Unit 1: Electric Field and Electric Potential

1. Electric Field and Field Lines

An Electric Field (E) is a vector field surrounding an electric charge that exerts a force on other charges. It is defined as the force (F) experienced by a small positive test charge (q₀) divided by the charge itself.

Formula: E = F / q₀

The unit of the electric field is Newtons per Coulomb (N/C) or Volts per meter (V/m).

Electric Field Lines

Electric field lines are imaginary lines used to visualize the electric field.

• They originate from positive charges and terminate on negative charges (or at infinity).
• The tangent to a field line at any point gives the direction of the electric field E at that point.
• The density of the field lines (how close they are) in a region is proportional to the strength (magnitude) of the electric field.
• Electric field lines can never cross each other. If they did, it would imply two different directions of the E-field at the same point, which is impossible.

2. Electric Flux (Φ_E)

Electric flux is the measure of the "flow" of the electric field through a given surface. It quantifies how many electric field lines pass through a specific area.

For a uniform electric field E passing through a flat area A with an angle θ between E and the area's normal vector:

Formula (Uniform Field): Φ_E = E A cos(θ)

In general, for a non-uniform field and a curved surface, the flux is calculated by integrating the dot product of the electric field (E) and the differential area vector (dA) over the entire surface.

Formula (General): Φ_E = ∫_S E ⋅ dA

3. Gauss's Law and its Applications

Gauss's Law relates the net electric flux (Φ_E) through a closed surface (called a "Gaussian surface") to the net electric charge (Q_enclosed) enclosed within that surface.

Gauss's Law: Φ_E = ∮ E ⋅ dA = Q_enclosed / ε₀

Where ε₀ is the permittivity of free space (8.854 × 10^-12 C²/N·m²).

Applications of Gauss's Law

a) Spherical Symmetry (e.g., Uniformly Charged Sphere)

To find the E-field from a sphere of radius R and total charge Q, we use a spherical Gaussian surface of radius r.

• Outside the sphere (r > R): The Gaussian surface encloses all the charge Q.

  E(4πr²) = Q / ε₀ => E = Q / (4πε₀r²)

  (The field is identical to that of a point charge Q at the center).

• Inside a conducting sphere (r < R): Q_enclosed = 0 (charge is on the surface). Therefore, E = 0.

b) Cylindrical Symmetry (e.g., Infinite Line of Charge)

For a line with charge per unit length λ, we use a cylindrical Gaussian surface of radius r and length L.

The flux is only through the curved wall (E is parallel to the end caps). Q_enclosed = λL.

E(2πrL) = λL / ε₀ => E = λ / (2πε₀r)

c) Planar Symmetry (e.g., Infinite Sheet of Charge)

For a sheet with charge per unit area σ, we use a cylindrical "pillbox" Gaussian surface that pokes through the sheet.

Flux is through the two end caps (Area A). Q_enclosed = σA.

E(A) + E(A) = σA / ε₀ => 2EA = σA / ε₀ => E = σ / (2ε₀)

Note: The field is constant and does not depend on the distance from the sheet.

4. Conservative Nature of Electrostatic Field

An electrostatic field is a conservative field. This means the work done by the field in moving a charge from one point to another is independent of the path taken.

A direct consequence of this is that the work done in moving a charge around any closed loop is zero.

∮ E ⋅ dl = 0

Mathematically, this is equivalent to saying the curl of the electrostatic field is zero:

∇ × E = 0

5. Electrostatic Potential (V)

Because the electrostatic field is conservative, we can define a scalar quantity called Electrostatic Potential (V). It is defined as the potential energy (U) per unit charge (q).

Definition: V = U / q

The potential difference (ΔV) between two points is the work done per unit charge (W) to move a charge from one point to the other.

ΔV = V_b - V_a = -∫_a^b E ⋅ dl

The unit of potential is the Volt (V), where 1 Volt = 1 Joule / Coulomb.

6. Relation between E-Field and Potential

The electric field E is the negative gradient of the potential V. The gradient is a vector operator (∇).

Formula: E = -∇V

This means the E-field points in the direction of the steepest decrease in potential.

In Cartesian coordinates, this breaks down into components:

• E_x = -∂V/∂x
• E_y = -∂V/∂y
• E_z = -∂V/∂z

7. Laplace's and Poisson's Equations

These two equations are fundamental in electrostatics. They are derived by combining Gauss's Law (in differential form) with the E-V relation.

1. Gauss's Law (differential): ∇ ⋅ E = ρ / ε₀ (where ρ is the volume charge density)
2. E-V Relation: E = -∇V

Substitute (2) into (1):

∇ ⋅ (-∇V) = ρ / ε₀ => -∇²V = ρ / ε₀

This gives Poisson's Equation, which relates the potential in a region to the charge density in that region.

Poisson's Equation: ∇²V = -ρ / ε₀

In a region of space where there is no charge (ρ = 0), Poisson's equation simplifies to Laplace's Equation.

Laplace's Equation: ∇²V = 0

The operator ∇² is called the Laplacian.

8. The Uniqueness Theorem

The syllabus requires the statement only.

Uniqueness Theorem Statement: If the potential (V) or its normal derivative (∂V/∂n) is specified on all boundaries (conductors, surfaces at infinity, etc.) of a region that obeys Poisson's equation, then the solution for the potential (V) within that region is unique.

9. Electric Dipole

An electric dipole consists of two equal and opposite charges (+q and -q) separated by a small vector distance (d, pointing from -q to +q).

The electric dipole moment (p) is a vector:

Formula: p = qd

Potential and Electric Field of a Dipole

At a point P far away from the dipole (r >> d), the potential V and field E can be calculated. In spherical coordinates (r, θ), where θ is the angle from the dipole axis:

• Potential (V): V(r, θ) = (p cos(θ)) / (4πε₀r²)
• Electric Field (E): Derived from E = -∇V. It has two components:
  • Radial component (E_r): E_r = (2p cos(θ)) / (4πε₀r³)
  • Tangential component (E_θ): E_θ = (p sin(θ)) / (4πε₀r³)

Force and Torque on a Dipole

When a dipole (p) is placed in an external electric field (E):

• In a UNIFORM E-field:
  • The net force on the dipole is zero.
  • It experiences a torque (τ) that tries to align it with the field.
  • Torque Formula: τ = p × E (Magnitude: τ = pE sin(θ))
  • Potential Energy (U): U = -p ⋅ E = -pE cos(θ)
• In a NON-UNIFORM E-field:
  • The dipole experiences both a torque *and* a net force.
  • The force is given by F = (p ⋅ ∇)E.