Recording-2025-01-27T23_48_14.015Z

Chapter 8: Electric Dipoles

Section 8.1: Behavior of Electric Dipoles

• Charging Basics

  • Most objects do not have a net charge due to the high energy requirement to produce one.

  • All objects contain charged particles; positive and negative charges may form quadrupoles, octopoles, etc., though these higher order moments are not covered in detail.

• Dipole Fields

  • Skills involved in drawing dipole fields include:

    • Deducing direction of dipole moments from fields.

    • Predicting dipole behavior in external fields.

  • For distributions with a non-zero net charge:

    • Electric field behaves radially far from the charge.

  • When total charge is zero, the dipole field is characterized by dipole moments.

Dipole Moment and Characteristics

• Definition of Dipole Moment (p)

  • Given by the formula:

    p = ∑ (qi * ri)

  • Indicates strength and direction of the dipole.

  • For dipoles formed from two equal and opposite charges:

    • Dipole moment points from negative to positive charge and calculated as:

    p = q * d

    where:

    • q = charge of the positive charge,

    • d = distance between the charges.

• Higher Order Moments

  • Monopole Moment: System's net charge.

  • Dipole Moment: The next significant characteristic if monopole moment is zero.

  • Other moments include quadrupoles, octopoles, etc.

Electric Fields and Dipoles

• Math Representation of Dipole Field

  • For a point dipole at the origin:

    • E(r) = k * (3(p·r)r - p) / r^3

    • k is Coulomb's constant.

  • Strength of dipole field drops off as 1/r^3, compared to 1/r^2 for net charge distributions.

• Calculating Dipole Fields

  • Example (Given a dipole formed by charges +1 nC and -1 nC at ±0.2 cm along the y-axis):

    • Dipole moment: p = q * d = (-2 * 10^-11 C m)

    • Electric fields calculated at 5 m along both axes, yielding:

      • Along x-axis: E = -2.88 * 10^-4 N/C in the y-direction.

      • Along y-axis: E = 5.75 * 10^-4 N/C in the y-direction.

Dipole Behavior in Fields

• Uniform vs Non-uniform Electric Fields

  • In a uniform electric field:

    • Dipole experiences zero net force, resulting in rotational motion towards alignment with the field.

    • Forces on dipole maintain equilibrium without rotating the center of mass.

  • In non-uniform fields:

    • This creates a net force on the dipole as the charges experience different magnitudes of force due to the field.

Example Problem 8.3: Electric Dipole in a Uniform Field

• Problem Setup:

  • Uniform electric field in positive x-direction.

  • Barbell dipole with dipole moment directed in the positive y-direction.

• Solution:

  1. Draw the uniform field and barbell dipole; dipole moment points from negative to positive charge.

  2. Forces acting on each charge are equal in magnitude but in opposite directions due to field uniformity.

  3. Resulting rotation dictated by these forces dictates clockwise direction of rotation, re-aligning the dipole with field lines.