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Intermediate

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Electrostatic Potential & Potential Energy

Master Electrostatic Potential and Potential Energy for JEE Main Advanced NEET and CBSE Class 12 Physics. Covers definition of electric potential work done per unit charge, V=kQ/r for point charge scalar superposition, potential due to dipole kp cosθ/r² axial equatorial equatorial always zero, potential energy of two charges kQ1Q2/r, system of three charges sum of pairs, work done W=q(Va-Vb), relation E=-dV/dr field as negative gradient of potential, equipotential surfaces properties E always perpendicular no work done, V inside shell equals surface value, with fully verified practice questions.

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When you lift a book against gravity, you store potential energy in it. Similarly, when you move a charge against an electric force, you store electrostatic potential energy. The Electric Potential at a point is the potential energy per unit positive charge at that point — a scalar quantity that is often much easier to work with than the vector electric field. The relationship between E and V (E = −dV/dr) links these two descriptions of the same physical reality. For JEE and NEET, potential calculations for point charges, dipoles, and systems of charges are tested every year, as are equipotential surfaces and the work-energy relationship.

1. Electric Potential — Definition

The Electric Potential V at a point is defined as the work done per unit positive test charge in bringing it from infinity to that point, without acceleration (quasi-statically).

$V = \frac{W_{\infty \to P}}{q_{0}} \Rightarrow W = q_{0} V$

SI unit: Volt (V) = J/C | Scalar quantity | Reference: V = 0 at infinity

Potential due to a Point Charge

$V = \frac{k Q}{r} = \frac{1}{4 π ε_{0}} \frac{Q}{r}$

V > 0 for positive charge; V < 0 for negative charge. V falls as 1/r (slower than E which falls as 1/r²).

Superposition Principle for Potential

Total potential = algebraic sum (scalar sum, not vector sum) of potentials due to individual charges:

$V_{t o t a l} = V_{1} + V_{2} + V_{3} + \dots = \sum \frac{k Q_{i}}{r_{i}}$

2. Potential due to a Dipole

$V = \frac{k p \cos θ}{r^{2}} (r ≫ a)$

Where θ is the angle from the dipole axis.

Position	θ	V
Axial (same side as +q)	0°	+kp/r²
Axial (same side as −q)	180°	−kp/r²
Equatorial (perp. bisector)	90°	0 (always, at all distances)

Example: p = 4 nC·m, r = 20 cm, θ = 60°: V = kp cosθ/r² = 9×10⁹ × 4×10⁻⁹ × 0.5 / (0.2)² = 450 V

3. Potential Energy of a System of Charges

Two Charges

$U = \frac{k Q_{1} Q_{2}}{r}$

U > 0: like charges (repulsive, must do work to bring together). U < 0: unlike charges (attractive, system releases energy).

System of Three Charges

$U = k (\frac{Q_{1} Q_{2}}{r_{12}} + \frac{Q_{2} Q_{3}}{r_{23}} + \frac{Q_{1} Q_{3}}{r_{13}})$

Sum over all distinct pairs — each pair counted once.

Work Done in Moving a Charge

$W = q (V_{A} - V_{B}) = q \cdot Δ V$

Moving from B to A. Work done by external agent (against electric force). If Va > Vb and q > 0: positive work must be done.

4. Relationship Between E and V

$E = - \frac{d V}{d r} \vec{E} = - \nabla V$

Electric field points in the direction of decreasing potential (from high V to low V). E is the negative gradient of V.

For uniform field: E = V/d (V = potential difference, d = distance between plates).

5. Equipotential Surfaces

An equipotential surface is a surface on which all points are at the same electric potential. Key properties:

E is always perpendicular to an equipotential surface — no work is done moving a charge along the surface.
Equipotential surfaces never cross each other.
For a point charge: equipotential surfaces are concentric spheres.
For a uniform field: equipotential surfaces are parallel planes ⊥ to E.
For a dipole: equipotential surfaces are more complex; V = 0 on the equatorial plane.
Closer equipotential surfaces → stronger E field (larger potential gradient).

Potential vs Field — Key Comparisons

Property	Electric Field E	Electric Potential V
Nature	Vector	Scalar
Point charge	kQ/r² (falls as 1/r²)	kQ/r (falls as 1/r)
Inside shell	E = 0	V = kQ/R (constant, non-zero)
Superposition	Vector sum	Algebraic (scalar) sum
Relation	E = −dV/dr; V = −∫E·dr

JEE/NEET Power Tips

V inside a charged shell = kQ/R (constant) even though E = 0. E = 0 means no change in V, so V is uniform inside — and equal to the value on the surface. A very common source of error: students say V = 0 inside the shell.
V = 0 at equatorial plane of a dipole — at ALL distances. Since cosθ = 0 at θ = 90°, V = kp cos90°/r² = 0 always on the equatorial plane regardless of r.
E = 0 does NOT imply V = 0. Example: inside a charged shell, E = 0 but V = kQ/R. Conversely, V = 0 does NOT mean E = 0 — at the midpoint between equal and opposite charges, V = 0 but E ≠ 0.
W = q(Va − Vb), not q(Vb − Va). Work done by external agent moving charge q from B to A is q(Va − Vb). If Va > Vb and q > 0, W > 0 (work done against repulsion). Memorise the sign convention carefully.

Practice Questions

Q1 (JEE Main): Three charges Q, −2Q and Q are placed at x = 0, x = d/2, x = d respectively. Find the potential at the midpoint x = d/4.

V = k[Q/(d/4) + (−2Q)/(d/4) + Q/(3d/4)]

= k[4Q/d − 8Q/d + 4Q/(3d)] = k[−4Q/d + 4Q/(3d)]

= k × (−12Q + 4Q)/(3d) = −8kQ/(3d)

Q2 (NEET): The work done in moving a charge of 3 μC from a point at 100 V to a point at 40 V is:

W = q(Va − Vb) = 3×10⁻⁶ × (100 − 40) = 3×10⁻⁶ × 60 = 1.8 × 10⁻⁴ J

Since Va > Vb and q > 0, positive work is done by the external agent (moving against the electric force).

Q3 (Board): What is an equipotential surface? Why is the electric field always perpendicular to an equipotential surface?

An equipotential surface is a surface where the electric potential V is the same at every point, so that no work is done in moving a charge along it (W = qΔV = q × 0 = 0).

E ⊥ equipotential surface: If E had a component along the surface, moving a charge along the surface would require work (W = qEd cos0° ≠ 0). But by definition, W = 0 along an equipotential. Therefore, E can have no component along the surface — it must be entirely perpendicular to it.

Q4 (MCQ): The potential inside a uniformly charged hollow sphere (shell) is:

A) Zero B) Equal to the potential at the surface C) Greater than at the surface D) Less than at the surface

Answer: B) Equal to the potential at the surface. E = 0 inside the shell means dV/dr = 0 → V is constant inside, equal to V at the surface = kQ/R.

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Electrostatic Potential & Potential Energy

1. Electric Potential — Definition

Potential due to a Point Charge

Superposition Principle for Potential

2. Potential due to a Dipole

3. Potential Energy of a System of Charges

Two Charges

System of Three Charges

Work Done in Moving a Charge

4. Relationship Between E and V

5. Equipotential Surfaces

Potential vs Field — Key Comparisons

Practice Questions

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