Electric Field and Potential

The relationship is: $\vec{E} = -\nabla V$, where $\nabla$ is the gradient operator

In Cartesian coordinates:

• $E_x = -\frac{\partial V}{\partial x}$
• $E_y = -\frac{\partial V}{\partial y}$
• $E_z = -\frac{\partial V}{\partial z}$

Key insights:

• Electric field points in direction of maximum potential decrease
• Field magnitude equals rate of potential change in that direction
• Perpendicular to equipotential surfaces (where V is constant)
• Units consistency: V/m = N/C
• Zero field where potential has zero gradient (local extrema)

Application: To find field from potential, calculate partial derivatives of V with respect to coordinates.

Distance dependencies:

1. Point charge (monopole):

   • Potential: V ∝ 1/r
   • Field: E ∝ 1/r²

2. Electric dipole:

   • Potential: V ∝ 1/r²
   • Field: E ∝ 1/r³

3. Quadrupole:

   • Potential: V ∝ 1/r³
   • Field: E ∝ 1/r⁴

Physical insight:

• Each higher-order multipole moment has faster spatial decay
• At sufficient distance, lowest non-zero multipole dominates
• Neutral objects with dipole moments influence surroundings less extensively than charged objects
• Complex charge distributions behave approximately as point charges at large distances

Principles determining field line shape and direction:

• Field lines start on positive charges and end on negative charges
• Direction indicates the force experienced by a positive test charge
• Lines never cross (would indicate multiple field directions at one point)
• The number of lines from a charge is proportional to the charge magnitude
• In free space, field lines extend to infinity unless terminated on opposite charges

Field strength determination:

• Higher density of field lines indicates stronger field
• Field strength is proportional to the number of lines crossing a unit area perpendicular to the lines
• Mathematically: $E \propto \frac{N}{A}$ where N is the number of lines and A is the area
• Field strength decreases with distance: $E \propto \frac{1}{r^2}$ for point charges
• Where field lines converge, field strength increases
• Where field lines diverge, field strength decreases

Example: Between two opposite charges, the field lines are closest together midway between charges of equal magnitude, indicating the strongest field in that region.

• 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.

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 $\frac{p}{4\pi\epsilon_0 r^3}\sqrt{5}$.

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

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°).

Original dipole: charges ±q separated by distance d Modified dipole: charges ±2q separated by distance d/2

1. Dipole moment:

   • Original: $p_{orig} = qd$
   • Modified: $p_{mod} = 2q \cdot \frac{d}{2} = qd$
   • Result: Dipole moment remains unchanged

2. Electric field at a distant point:

   • Field is proportional to dipole moment: $E \propto \frac{p}{r^3}$
   • Since p is unchanged, the electric field remains identical at all points

3. Potential energy in uniform field:

   • Potential energy: $U = -\vec{p}\cdot\vec{E}$
   • Since p is unchanged, the potential energy in any given field remains identical

This demonstrates that different charge configurations can produce identical external electromagnetic effects if they have the same dipole moment.

The torque acting on an electric dipole in a uniform electric field is:

τ = p⃗ × E⃗

Where:

• p⃗ is the dipole moment vector
• E⃗ is the uniform electric field vector

The magnitude of the torque is: τ = pE sin θ

Where:

• p is the magnitude of the dipole moment
• E is the magnitude of the electric field
• θ is the angle between the dipole axis and the field

The direction of the torque is perpendicular to the plane containing the dipole axis and the electric field.

When a conductor is exposed to an external electric field:

1. Initial state: Free electrons are uniformly distributed throughout the conductor
2. When field is applied: Free electrons experience a force opposite to the field direction
3. Electron movement:
   • Electrons drift toward the side facing the positive direction of the field
   • This creates a negative charge density on that surface
   • The side facing the field's negative direction develops a positive charge density due to electron deficiency
4. Result: Surface charge accumulation creates an internal opposing field
5. Equilibrium: Reached when the internally generated field exactly cancels the external field

This process demonstrates how conductors shield their interior from external electric fields.