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Potential and potential energy share a name, a symbol and units that differ by one coulomb, so students mix them up. Dipoles add a second trap: torque uses sin θ and energy uses cos θ. This post separates the ideas, works through a dipole with real numbers, and gives you a routine to make the distinctions automatic.
What does the syllabus ask for?
The JEE (Main) 2026 syllabus lists electric potential for a point charge, a dipole and a system of charges, equipotential surfaces, the potential energy of two point charges and of a dipole in an electrostatic field, and the torque on a dipole in a uniform field. The JEE (Advanced) 2026 brochure lists electric potential and the potential energy of point charges and dipoles in a uniform field. NCERT covers torque in Class 12 Chapter 1, section 1.11, and potential, equipotentials and energy in Chapter 2, where section 2.8.3 derives U = −p·E. Check the bulletin for your year.
Why are these ideas hard even after a full course?
The Conceptual Survey of Electricity and Magnetism, reported by Maloney and colleagues (2001), was given to more than 5000 introductory students at 30 institutions. Students in calculus-based courses averaged 31% before instruction and only 47% after; algebra-based courses went from 25% to 44%. Lindsey (2014) found that the same wrong reasoning patterns about potential energy appeared in electrostatics and in gravitation, both after instruction. Potential energy belongs to a pair of interacting charges, not to one charge alone; that idea is often missing.
Potential versus potential energy: a quick test
Potential V is a property of a point in space: work per unit charge to bring a test charge there from infinity. Units: volts.
Potential energy U belongs to a system of charges. For two point charges, U = kq₁q₂/r.
Example: +3 μC and −2 μC sit 0.30 m apart. U = (9 × 10⁹)(3 × 10⁻⁶)(−2 × 10⁻⁶)/0.30 = −0.054/0.30 = −0.18 J. The negative sign means you must supply +0.18 J to pull them apart to infinity.
Worked example: a dipole in a uniform field
Charges of ±2.0 μC are 2.0 cm apart, so the dipole moment is p = q × 2a = 2.0 × 10⁻⁶ × 0.020 = 4.0 × 10⁻⁸ C m. It sits in a uniform field E = 1.0 × 10⁵ N/C at θ = 30°.
Torque: τ = pE sin θ = 4.0 × 10⁻⁸ × 1.0 × 10⁵ × 0.5 = 2.0 × 10⁻³ N m.
Potential energy: U = −pE cos θ = −4.0 × 10⁻³ × 0.866 ≈ −3.46 × 10⁻³ J.
Work to turn it from θ = 0° to 180°: W = U(180°) − U(0°) = pE − (−pE) = 2pE = 8.0 × 10⁻³ J.
Equilibrium: at 0°, U is lowest, so it is stable. At 180°, torque is also zero but U is highest, so it is unstable.
Potential of the dipole. On the axis, far away, V ≈ kp/r². At r = 0.20 m, V ≈ (9 × 10⁹ × 4.0 × 10⁻⁸)/0.04 = 9000 V. The exact sum kq[1/(r − a) − 1/(r + a)] with a = 0.010 m gives about 9020 V, so the approximation is good when r is much larger than a. On the perpendicular bisector, V = 0 at every point, even though the field there is not zero.
What are equipotential surfaces telling you?
An equipotential surface joins points of equal potential. NCERT section 2.6 shows the field is always perpendicular to it, which is why moving a charge along it takes no work. Around a point charge they are spheres; in a uniform field they are planes perpendicular to the field. Closely spaced equipotentials mean a strong field, because E = −dV/dr. If V = 4x² volts along x, then E = −8x V/m, so at x = 0.5 m the field is 4 V/m pointing in the −x direction.
Common mistakes and how to avoid them
"V = 0 so E = 0." On a dipole's bisector V is zero but E is not.
Dropping signs in kq₁q₂/r. Keep the charges' signs; a negative U means a bound pair.
Using r instead of r² for dipole potential. Dipole potential falls as 1/r²; the field falls as 1/r³.
Treating p as q. The dipole moment is charge times separation, pointing from − to +.
For sign and superposition errors in the same chapter, see our post on common electrostatics mistakes. Capacitors come next in the syllabus; our capacitors and dielectrics guide builds on these ideas.
A practice routine
Write V, U, E and τ for one dipole at five angles (0°, 30°, 90°, 150°, 180°) in a table on paper.
Solve four two-charge and three-charge energy problems, keeping every sign.
Sketch equipotentials for a point charge, a dipole and a uniform field, and add field arrows at right angles.
Field and potential are often easier to see than to read about. Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read.
Start with the five-angle table tonight; it takes ten minutes and fixes the sine-cosine mix-up.
