Field due to an electric dipole (Halliday & Resnick, Ch. 27 problem 23)

In summary, the problem discusses the electric field generated by an electric dipole, which consists of two equal and opposite charges separated by a distance. The electric field at a point in space depends on the orientation of the dipole and the distance from it. The dipole moment, defined as the product of the charge and the separation distance, plays a crucial role in determining the field's strength and direction. The problem typically involves calculating the electric field at a specific point, using the principles of superposition and the dipole's geometric arrangement.
  • #1
Ben2
37
9
Homework Statement
"Show that the components of ##\mathbf{E}## are given, at distant points, by [expressions below], where x and y are [coordinates of a point in the first quadrant...]." Here the dipole has charge q at (0,a) and charge -q are (0,-a).
Relevant Equations
$$E_x=\frac{1}{4\pi\epsilon_0}\frac{3pxy}{\left(x^2+y^2\right)^{(5/2)}}$$
$$E_y=\frac{1}{4\pi\epsilon_0}\frac{p\left(2y^2-x^2\right) }
{\left(x^2+y^2\right)^{(5/2)}}$$
Using either H&R's Chapter 27 Example 3 or Problem 590 of the ##\mathbf{Physics Problem Solver}##, I've been unable to get the component ##E_x## or ##E_y##. There are now different angles at the charges. My thanks to berkeman for LaTeX advice, but any errors are of course my own. Thanks in advance to all contributors!
 
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  • #2
I see no errors mainly because I see no work. Please edit your post to fix the LaTeX.
Hint: Click the "Preview" button, last one on the right to see what your LaTeX will look like before committing yourself to posting.
Screen Shot 2024-04-12 at 2.22.58 PM.png
 
  • #3
@Ben2 : Please wrap your last '5/2' term with {}.
 
  • #4
Also, in a "Show that" type of question, in addition to what you are asked to "show" you must include the starting expression.
 

FAQ: Field due to an electric dipole (Halliday & Resnick, Ch. 27 problem 23)

What is an electric dipole?

An electric dipole consists of two equal and opposite charges separated by a small distance. It is characterized by its dipole moment, which is a vector quantity defined as the product of the charge and the separation distance. The dipole moment is directed from the negative charge to the positive charge.

How do you calculate the electric field due to an electric dipole?

The electric field \(\mathbf{E}\) due to an electric dipole at a point in space can be calculated using the formula: \[\mathbf{E} = \frac{1}{4\pi \epsilon_0} \cdot \frac{2\mathbf{p} \cos \theta}{r^3} \hat{r} + \frac{1}{4\pi \epsilon_0} \cdot \frac{\mathbf{p} \sin \theta}{r^3} \hat{\theta}\] where \(\mathbf{p}\) is the dipole moment, \(r\) is the distance from the dipole center to the point of interest, \(\theta\) is the angle between the dipole moment and the line connecting the dipole to the point, and \(\epsilon_0\) is the permittivity of free space.

What is the significance of the dipole moment?

The dipole moment is a measure of the strength and orientation of the dipole. It determines how the dipole interacts with external electric fields and how it produces an electric field in its surroundings. A larger dipole moment indicates a stronger dipole, which can have significant effects on the electric field it generates and how it affects nearby charges.

How does the electric field of a dipole vary with distance?

The electric field of a dipole decreases with the cube of the distance from the dipole, specifically as \(1/r^3\). This means that as you move further away from the dipole, the strength of the electric field diminishes rapidly, making dipoles have a localized effect in their vicinity.

What is the direction of the electric field produced by an electric dipole?

The direction of the electric field produced by an electric dipole is along the axis of the dipole moment vector. At points along the axis (the line extending from the positive charge to the negative charge), the field points away from the dipole if the point is outside the dipole. At points along the perpendicular bisector of the dipole, the field points towards the dipole, indicating that the field lines converge towards the dipole's center.

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