Electric Forces and Coulomb's Law

Forces Between Point Charges

The interaction between stationary electric charges is governed by Coulomb's Law, which quantifies the force between two point charges.

• Coulomb's Law: The magnitude of the force between two point charges, and , separated by a distance , is given by:

• Where is Coulomb's constant.

• The force acts along the line joining the two charges.

• Like charges repel; unlike charges attract.

• Vector Form:

• Where is the unit vector from to .

• Superposition Principle: The net force on a charge is the vector sum of the forces exerted by all other charges.

Example: Three charges at the vertices of a triangle: Calculate the net force on one charge by summing the vector forces from the other two.

Electric Field

Electric Field Due to Point Charges

The electric field at a point in space is defined as the force per unit positive charge placed at that point.

• Definition:

• Where is a small positive test charge.

• Field of a Point Charge:

• Direction: Away from positive charges, toward negative charges.

• Superposition Principle: For multiple charges, the total electric field is the vector sum of the fields due to each charge.

Example: Find the electric field at a point due to two charges located at different positions by summing their individual fields as vectors.

Motion of a Point Charge in a Uniform Electric Field

A charged particle in a uniform electric field experiences a constant force and thus undergoes constant acceleration.

• Force:

• Acceleration:

• The equations of motion are analogous to those for constant acceleration in mechanics.

Example: An electron released from rest in a uniform electric field will accelerate opposite to the field direction (since is negative).

Electric Field Lines and Visualization

Rules for Drawing Electric Field Lines

• Field lines begin on positive charges and end on negative charges, or at infinity.

• The number of lines is proportional to the magnitude of the charge.

• At any point, the tangent to a field line gives the direction of the electric field.

• The density of lines (lines per unit area) is proportional to the field's magnitude.

• Field lines never cross.

• On conductors, field lines meet the surface perpendicularly and do not penetrate the conductor (in electrostatics).

Example: Sketch field lines for:

• A single point charge (radial lines).

• A dipole (lines emerge from positive, curve to negative).

• Two like charges (lines repel, no lines between them).

• A charge near a conducting surface (lines bend to meet the surface perpendicularly).

Electric Flux and Gauss's Law

Electric Flux

Electric flux measures the number of electric field lines passing through a surface.

• Definition:

• For a uniform field and flat surface:

• is a vector normal to the surface element, with magnitude equal to the area.

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the net charge enclosed by that surface.

• is the permittivity of free space.

• Gauss's Law is especially useful for calculating electric fields with high symmetry (spherical, cylindrical, planar).

Example: The electric field outside a uniformly charged sphere can be found using Gauss's Law and is identical to that of a point charge at the center.

Electric Potential and Potential Difference

Electric Potential: Definition and Properties

The electric potential provides a scalar description of the electric field, simplifying calculations involving energy.

• Potential Difference (): The negative of the work done by the electric field on a unit positive charge moving from A to B.

• The potential difference is independent of the path taken (for electrostatic fields).

• Absolute Potential: The potential at a point relative to a reference point (often taken at infinity).

Potential in a Uniform Electric Field

• For a uniform field , the potential difference between two points separated by distance along the field is:

• Potential decreases in the direction of the electric field.

Potential Due to Point Charges

• The potential at a distance from a point charge is:

• For multiple charges, use superposition:

Example: Find the potential at a point due to two charges by summing their individual potentials (scalars).

Equipotential Surfaces

• An equipotential surface is a surface on which the electric potential is constant.

• Electric field lines are always perpendicular to equipotential surfaces.

• No work is done by the electric field when moving a charge along an equipotential surface.

Example: For a point charge, equipotential surfaces are concentric spheres centered on the charge.

Energy Considerations

• The change in electric potential energy of a charge moving through a potential difference is:

• For a positive charge, moving to a lower potential decreases its potential energy.

Summary Table: Key Concepts

Additional info: The above notes expand on the study guide by providing definitions, equations, and examples for each concept, ensuring a self-contained summary suitable for exam preparation.