Electric Potential

• Electric potential V as work per unit charge bringing test charge from infinity.
• Relation E = −dV/dr (potential gradient) — field points from high to low potential.
• Potential due to point charge V = kQ/r; superposition for multiple charges.
• Equipotential surfaces — perpendicular to field lines; no work moving charge on equipotential.
• Potential energy of charge system U = qV and dipole in field; connection to conservation of energy.

Level 1 — Foundations

Verbal: Electric potential at a point is the work done per unit positive charge in bringing a test charge slowly from infinity to that point against electrostatic forces.

Scalar definition: V = W/q₀ (scalar — easier than vector field).

Point charge: V = (1/4πε₀)Q/r. Sign of Q matters (+Q gives positive V).

Reference: V(∞) = 0 convention.

Relation to field: ΔV = −∫ E·dl along path; for uniform field E = −ΔV/d.

Units: volt (V) = J/C.

Level 2 — JEE / NEET depth

Equipotential surfaces: V constant on surface; E⃗ perpendicular to surface; moving charge on surface does zero work.

Potential energy: U = qV for charge q at potential V. Two charges: U = k q₁q₂/r.

Dipole potential: V ∝ p cos θ / r² on axis/equator expansion.

Capacitor link (preview): Potential difference between plates ΔV drives charge storage Q = CV.

Conservative nature: Electrostatic field is conservative; work around closed loop zero; potential independent of path.

Graph skills: Slope of V–r graph gives −E_r. Uniform field → linear V variation.

JEE problems: Find V at centroid of charge triangle; energy to assemble charge configuration; electron accelerated through ΔV gains eΔV kinetic energy.

Potential due to point charge

Q = −5 μC. Find V at r = 0.1 m.

Step 1 — V = kQ/r = (9×10⁹)(−5×10⁻⁶)/0.1.
Step 2 — V = −45×10³/0.1 = −4.5×10⁵ V.
Step 3 — Negative potential near negative charge.
Step 4 — Electron (q = −e) has higher potential energy where V is more negative if U = qV.

Work to move charge between potentials

Move +2 μC from A (V_A = 100 V) to B (V_B = 40 V).

Step 1 — ΔV = V_B − V_A = −60 V.
Step 2 — Work by external agent W_ext = qΔV = (2×10⁻⁶)(−60) = −1.2×10⁻⁴ J.
Step 3 — Negative W_ext: field does positive work; charge moves toward lower potential naturally.
Step 4 — ΔU = qΔV = −1.2×10⁻⁴ J potential energy decrease.

• Define 1 volt.
• Relation between E and V for uniform field?
• Potential at midpoint between +Q and −Q (equal magnitude)?
• Energy gained by electron through 100 V potential difference?
• Stretch: Derive V = kQ/r from work integral definition.

NCERT Chapter 2 link: Potential is scalar — add algebraically without angle headaches. Equipotential surfaces perpendicular to field lines — draw both together in revision diagrams.

Exam connections: Work-energy problems: electron accelerated through V volts gains eV joules kinetic energy. Relation E = −dV/dr for uniform field between plates. Potential of dipole and system assembly energy appear in NCERT Exercises 2.11–2.15.

Study strategy: Potential adds with signs; field requires vectors. Zero potential reference at infinity — state convention. For conductors in equilibrium, entire surface is equipotential — explains why field inside is zero.

When studying Electric Potential within Electrostatics, start by listing every formula and definition on one page without looking at the textbook. Compare your list to NCERT — missing items indicate gaps to fix immediately. Work through at least two NCERT Examples for this section with steps written in full; examiners award method marks even when arithmetic slips.

For board exams (CBSE), long answers benefit from a clear structure: definition → explanation → diagram or formula → example → brief conclusion. Underline key terms. For JEE Main and NEET, prioritise conceptual traps and quick calculation paths; timed mixed quizzes of 10 questions after revision simulate exam pressure.

Cross-topic link: Calculus-based derivations assume differentiation comfort; units and dimensional analysis prevent numerical errors.

Spaced revision: Review this note at 1 day, 3 days, and 7 days after first study. Attempt the Quick check questions closed-book, then open the Practice tab for graded reinforcement. Maintain an error log — repeated mistake patterns reveal whether the issue is concept, formula recall, or careless reading.

Diagram and terminology drill: For Physics, redraw key figures from memory and define every labelled part in one sentence. Vocabulary precision prevents mark loss in descriptive answers — use NCERT terms exactly as printed in the textbook.

Revision tip: Link this topic to adjacent Class 12 chapters before attempting mixed practice.

Open the Practice tab for graded questions on Electric Potential.