Q.A molecule of a substance has a permanent electric dipole moment of magnitude 10−29C m. A mole of this substance is polarised (at low temperature) by applying a strong electrostatic field of magnitude 106V m−1. The direction of the field is suddenly changed by an angle of 60∘. Estimate the heat released by the substance in aligning its dipoles along the new direction of the field. For simplicity, assume 100% polarisation of the sample.
Dipole Alignment Energy — From Intuition to Formula
Imagine you have a tiny bar magnet — a compass needle. You know it always turns to point north. But what if you try to hold it pointing east? You feel a torque, a twisting force that wants to rotate it back. If you let go, it snaps to align with the field.
That "snap" releases energy. The energy that was stored in the misaligned configuration is called dipole alignment energy (or potential energy of a dipole in an external field).
The Core Intuition
A dipole (like a compass needle or a polar molecule) has two opposite "poles" — a north and a south, or a positive and a negative charge. When placed in an external field:
Aligned (parallel to the field): the dipole is in its lowest energy state — like a ball at the bottom of a valley.
Anti-aligned (opposite to the field): the dipole is in its highest energy state — like a ball balanced at the top of a hill.
Perpendicular: the energy is somewhere in between.
The energy depends on how much the dipole is twisted away from the field direction. The more you force it to point against the field, the more energy you store — like winding a spring.
The Precise Statement
For an electric dipole with dipole moment p placed in a uniform external electric field E, the potential energy of alignment is:
U=−p⋅E=−pEcosθ
where θ is the angle between p and E.
For a magnetic dipole (like a current loop or a compass needle) with magnetic moment μ in a magnetic field B:
U=−μ⋅B=−μBcosθ
Why the Negative Sign?
This is the part that confuses most students. Let's break it down.
When θ=0∘ (aligned), cosθ=1, so U=−pE. This is the minimum energy — the most stable configuration.
When θ=180∘ (anti-aligned), cosθ=−1, so U=+pE. This is the maximum energy — the least stable.
The negative sign is a convention that makes the aligned state the lowest energy. Think of it this way: the field does positive work to rotate the dipole from anti-aligned to aligned, so the dipole loses potential energy. The formula captures that loss as a negative value relative to the zero-energy reference (which is usually taken at θ=90∘, where U=0).
Watch out
A common mistake: thinking U=p⋅E (without the minus sign). That would make the aligned state highest energy — physically wrong. The dipole wants to align, so aligned must be lowest energy.
What It Physically Means
The alignment energy tells you:
How much work an external agent must do to rotate the dipole from aligned to some angle θ.
How stable the dipole is in a given orientation — the deeper the energy well (larger p or E), the harder to knock it out of alignment.
The torque on the dipole: τ=−dθdU=−pEsinθ, which matches the familiar τ=p×E.
A Quick Example
A water molecule has a permanent electric dipole moment p=6.2×10−30 C⋅m. In an electric field of 106 N/C (a strong laboratory field): …
Concept: Dipole Alignment Energy, U=−pEcosθ. The dipoles start aligned with the old field, i.e. at 60∘ to the new field, then relax to 0∘; the released energy appears as heat.
Heat per dipole =U60∘−U0∘=(−pEcos60∘)−(−pEcos0∘)=pE(1−21)=21pE.
The dipoles begin aligned with the old field (so 60∘ from the new one) and relax to alignment; the released energy is Q=21NApE=21×6.022×1023×10−29×106≈3.0J.
The physics
A permanent dipole in a field has potential energy U=−p⋅E=−pEcosθ, minimum (−pE) when aligned. When the field direction is suddenly turned by 60∘, the dipoles — still pointing the old way — are now at 60∘ to the new field. As they swing round to align with it, their potential energy drops, and that energy is dissipated as heat.
Watch out
The dipoles do not start aligned with the new field; they start 60∘ from it (their old alignment direction).
Method: Heat Released When a Field Reorients a Population of Dipoles
This method applies whenever a strong external field suddenly changes direction and a collection of permanent dipoles — already aligned with the old field direction — relaxes to align with the new one, releasing energy as heat.
Steps
Step 1: Identify the dipole's initial and final angle relative to the new field direction
The dipole is not initially aligned with the new field — it is still pointing along the old field direction, which now makes some angle θ with the new direction (equal to the angle through which the field was rotated). The final angle, once the dipole has settled, is 0∘ (fully aligned).
Step 2: Write the potential energy at each angle using U=−pEcosθ
Ui=−pEcosθ,Uf=−pEcos0∘=−pE
Step 3: The heat released per dipole is the drop in potential energy
Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.
CBSE 2020Set 55/1/11 markMCQ
Q.An electric dipole consisting of charges +q and −q separated by a distance L is in stable equilibrium in a uniform electric field E. The electrostatic potential energy of the dipole is (A) qLE (B) zero (C) −qLE (D) −2qEL
›Reveal solutionSolution
In stable equilibrium, the dipole moment p aligns with the electric field E, giving the minimum potential energy U=−p⋅E=−qLE. The correct option is (C).
Potential Energy of a Dipole — The Concept First
An electric dipole placed in a uniform field experiences a torque that tries to align its dipole moment p with the field E — much like a compass needle turning to align with a magnetic field. The stable equilibrium position is when the dipole points along the field direction.
The electrostatic potential energy of a dipole in a uniform field is not just any number — it depends on the angle between the dipole moment and the field. The formula is:
U=−p⋅E=−pEcosθ
where p=qL (from −q to +q), and θ is the angle between p and E.
Step-by-Step Reasoning
Identify the dipole moment.
The dipole moment vector p points from the negative charge to the positive charge, with magnitude p=qL. So p=qL.
Recall the potential energy formula.
For any dipole in a uniform electric field, the potential energy is U=−p⋅E. This is derived from the work done to rotate the dipole from a reference angle (usually θ=90∘, where U=0) to its current orientation.
Understand "stable equilibrium".
Stable equilibrium means the system returns to that position if slightly disturbed. For a dipole, this happens when the torque τ=p×E is zero and the energy is at a minimum.
Torque is zero when p is parallel or antiparallel to E.
The energy U=−pEcosθ is minimum when cosθ is maximum, i.e., cosθ=+1, which means θ=0∘ — the dipole points along the field.
Q.State the condition (in terms of the angle between the dipole moment p and the field E) at which the potential energy of the electric dipole placed in a uniform electric field is minimum.
›Reveal solutionSolution
U=−pEcosθ is most negative when cosθ is largest, i.e. θ=0.
Q.State the condition (in terms of the angle between the dipole moment p and the field E) at which the potential energy of the electric dipole placed in a uniform electric field is maximum.
›Reveal solutionSolution
U=−pEcosθ is largest when cosθ is most negative, i.e. θ=180∘.
Q.An electric dipole of dipole moment p is placed in a uniform electric field of strength E. If θ is the angle between the positive direction of p and E, then potential energy of the dipole is largest when θ is
(a) 0°
(b) 90°
(c) 180°
(d) 45°
›Reveal solutionSolution
U=−pEcosθ is largest when cosθ is most negative, i.e. at θ=180∘ (dipole anti-parallel to the field).