Electric Field and Potential

• Electric force between two electrons at 1 cm = 2.3 × 10⁻²⁴ N (repulsive)
• Gravitational force between them = 5.5 × 10⁻⁶⁷ N (attractive)
• Electric force is approximately 10⁴³ times stronger than gravitational force
• This enormous difference explains why:
  • Electric forces dominate particle interactions at atomic scale
  • Gravitational effects between elementary particles are negligible
  • Charge neutrality is crucial for macroscopic objects

Continuous charge approximation works because:

• Elementary charge (e = 1.60218 × 10⁻¹⁹ C) is extremely small compared to macroscopic charges
• Even 1 μC contains approximately 6 × 10¹² elementary charges
• Quantization step size (e) becomes negligible with large numbers
• Charge variation can be treated as continuous for most practical purposes

Physical constraint: The approximation is valid when the system size (L) and observational distance (r) are much larger than the average separation between elementary charges (s): L,r >> s

Example: In metals, the electron spacing is ~10⁻¹⁰ m, so continuity is excellent for objects larger than ~10⁻⁷ m

When two electric field vectors of equal magnitude meet at a point:

1. Determine the angle between the vectors (often geometrically based on the point's position)
2. Resolve each vector into components
3. Add corresponding components together
4. Find the resultant magnitude using: $E = \sqrt{E_x^2 + E_y^2}$
5. Find the direction using: $\theta = \tan^{-1}(E_y/E_x)$

For special cases like perpendicular bisectors between opposite charges:

• The field vectors often have symmetrical orientations
• The resultant field points along the bisector of the angle between the individual fields
• When angles are equal, resultant magnitude = 2E·cos(θ), where E is the magnitude of individual fields

Example: For equal magnitude fields at angle θ from horizontal axis, the resultant points horizontally with magnitude 2E·cos(θ).

To compare electric field strengths using field line density:

• Draw equal-sized small areas perpendicular to the field lines at the points of interest
• Count the number of field lines passing through each area
• The ratio of these counts equals the ratio of field strengths at those points
• More lines passing through an area indicates stronger field at that location
• Fewer lines passing through an area indicates weaker field at that location

Example: If an area at distance r₁ from a point charge has 4 times more field lines passing through it than an equal area at distance r₂, then the electric field at r₁ is 4 times stronger than at r₂. This aligns with the inverse square law since if r₂ = 2r₁, then E₁/E₂ = (r₂/r₁)² = 4.

• Electric field lines and equipotential surfaces are perpendicular to each other at every point • For a point charge:

• Electric field lines radiate directly outward (or inward) from the charge
• Equipotential surfaces form concentric spheres around the charge • The electric field points in the direction of maximum potential decrease • No work is done moving a test charge along an equipotential surface • The closer the equipotential surfaces are to each other, the stronger the electric field

Example: Moving a test charge in a circular path around a point charge requires zero work because you stay on the same equipotential surface.

$E = \frac{1}{4\pi\epsilon_0}\frac{p}{r^3}\sqrt{3\cos^2\theta + 1}$

Where:

• $p$ is the dipole moment
• $r$ is the distance from dipole center to the point
• $\theta$ is the angle between dipole axis and the line to the point

This expression combines both radial and angular components of the field into a single magnitude formula.

$\vec{\tau} = \vec{p} \times \vec{E}$

Derivation:

1. For a dipole with charges +q at B and -q at A with midpoint O:
2. Force on +q: $\vec{F}_1 = q\vec{E}$
3. Force on -q: $\vec{F}_2 = -q\vec{E}$
4. Torque from +q: $\vec{\tau}_1 = \vec{OB} \times q\vec{E}$
5. Torque from -q: $\vec{\tau}_2 = \vec{OA} \times (-q\vec{E}) = q(\vec{AO} \times \vec{E})$
6. Net torque: $\vec{\tau} = q(\vec{OB} \times \vec{E}) + q(\vec{AO} \times \vec{E}) = q(\vec{OB} + \vec{AO}) \times \vec{E} = q\vec{AB} \times \vec{E} = \vec{p} \times \vec{E}$

Magnitude: $\tau = pE\sin\theta$, where θ is the angle between dipole axis and field.

Derivation:

1. The torque on a dipole is $\vec{\tau} = \vec{p} \times \vec{E}$ with magnitude $\tau = pE\sin\theta$
2. For a small rotation dθ, work done by the torque is $dW = \tau d\theta = pE\sin\theta\,d\theta$
3. This work is negative as rotation dθ opposes the torque
4. Change in potential energy: $dU = -dW = pE\sin\theta\,d\theta$
5. For change from θ = 90° to some angle θ: $U(\theta) - U(90°) = \int_{90°}^{\theta}pE\sin\theta\,d\theta = pE[-\cos\theta]_{90°}^{\theta} = -pE\cos\theta$
6. If we set $U(90°) = 0$ (zero potential energy when dipole perpendicular to field): $U(\theta) = -pE\cos\theta = -\vec{p}\cdot\vec{E}$

This shows the potential energy is minimized when the dipole aligns with the field (θ = 0°).

Forces and torque analysis:

1. Forces on the charges:

   • Force on +q: F₁ = qE⃗ (parallel to field)
   • Force on -q: F₂ = -qE⃗ (antiparallel to field)

2. Net force calculation:

   • F₁ + F₂ = qE⃗ + (-qE⃗) = 0
   • The net force on the dipole is zero

3. Torque calculation:

   • Torque about center O is τ = r₁ × F₁ + r₂ × F₂
   • Where r₁ is the vector from O to +q and r₂ is the vector from O to -q
   • This simplifies to τ = q(AB⃗) × E⃗ = p⃗ × E⃗

4. Magnitude of torque:

   • τ = pE sin θ
   • Where θ is the angle between the dipole axis and the field

This creates a rotational effect that tends to align the dipole moment with the electric field.

Electric Field Line Representation

Shared Characteristics:

• Field lines are straight for isolated point charges
• Field strength is proportional to the density of field lines (number of lines per unit area)
• Field strength decreases with distance following an inverse square relationship (E ∝ 1/r²)
• The tangent to a field line at any point gives the direction of the electric field at that point
• Closer field lines indicate stronger fields; more distant lines indicate weaker fields

Differences Between Charges:

| Positive Point Charge | Negative Point Charge | |----------------------|----------------------| | Field lines radiate outward from the charge | Field lines point inward toward the charge | | Test charges experience force away from the charge | Test charges experience force toward the charge |

Visualization Tip: Electric field lines always start on positive charges and end on negative charges (or extend to infinity).

Application: When analyzing electric field patterns, the pattern and density of field lines provide immediate insight into both the direction and relative strength of the electric field at any point.