Foundations and Constants

Physical Constants

Physics problems involving electric fields and potentials often require the use of fundamental constants:

• Elementary charge (e):

• Coulomb's constant (k):

Electric Fields and Potentials

Definitions and Properties

Electric fields and electric potentials are fundamental concepts in electromagnetism:

• Electric Field (\(\vec{E}\)): A vector field representing the force per unit charge at each point in space.

• Electric Potential (V): A scalar quantity representing the potential energy per unit charge at a point in space.

• Properties: Electric fields point from positive to negative charges; electric potential decreases in the direction of the electric field.

• Conductors and Insulators: Conductors allow free movement of charge; insulators do not.

General Relations

• Force on a Charge: A charged particle in an external electric field experiences a force:

• Change in Potential Energy: For a charge moving across a potential difference:

Uniform Electric Field

• A uniform electric field has the same magnitude and direction at all points.

• The potential difference between two points separated by distance \(d\) in a uniform field is:

Point Charges

• Electric Field from a Point Charge: where \(q\) is the charge and \(r\) is the distance from the charge.

• Electric Potential from a Point Charge:

• Superposition Principle: The net electric field or potential from multiple charges is the sum of the individual contributions.

Force Between Point Charges

• The force between two point charges \(q_1\) and \(q_2\) separated by distance \(r\):

• Like charges repel; unlike charges attract.

Potential Energy of Point Charges

• The potential energy of two point charges separated by distance \(r\):

Uniformly Charged Sphere

• A sphere of radius \(R\) with total charge \(Q\) uniformly distributed:

• Outside the sphere (\(r \geq R\)):

• Inside the sphere (\(r < R\)):

Uniformly Charged Infinite Sheet

• An infinite sheet with surface charge density \(\sigma\) produces a uniform electric field:

Charged Parallel Plates

• Two parallel plates with equal but opposite charges create a uniform electric field between them:

• The electric potential decreases in the direction of the electric field.

Isolated Conductor

• Interior: The electric field inside an isolated conductor is always zero.

• Surface: Any excess charge resides on the surface, and the electric field just outside is perpendicular to the surface:

Capacitance

Definition and Formula

• Capacitance (\(C\)) measures the ability of a system to store electric charge per unit potential difference:

• where \(Q\) is the charge and \(V\) is the potential difference.

Parallel Plate Capacitor

• For a parallel plate capacitor: where \(A\) is the plate area and \(d\) is the separation.

Stored Energy in a Capacitor

• The energy stored in a capacitor is:

Summary Table: Key Equations