\]
\( \textbf{Result:} \)
\[
W_{\text{ext}}=2pE
\]
\( \textbf{Final answer:} \) The required external work is \(2pE\).
The rotation takes the dipole from its lowest-energy orientation to its highest-energy orientation.
211. Assertion: A dipole placed at \(\theta=180^\circ\) in a uniform electric field is in unstable equilibrium.
Reason: At \(\theta=180^\circ\), \(U=-pE\cos\theta\) has its maximum value.
ⓐ. Assertion is true, but Reason is false
ⓑ. Both Assertion and Reason are true, but Reason does not explain Assertion
ⓒ. Assertion is false, but Reason is true
ⓓ. Both Assertion and Reason are true, and Reason explains Assertion
Correct Answer: Both Assertion and Reason are true, and Reason explains Assertion
Explanation: At \(\theta=180^\circ\), the dipole moment is opposite to the electric field. The energy is \(U=-pE\cos180^\circ=+pE\), which is the maximum value of the dipole’s potential energy. A small displacement from this orientation lowers the energy, so the dipole does not tend to return to \(\theta=180^\circ\). That is the meaning of unstable equilibrium here. The Reason correctly explains the Assertion through the energy maximum. Stable equilibrium corresponds instead to the energy minimum at \(\theta=0^\circ\).
212. The torque on a dipole in a uniform electric field has magnitude:
ⓐ. \(\tau=pE\sin\theta\)
ⓑ. \(\tau=-pE\cos\theta\)
ⓒ. \(\tau=pE\cos\theta\)
ⓓ. \(\tau=\frac{p}{E}\sin\theta\)
Correct Answer: \(\tau=pE\sin\theta\)
Explanation: The torque on an electric dipole in a uniform electric field is \(\vec{\tau}=\vec{p}\times\vec{E}\). Its magnitude is therefore \(\tau=pE\sin\theta\). This torque tends to rotate the dipole toward the lower-energy alignment with the field. The expression \(U=-pE\cos\theta\) is for potential energy, not torque. The torque is zero at \(\theta=0^\circ\) and \(\theta=180^\circ\), even though the energies at those two orientations are different. The sine dependence belongs to rotational tendency, while the cosine dependence belongs to orientation energy.
213. A dipole in a uniform electric field has zero torque at \(\theta=0^\circ\) and \(\theta=180^\circ\). These two cases differ because:
ⓐ. \(\theta=0^\circ\) is stable equilibrium, while \(\theta=180^\circ\) is unstable equilibrium
ⓑ. Both are stable because torque is zero in both cases
ⓒ. \(\theta=0^\circ\) is unstable equilibrium, while \(\theta=180^\circ\) is stable equilibrium
ⓓ. Both are unstable because torque is zero in both cases
Correct Answer: \(\theta=0^\circ\) is stable equilibrium, while \(\theta=180^\circ\) is unstable equilibrium
Explanation: The torque magnitude on a dipole is \(\tau=pE\sin\theta\), so it is zero at both \(\theta=0^\circ\) and \(\theta=180^\circ\). Zero torque alone does not decide whether equilibrium is stable or unstable. The potential energy is \(U=-pE\cos\theta\). At \(\theta=0^\circ\), \(U=-pE\), the minimum value, so a small displacement tends to be restored. At \(\theta=180^\circ\), \(U=+pE\), the maximum value, so a small displacement lowers the energy and the dipole tends to move away from that orientation. Stability is decided by the energy behaviour near the equilibrium angle, not only by \(\tau=0\).
214. A dipole is slowly rotated in a uniform electric field from \(\theta=90^\circ\) to \(\theta=0^\circ\). The work done by the external agent is:
ⓐ. \(+pE\)
ⓑ. \(-pE\)
ⓒ. \(0\)
ⓓ. \(+2pE\)
Correct Answer: \(-pE\)
Explanation: \( \textbf{Potential-energy relation:} \)
\[
U=-pE\cos\theta
\]
\( \textbf{Initial angle:} \) \(\theta_i=90^\circ\).
\[
U_i=-pE\cos90^\circ=0
\]
\( \textbf{Final angle:} \) \(\theta_f=0^\circ\).
\[
U_f=-pE\cos0^\circ=-pE
\]
\( \textbf{Slow rotation condition:} \)
\[
W_{\text{ext}}=\Delta U=U_f-U_i
\]
\( \textbf{Substitution:} \)
\[
W_{\text{ext}}=(-pE)-0
\]
\( \textbf{Result:} \)
\[
W_{\text{ext}}=-pE
\]
\( \textbf{Final answer:} \) The external work is \(-pE\).
The negative value means the field helps the dipole rotate toward its lower-energy aligned orientation.
215. A dipole of moment \(3.0\times10^{-8}\,\text{C m}\) is placed in a uniform electric field of \(4.0\times10^5\,\text{N C}^{-1}\). At \(\theta=30^\circ\), the torque magnitude is nearest to:
ⓐ. \(3.0\times10^{-3}\,\text{N m}\)
ⓑ. \(2.4\times10^{-2}\,\text{N m}\)
ⓒ. \(6.0\times10^{-3}\,\text{N m}\)
ⓓ. \(1.2\times10^{-2}\,\text{N m}\)
Correct Answer: \(6.0\times10^{-3}\,\text{N m}\)
Explanation: \( \textbf{Given data:} \) \(p=3.0\times10^{-8}\,\text{C m}\), \(E=4.0\times10^5\,\text{N C}^{-1}\), and \(\theta=30^\circ\).
\( \textbf{Required quantity:} \) Torque magnitude \(\tau\).
