JoVE Logo

Sign In

24.13 : Energy Associated With a Charge Distribution

The work done to bring a charge through a distance r is given by the potential difference between the initial and the final position. To assemble a collection of point charges, the total work done can be expressed in terms of the product of each pair of charges divided by their separation distance, defined with respect to a suitable origin. Solving this expression gives the energy stored in a point charge distribution.

Hydrogen bonding in water molecules, diagram showing H2O interaction via dashed lines, chemistry.

Consider an infinitesimal charge element in a configuration of continuous charge distribution enclosed in a definite volume. The product of the volume charge density and the volume of the element gives the total charge in this element. The energy stored in this configuration of continuous charge distribution is given by integrating volume charge density and the corresponding potential.   

Applying Gauss's law in its differential form, the volume charge density can be written in terms of the electric field. Using the product rule in this expression gives the divergence of the electric field. The volume integral can be written as a surface integral using Gauss's divergence theorem. Rewriting the potential in terms of the electric field gives the energy stored in this configuration.

DNA replication diagram showing polymerase chain reaction steps; PCR process illustration.

Recall that to obtain the expression for work done, the integration must be performed over the region where the charge is located. Even if the integration is performed over a larger volume, the work done remains conserved as the charge density in the extra volume is zero.

The surface integral of an electric field, which relates to electric potential energy, depends on factors beyond distance, such as charge distribution and system geometry. To calculate total energy, integration over all space, considering the entire volume, is necessary, as the electric field alone at the surface does not provide the complete picture.

Molecular orbitals of dihydrogen ion diagram, illustrating bonding and antibonding interactions.

Tags

EnergyCharge DistributionPotential DifferenceWork DonePoint ChargesVolume Charge DensityElectric FieldGauss s LawDivergence TheoremElectric Potential EnergyIntegrationSurface IntegralTotal Energy

From Chapter 24:

article

Now Playing

24.13 : Energy Associated With a Charge Distribution

Electric Potential

1.5K Views

article

24.1 : الطاقة الكامنة الكهربائية

Electric Potential

5.7K Views

article

24.2 : الطاقة الكامنة الكهربائية في مجال كهربائي موحد

Electric Potential

4.6K Views

article

24.3 : الطاقة الكامنة الكهربائية لشحنتين من نقطتين

Electric Potential

4.4K Views

article

24.4 : الجهد الكهربائي وفرق الجهد

Electric Potential

4.3K Views

article

24.5 : إيجاد الجهد الكهربائي من المجال الكهربائي

Electric Potential

4.0K Views

article

24.6 : حسابات الجهد الكهربائي I

Electric Potential

1.9K Views

article

24.7 : حسابات الجهد الكهربائي II

Electric Potential

1.6K Views

article

24.8 : الأسطح متساوية الجهد وخطوط المجال

Electric Potential

3.6K Views

article

24.9 : الأسطح والموصلات متساوية الجهد

Electric Potential

3.3K Views

article

24.10 : تحديد المجال الكهربائي من الجهد الكهربائي

Electric Potential

4.3K Views

article

24.11 : معادلة بواسون ولابلاس

Electric Potential

2.6K Views

article

24.12 : مولد فان دي جراف

Electric Potential

1.7K Views

article

24.14 : شروط الحدود الكهروستاتيكية

Electric Potential

409 Views

article

24.15 : نظرية التفرد الثانية

Electric Potential

970 Views

JoVE Logo

Privacy

Terms of Use

Policies

Research

Education

ABOUT JoVE

Copyright © 2025 MyJoVE Corporation. All rights reserved