Overview
This lecture explains how to calculate the electric field of a dipole—two equal and opposite point charges separated by a distance—and introduces the concept of the dipole moment.
Electric Field of a Dipole: Setup
- A dipole consists of charges +Q and -Q separated by distance d along the x-axis.
- The electric field is calculated at a point a distance s from the midpoint, perpendicular to the dipole axis.
Calculating the Field: Perpendicular Bisector
- The electric field from each charge is ( E = \frac{kQ}{r^2} ) in the direction from charge to field point.
- By geometry, ( r = \sqrt{s^2 + (d/2)^2} ), with angle θ defined by triangle relationships.
- The field components: ( E_x = E \cos\theta ), ( E_y = E \sin\theta ).
- ( \cos\theta = \frac{d/2}{r} ), ( \sin\theta = \frac{s}{r} ).
- Adding fields from both charges: y-components cancel, x-components add, yielding ( E = \frac{k Q d}{r^3} \hat{x} ).
Dipole Moment Introduction
- The dipole moment ( \vec{p} ) is defined as ( Q \times d ), pointing from negative to positive charge.
- The field simplifies to ( E = \frac{k p}{r^3} ) at the perpendicular bisector.
Field Along the Dipole Axis (End-On)
- Analyze field at distance x along the dipole axis from the center.
- The distances are ( x + d/2 ) and ( x - d/2 ) for each charge.
- Sum the contributions, simplify algebraically: numerator reduces to (-2Qdx), denominator to ((x^2 - d^2/4)^2).
- For ( x \gg d ), denominator approximates to ( x^4 ); final field ( E = -\frac{2kp}{x^3} \hat{x} ).
Dipole Field Patterns
- Field at the perpendicular bisector points horizontally; along the axis, it points toward the negative charge.
- Mapping various points shows classic dipole field lines: out of positive, into negative charge.
Key Terms & Definitions
- Dipole — Two equal and opposite charges separated by a distance.
- Dipole Moment (p) — ( Q \times d ), measures the strength and orientation of a dipole.
- Electric Field (E) — Force per unit charge produced by other charges.
Action Items / Next Steps
- Practice calculating dipole fields at different points.
- Review algebraic simplification of field expressions.
- Read about applications of dipoles in Physics 350.