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Potential Energy in an External Field - Potential Energy of a Single Charge

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Estimated time: 8 minutes
CBSE: Class 12

Introduction

In earlier sections, the source of the electric field was known — the charges and their locations were specified. Now the focus shifts to a charge q placed in an externally created electric field.

Key distinction:

  • The external field E and potential V are created by other source charges (not by q itself)
  • Charge q is assumed to be small enough that it does not disturb the original field
  • The source charges are fixed (not affected by q)
CBSE: Class 1

Definition: Potential Energy of a Single Charge

The potential energy of a charge q placed at a point with external electric potential V(r) is equal to the work done in bringing the charge from infinity to that point against the external field.

CBSE: Class 12

Derivation (Step-by-Step)

Step Statement Reasoning
1 Place an external potential V in space Other charges create the field; it is not disturbed by q.
2 Potential at infinity = 0 Standard reference convention
3 Work done to bring a unit positive charge from ∞ to (P = V(P)) Definition of electric potential
4 Work done to bring a charge (q) from ∞ to point r = qV(r) Scales by a factor of q
5 This work is stored as potential energy Electrostatic force is conservative
Result U(r) = qV(r) Final expression
CBSE: Class 12

Formula: Potential Energy of a Single Charge

U(r) = qV(r)

where:

  • U(r) = Potential energy of the charge at position r (in Joules, J)
  • q = Charge of the particle (in Coulombs, C)
  • V(r) = External electric potential at position r (in Volts, V)
  • r = Position vector of the point from the origin
Quantity Symbol SI Unit Dimensional Formula
Potential Energy U Joule (J) [ML2T−2]
Charge q Coulomb (C) [AT]
Electric Potential V Volt (V) [ML2T−3A−1]
CBSE: Class 12

Special Case: The Electron Volt (eV)

When the charge is an electron (q = e = 1.6 × 10−19 C), and it is accelerated through a potential difference of 1 Volt, the energy gained is:

U = qV = (1.6 × 10−19 C) × (1 V) = 1.6 × 10−19 J = 1 eV

eV Conversion Table

Unit Relation to eV Value in Joules
1 eV 1 electron volt 1.6 × 10−19 J
1 keV 103 eV 1.6 × 10−16 J
1 MeV 106 eV 1.6 × 10−13 J
1 GeV 109 eV 1.6 × 10−10 J
1 TeV 1012 eV 1.6 × 10−7 J
CBSE: Class 12

Real-Life Analogy

Gravity vs. Electricity (Analogy)

Just as a mass m has gravitational PE = mgh in a gravitational field, a charge q has electrostatic PE = qV in an electric field.

Concept Gravity Electrostatics
Field agent Gravitational field g Electric field E
Potential Gravitational potential Electric potential V
Potential Energy U=mghU=mgh U = qV
Reference level Ground (h = 0) Infinity (V = 0)
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