Question
State the properties of equipotential surfaces and explain why the electric field is always perpendicular to them. Also identify the shape of equipotential surfaces around a point charge.
Solution — Step by Step
An equipotential surface is a surface where every point is at the same electric potential. No matter which path you take between two points on this surface, the potential difference is zero — so throughout.
Work done in moving a charge between two points is . Since on an equipotential surface, for any displacement along it. This is the defining property — zero work done, always.
The relation between electric field and potential is:
Along the equipotential surface, , so the component of along the surface must be zero. This forces to be entirely in the direction normal (perpendicular) to the surface. If had any tangential component, it would do work on a charge moving along the surface — contradicting our result in Step 2.
For a point charge at the origin, the potential at distance is:
Since depends only on (not on direction), all points at the same distance have the same potential. These surfaces are concentric spheres centred on the charge. Closer spheres have higher potential (for positive charge).
- No two equipotential surfaces can intersect (two different potentials at one point is a contradiction)
- They are always denser (closer together) where is stronger
- The field lines are always perpendicular to them
- Answer: Spherical, concentric surfaces for a point charge; the electric field is perpendicular to every equipotential surface.
Why This Works
The perpendicularity rule is essentially a statement about energy. Moving a charge along an equipotential requires zero work from the electric field, which means the field exerts no force component in that direction. Force and field are parallel, so the field itself has no component along the surface — it must point normal to it.
This is why field lines and equipotentials are always drawn as a perpendicular grid. In JEE Main and CBSE boards, one standard question asks you to sketch both together for common charge configurations — a skill worth practising once for each: point charge, uniform field, electric dipole.
For a uniform electric field , the equipotentials are flat planes perpendicular to the x-axis. For a dipole, they’re more complex — but the perpendicularity rule still holds everywhere.
Alternative Method
Using the gradient definition directly:
The electric field is the negative gradient of potential:
The gradient of any scalar field points in the direction of maximum rate of change. On an equipotential surface, is constant, so its rate of change along the surface is zero — the gradient has no component along the surface. Therefore is entirely perpendicular to the surface. Same conclusion, more compact reasoning — useful to write in JEE Advanced proofs.
Common Mistake
Students often think equipotential means zero electric field. Wrong. on the surface, but changes as you move perpendicular to it — and that change is what gives a non-zero . The field is zero only at specific points (like the midpoint between equal and opposite charges), not across the whole surface. Mixing up (along surface) with (everywhere) costs marks in CBSE short-answer questions.
In NCERT Class 12 Chapter 2, the standard examples to memorise are: point charge → spheres, uniform field → parallel planes, dipole → distorted ovals. CBSE boards have asked “draw the equipotential surfaces for X” at least once every two years. Sketch them with field lines crossing at 90° and you’ll get full marks.