Electric Field and Potential

Starting with dipole potential: $V = \frac{1}{4\pi\varepsilon_0}\frac{p\cos\theta}{r^2}$

For radial component $E_r$:

• $E_r = -\frac{\partial V}{\partial r} = -\frac{\partial}{\partial r}\left(\frac{1}{4\pi\varepsilon_0}\frac{p\cos\theta}{r^2}\right)$
• $E_r = \frac{1}{4\pi\varepsilon_0}\frac{2p\cos\theta}{r^3}$

For angular component $E_\theta$:

• $E_\theta = -\frac{1}{r}\frac{\partial V}{\partial \theta} = -\frac{1}{r}\frac{\partial}{\partial \theta}\left(\frac{1}{4\pi\varepsilon_0}\frac{p\cos\theta}{r^2}\right)$
• $E_\theta = \frac{1}{4\pi\varepsilon_0}\frac{p\sin\theta}{r^3}$

Field magnitude: $E = \sqrt{E_r^2 + E_\theta^2} = \frac{p}{4\pi\varepsilon_0 r^3}\sqrt{3\cos^2\theta + \sin^2\theta}$

$-\nabla \cdot \nabla V$ represents Poisson's equation:

$-\nabla \cdot \nabla V = \nabla^2 V = -\frac{\rho}{\varepsilon_0}$

Physical meaning:

• $\nabla^2V$ is the Laplacian of potential (sum of second derivatives)
• It measures the "curvature" or "concentration" of potential in space
• Proportional to charge density $\rho$ at that point

Significance:

• Connects potential (scalar field) to charge distribution
• In charge-free regions: $\nabla^2V = 0$ (Laplace's equation)
• A non-zero Laplacian indicates presence of charge
• Allows calculation of charge density from potential measurements
• Fundamental relation in electrostatics used for boundary value problems

Example: Spherical charge has positive $\nabla^2V$ inside (potential curves upward), zero outside.

Electric Field vs. Electric Potential from a Point Charge:

Electric Field (E):

• Varies as 1/r²
• Vector quantity with direction
• Formula: E = Q/(4πε₀r²)
• Direction: radially outward (Q positive) or inward (Q negative)
• Units: N/C

Electric Potential (V):

• Varies as 1/r
• Scalar quantity
• Formula: V = Q/(4πε₀r)
• No direction, just magnitude
• Units: J/C or Volts

Key insight: This difference in distance dependence (1/r² vs 1/r) explains why electric field weakens more rapidly than electric potential as you move away from a point charge.

For two like positive charges:

• Field lines radiate outward from each positive charge
• Lines never cross and repel each other in the region between charges
• Field strength is highest near each charge (indicated by line density)
• Field lines extend to infinity and never form closed loops

For an electric dipole (negative and positive charge pair):

• Field lines originate at the positive charge and terminate at the negative charge
• Lines curve from positive to negative, showing the attractive force
• Field pattern is asymmetric, with high density between charges
• At large distances, field decreases more rapidly (as $1/r^3$) compared to single charges ($1/r^2$)

The fundamental difference is that like charges create repulsive field patterns, while opposite charges create attractive field patterns that connect the charges.

The electric field strength along the axis is:

$E(x) = \frac{1}{4\pi\epsilon_0} \frac{xq}{(a^2+x^2)^{3/2}}$

Key characteristics:

• At x = 0 (center of ring): E = 0 (field components cancel completely)
• As x → ∞: E approaches zero, proportional to 1/x²
• Maximum field occurs at x = a/√2 ≈ 0.707a

For the given ring (a = 10 cm, q = 5 nC):

• Maximum field occurs at x = 7.07 cm from center
• This maximum value is E_max = 0.385q/(4πε₀a²)
• At large distances, the field approximates that of a point charge

This non-intuitive maximum location (at x = a/√2) can be found by setting dE/dx = 0 and solving for x.

• Moving along an equipotential surface means ΔV = 0 • The component of electric field parallel to an equipotential surface must be zero:

• Since E = -∇V (electric field is negative gradient of potential)
• If V doesn't change along a path, E has no component in that direction • Therefore:
• Electric field vectors must be perpendicular to equipotential surfaces
• Moving perpendicular to field lines causes maximum potential change
• Moving parallel to field lines causes the most rapid potential change

Note: This orthogonality principle applies to all electrostatic fields, not just point charges, and is a fundamental property of conservative vector fields.

The electric field components due to a dipole are derived by taking the negative gradient of the potential:

For a dipole with potential $V = \frac{1}{4\pi\epsilon_0}\frac{p\cos\theta}{r^2}$:

1. Radial component (along r-direction): $E_r = -\frac{\partial V}{\partial r} = \frac{1}{4\pi\epsilon_0}\frac{2p\cos\theta}{r^3}$

2. Transverse component (perpendicular to r): $E_\theta = -\frac{1}{r}\frac{\partial V}{\partial \theta} = \frac{1}{4\pi\epsilon_0}\frac{p\sin\theta}{r^3}$

These expressions show that both field components:

• Decrease as $1/r^3$ (faster than the $1/r^2$ for a point charge)
• Have different angular dependencies
• Vary in strength with the dipole moment p

Example: At a point directly along the dipole axis ($\theta=0°$), the field is purely radial and twice as strong as at the same distance along the equator ($\theta=90°$).

In analyzing the electric field of a dipole, small displacements reveal the field components:

1. Relationship between displacement and field:

   • For any small displacement $\vec{dl}$, the potential change is $dV = -\vec{E} \cdot \vec{dl}$
   • This means $E_l = -\frac{dV}{dl}$ for any direction $l$

2. Key displacements for a dipole:

   • Radial displacement ($dr$): Changes only the distance $r$ while keeping angle $\theta$ constant
   • Transverse displacement ($rd\theta$): Changes only the angle $\theta$ while keeping $r$ constant

3. Resulting field components:

   • Radial: $E_r = -\frac{\partial V}{\partial r}$
   • Transverse: $E_\theta = -\frac{1}{r}\frac{\partial V}{\partial \theta}$

This method is particularly powerful because it allows us to derive vector field components by taking derivatives of a scalar potential function along different directions.

Example: At a 45° angle from a dipole, both components contribute equally to the total field, which has a magnitude of $\frac{p}{4\pi\epsilon_0 r^3}\sqrt{5}$.

$\alpha = \tan^{-1}\left(\frac{1}{2}\tan\theta\right)$

This relationship shows how the field direction varies with position around the dipole. The field is not generally parallel to the radial direction except in special cases.

Note: As θ approaches 90°, α approaches 90° as well, meaning the field becomes perpendicular to the radial line (and antiparallel to the dipole axis) in the broadside-on position.

From $U(\theta) = -\vec{p}\cdot\vec{E} = -pE\cos\theta$ and $\tau = pE\sin\theta$:

1. Minimum potential energy:

   • θ = 0° (dipole aligned parallel with field)
   • $U_{min} = -pE$
   • Stable equilibrium position

2. Maximum potential energy:

   • θ = 180° (dipole aligned antiparallel to field)
   • $U_{max} = +pE$
   • Unstable equilibrium position

3. Maximum torque:

   • θ = 90° (dipole perpendicular to field)
   • $\tau_{max} = pE$
   • Zero potential energy
   • Maximum rotational tendency

Note: The dipole naturally rotates to align with the field (θ = 0°) to minimize potential energy.