\( \textbf{Torque formula:} \)
\[
\tau=pE\sin\theta
\]
\( \textbf{Angular value:} \)
\[
\sin30^\circ=\frac{1}{2}
\]
\( \textbf{Product} \) \(pE\):
\[
pE=(3.0\times10^{-8})(4.0\times10^5)=12.0\times10^{-3}\,\text{N m}
\]
\( \textbf{Apply angular factor:} \)
\[
\tau=(12.0\times10^{-3})\left(\frac{1}{2}\right)
\]
\( \textbf{Calculation:} \)
\[
\tau=6.0\times10^{-3}\,\text{N m}
\]
\( \textbf{Final answer:} \) The torque magnitude is \(6.0\times10^{-3}\,\text{N m}\).
The sine factor is used for torque, while the cosine factor belongs to potential energy.
216. The row that correctly compares torque and potential energy of a dipole in a uniform electric field is:
| Row | Quantity | Expression | Zero at |
|---|
| P | Torque magnitude | \(\tau=pE\sin\theta\) | \(\theta=0^\circ,180^\circ\) |
| Q | Torque magnitude | \(\tau=pE\cos\theta\) | \(\theta=0^\circ,180^\circ\) |
| R | Potential energy | \(U=-pE\sin\theta\) | \(\theta=0^\circ,180^\circ\) |
| S | Potential energy | \(U=pE\cos\theta\) | \(\theta=90^\circ\) |
ⓐ. Row Q
ⓑ. Row R
ⓒ. Row P
ⓓ. Row S
Correct Answer: Row P
Explanation: The torque on a dipole is given by \(\vec{\tau}=\vec{p}\times\vec{E}\), so its magnitude is \(\tau=pE\sin\theta\). This torque is zero when \(\theta=0^\circ\) and \(\theta=180^\circ\), because \(\sin0^\circ=\sin180^\circ=0\). Potential energy is \(U=-pE\cos\theta\), not a sine expression. The zero value of \(U\) occurs at \(\theta=90^\circ\), but that alone is not the torque formula. The negative sign in energy is essential for identifying the stable orientation. Torque and energy are related, but their angle dependences are not the same.
217. A dipole initially at \(\theta=60^\circ\) is allowed to rotate freely in a uniform electric field. Ignoring energy losses, it tends first to rotate toward:
ⓐ. \(\theta=90^\circ\), because torque is maximum there
ⓑ. No direction, because the net force on the dipole is zero
ⓒ. \(\theta=180^\circ\), because this gives maximum potential energy
ⓓ. \(\theta=0^\circ\), because this lowers \(U=-pE\cos\theta\)
Correct Answer: \(\theta=0^\circ\), because this lowers \(U=-pE\cos\theta\)
Explanation: A dipole in a uniform electric field experiences no net force, but it can experience a torque. The torque tends to align \(\vec{p}\) with \(\vec{E}\). The potential energy \(U=-pE\cos\theta\) decreases as \(\theta\) moves from \(60^\circ\) toward \(0^\circ\). The orientation \(\theta=0^\circ\) is the minimum-energy stable orientation. Although the torque magnitude is maximum at \(\theta=90^\circ\), that angle is not the final stable orientation. Zero net force does not mean zero rotational tendency.
218. Assertion: The torque on a dipole in a uniform electric field is maximum at \(\theta=90^\circ\).
Reason: At \(\theta=90^\circ\), the potential energy of the dipole is minimum.
ⓐ. Assertion is true, but Reason is false
ⓑ. Both Assertion and Reason are true, but Reason does not explain Assertion
ⓒ. Assertion is false, but Reason is true
ⓓ. Both Assertion and Reason are true, and Reason explains Assertion
Correct Answer: Assertion is true, but Reason is false
Explanation: The torque magnitude is \(\tau=pE\sin\theta\), so it is maximum when \(\sin\theta=1\), which occurs at \(\theta=90^\circ\). The Assertion is therefore true. The potential energy is \(U=-pE\cos\theta\). At \(\theta=90^\circ\), \(U=0\), not the minimum value. The minimum value is \(U=-pE\) at \(\theta=0^\circ\). The angle of maximum torque is not the same as the angle of minimum energy.
219. Inside a conductor in electrostatic equilibrium, the electric field is:
ⓐ. Equal to the surface charge density
ⓑ. Zero
ⓒ. Directed from lower potential to higher potential
ⓓ. Maximum at the centre
Correct Answer: Zero
Explanation: In electrostatic equilibrium, free charges inside a conductor are at rest. If an electric field existed inside the conducting material, the free charges would experience force and start moving. That would contradict the equilibrium condition. Therefore, the electric field inside the conductor must be zero. This result applies to the conducting material, not necessarily to the region outside the conductor. The surface may still carry charge, and the field just outside the surface can be non-zero.
220. A conductor is in electrostatic equilibrium. The potential throughout its volume and surface is:
ⓐ. Different at every point because charges are present
ⓑ. Increasing linearly from centre to surface
ⓒ. Zero only at the surface
ⓓ. Constant
Correct Answer: Constant
Explanation: In electrostatic equilibrium, the electric field inside the conducting material is zero. Since \(\vec{E}=-\vec{\nabla}V\), zero electric field means there is no spatial variation of potential inside the conductor. Therefore, the whole conductor is at one constant potential. This does not require the potential value to be \(0\,\text{V}\); it depends on the chosen reference and charge state. The surface of the conductor is also an equipotential surface. Charges may be present on the surface, but they do not make different points of the conductor remain at different potentials in equilibrium.
