Derive Equation 23.11 for the field Ē dipole in the plane that bisects an electric dipole.

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Start by recalling the configuration of an electric dipole. An electric dipole consists of two charges, +q and -q, separated by a distance d. The dipole moment $ \mathbf{p} $ is defined as $ \mathbf{p} = q \cdot \mathbf{d} $, where $ \mathbf{d} $ is the vector pointing from the negative charge to the positive charge.

In the plane that bisects the dipole (the perpendicular bisector), the contributions to the electric field from the positive and negative charges will have components that cancel along the axis of the dipole. Only the components perpendicular to the dipole axis will add up.

The electric field due to a point charge is given by $ \mathbf{E} = \frac{1}{4 \pi \epsilon_0} \frac{q}{r^2} \hat{r} $, where $ r $ is the distance from the charge and $ \hat{r} $ is the unit vector pointing from the charge to the observation point. For the dipole, calculate the field contributions from both charges at a point on the bisector.

Using geometry, express the distance from each charge to the observation point in terms of the dipole separation $ d $ and the perpendicular distance $ r $ from the dipole axis. Approximate for $ r \gg d $ (far-field approximation) to simplify the expressions for the electric field components.

Combine the contributions from both charges, keeping only the perpendicular components. Simplify the resulting expression to derive the electric field on the bisector plane: $ \mathbf{E} = -\frac{1}{4 \pi \epsilon_0} \frac{p}{r^3} \hat{r} $, where $ p $ is the magnitude of the dipole moment and $ r $ is the distance from the dipole center to the observation point.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a distance. It is characterized by its dipole moment, which is a vector quantity pointing from the negative to the positive charge. The dipole moment is crucial for understanding the electric field generated by the dipole, especially in the context of deriving equations related to its field distribution.

Electric Field (E-field)

The electric field is a vector field that represents the force exerted by an electric charge on other charges in its vicinity. For a dipole, the electric field can be derived from the superposition of the fields due to each charge. Understanding how to calculate the electric field in different regions, particularly in the plane bisecting the dipole, is essential for deriving the relevant equations.

Field Lines and Symmetry

Field lines are a visual representation of the electric field, indicating the direction and strength of the field. The symmetry of the dipole configuration allows for simplifications in calculations, particularly in the plane that bisects the dipole. Recognizing the symmetry helps in deriving equations by focusing on the contributions of the dipole's charges and their resultant field in specific regions.

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Textbook Question

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Textbook Question

FIGURE P23.44 shows a thin rod of length L with total charge Q. Evaluate E at r=3.0 cm if L=5.0 cm and Q=3.0 nC.

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A ring of radius R has total charge Q. At what distance along the z-axis is the electric field strength a maximum?