In Fig. 26-5, consider a point that is a distance z from the center of a dipole along its axis. (a) Show that, at large values of z, the magnitude of the electric field is given by E=1/2piepselon 0(p/z^3), (Compare with the field at a point on the perpendicular bisector.) (b) What is the direction of E?

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In Fig. 26-5, consider a point that is a distance z from the center of a dipole along its axis. (a) Show that, at large values of z, the magnitude of the electric field is given by E=1/2piepselon 0(p/z^3),

To analyze the electric field along the axis of a dipole, let’s break the problem into parts:

(a) Derive the Electric Field Expression at Large z

A dipole consists of two charges and , separated by a distance d. The dipole moment is defined as:

[math]\vec{p} = q \vec{d}[/math]

In Fig. 26-5, consider a point that is a distance z from the center of a dipole along its axis. (a) Show that, at large values of z, the magnitude of the electric field is given by E=1/2piepselon 0(p/z^3),
In Fig. 26-5, consider a point that is a distance z from the center of a dipole along its axis. (a) Show that, at large values of z, the magnitude of the electric field is given by E=1/2piepselon 0(p/z^3),

so its proved

[math]E_z = \frac{1}{4\pi\epsilon_0} \frac{2p}{z^3}[/math]

Comparison with the Field Along the Perpendicular Bisector

  1. Axial Field:

    [math]E_\text{axis} = \frac{2p}{4\pi\epsilon_0 z^3}[/math].

  2. Perpendicular Bisector Field:

    [math]E_\text{bisector} = \frac{p}{4\pi\epsilon_0 r^3}[/math]

The axial field is twice as strong as the perpendicular field at the same distance:

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